Spatio-temporal dynamics of electrons and helium metastables in uniform dielectric barrier discharges formed in He/N2

A uniform atmospheric pressure dielectric barrier discharge is operated in helium with an admixture (0.45%) of nitrogen. The discharge is ignited in the gas gap between a driven and a grounded electrode and propagates along the dielectric surface outside the gap. Plasma conditions are characterised with current and voltage measurements and by application of absolutely calibrated optical emission spectroscopy, with a focus on nitrogen molecular emission. Plasma parameters, namely electron density and reduced electric field, are determined with spatial and temporal resolution in the frame of a collisional-radiative model using a calibrated charge coupled device camera and Abel inversion of measured images. The density of an effective helium metastable state is calculated using the measured plasma parameters and compared with values of the He(23S) state density measured with tunable diode laser absorption spectroscopy.


Introduction
Excited species and chemically active radicals are effectively produced by atmospheric pressure non-thermal plasmas, which find extensive usage in medical and technical applications, such as the inactivation of bacteria and viruses [1], wound healing [2], catalytic gas conversion [3] or surface Original Content from this work may be used under the terms of the Creative Commons Attribution 4.0 licence.Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI.modification [4].Among these plasmas, the dielectric barrier discharge (DBD) represents a valuable source for most of the above-mentioned applications [5,6].As with other nonthermal plasma sources, one of the advantages of DBD sources lies in the production of electrons with a kinetic energy higher than the dissociation energy of molecules and higher than the excitation energy of metastable states, leading to the formation of a variety of reactive species.Further, in biomedical applications, the low electric current through the treated object and a low plasma conductivity associated with DBDs are also favourable properties [7].
In general, the outcome of a plasma application is defined by the fundamental properties of the plasma source, namely the reduced electric field E/N, the electron density n e and the gas temperature T g that can be influenced by the exact operating conditions of the device, e.g. the gas composition or the ignition characteristics.These plasma parameters characterise the discharge properties and are therefore often studied using different techniques, as they allow to further estimate other relevant quantities such as reaction rates or species densities and are used to compare plasma reactors presented in the literature.This insight contributes to a better understanding of various other plasma processes such as excitation, ionisation and chemical reactions.Due to the low mean free path of particles at atmospheric pressure and the transient pulse waveforms often used to ignite discharges, their properties vary strongly at temporal and spatial scales in the range of nanoseconds and micrometres, respectively.As a result, temporal and spatial averaging during measurements can lead to errors due to the non-linear nature of the measured properties such as molecular emission and its derived parameters.To understand the physical and chemical properties of these transient discharges, diagnostics with nanosecond and micrometre resolution are required.Further, traditional low-pressure plasma diagnostics, such as Langmuir or multipole-resonance probes, which must be placed inside the plasma region [8,9], suffer from major challenges and significant limitations at atmospheric pressure.
In this context, under atmospheric pressure conditions, the combination of optical emission spectroscopy (OES) and numerical simulations has been presented as a versatile diagnostic tool to obtain information about plasma conditions, such as E/N and n e [10][11][12] or reactive species densities [13][14][15] at sufficient temporal and spatial resolution.While there are a variety of methods to combine experiment and simulation for this purpose, typical schemes involve the calculation of electron energy distribution functions using a numerical solution of the Boltzmann equation at variable reduced electric fields.From these distribution functions, rate constants of electron impact excitation reactions, or ratios thereof, can be calculated and compared with measured intensity ratios.Using a fitting or interpolation procedure, the actual electric field strength, for example, can then be determined.Further, the application of absolutely calibrated spectrometers and intensified charge coupled device (ICCD) cameras allows absolute intensities of photoemission (in photons cm −3 s −1 ) to be measured, which can be used to determine electron densities [16].An important aspect of such diagnostics is the selection of the utilised atomic or molecular transitions.For this, nitrogen is a molecule that offers several advantages compared to other molecules or atoms: there are multiple intense emission bands such as N 2 (C-B) (second positive system) and N + 2 (B-X) (first negative system) available in the ultraviolet (UV) spectral range, the molecular data like emission coefficients, electron impact excitation cross sections and the rate constants for important reactions are well-known and reliably determined [17][18][19][20], and the different excitation processes can be highly sensitive to variations of the electron energy distribution due to large energy differences between the available transitions.Moreover, excitation models including nitrogen transitions are well established [21].In addition, determining the gas temperature under the studied plasma conditions is crucial for plasma characterisation.Nitrogen allows this with sufficient precision by comparing measured and simulated emission spectra, assuming that the translational and rotational degrees of freedom are in equilibrium at atmospheric pressure and electron impact excitation is the main mechanism for populating the corresponding upper states [16,22].
When a collisional-radiative model is used in combination with OES diagnostics of transient, non-uniform discharges, such as DBDs, a major limitation is the low intensity of the observed emission associated with insufficient sensitivity of available detectors.Further, due to the stochastic distribution of the plasma, the discharges do not allow averaging or accumulating multiple images to increase the sensitivity.That is a critical point for the characterisation of filamentary DBDs formed from thin (about 100 µm) channels with durations in the range of 100 ns, that are stochastically distributed in the volume between the electrodes.This problem was solved by Kozlov et al [23] using time-correlated singlephoton counting in a special electrode arrangement, which causes a repeatable ignition of micro-discharges at a constant position.There, the temporal and spatial distribution of molecular nitrogen photoemission was measured and a system of differential equations for molecular nitrogen emission intensities was solved [23].Hoder et al [24] then investigated the reduced electric field of the propagating steamer head in a DBD ignited in synthetic air using the temporal behaviour of the ratio of N 2 (C-B) and N 2 + (B-X) intensities [24].Alternatively, time integrated spectra of transient discharges can also be analysed to obtain some insight into the underlying plasma parameters [25,26].In such cases, the transient nature of the plasma cannot be accounted for directly as the measurements are averaged across the discharge time and the observed volume.Such methods therefore lead to 'effective' plasma parameters, that can be used to compare the same discharge under different operating conditions, but can be difficult to compare to other discharges or time resolved simulations.This method is likewise based on the determination of the intensity ratio in the emission spectrum of molecular nitrogen, which provides information about the reduced electric fields in the plasma.The electron density is determined using the rate constant of electron impact excitation, calculated for the measured effective reduced electric field, and the intensity of N 2 (C-B) emission measured using an absolutely calibrated photodetector, like a spectrometer, or an ICCD camera provided with band-pass optical filters for the wavelengths of the respective transitions.Additionally, applying inverse Abel transforms to measured ICCD images of radially symmetric discharges can provide information on effective plasma parameters with spatial resolution down to several micrometres [27,28].Collisional-radiative models based on these approaches have been used to study a variety of plasma reactors and compared with other methods where the same plasma properties can be determined [24,[29][30][31][32].
In the present study, spatially and temporally resolved plasma parameters of a uniform atmospheric pressure DBD are determined using an absolutely calibrated ICCD camera equipped with band-pass filters in combination with a collisional-radiative model developed for helium with a small admixture (0.45%) of nitrogen.Based on the determined plasma parameters, the spatially resolved densities of an effective helium metastable state are calculated.These are used to validate the overall model performance with the densities of the He(2 3 S 1 ) measured using temporally resolved tunable diode laser absorption spectroscopy (TDLAS).

Experimental setup
A scheme of the setup is shown in figure 1. Basically, the uniform DBD is operated in a sealed vacuum vessel made of aluminium.As process gases, helium and nitrogen (Air Liquide, Germany) with purity 5.0 are used and their flow is controlled using two mass flow controllers (Bronkhorst EL-FLOW Prestige, Bronkhorst, Germany), a vacuum pump (TRIVAC D 16 BCS, Leybold, Germany) and a needle valve to control the pump speed.This setup allows precise adjustment of the gas composition and pressure which ensures stable and reproducible operating conditions.The gas flows are set to 2 slm helium with an admixture of 9 sccm of nitrogen, resulting in a concentration of ≈ 4500 ppm N 2 , the pressure is set to (1000 ± 5)mbar.Optical access is allowed via two quartz windows transparent in the spectral range from 200 nm to 1100 nm (Viosil SQ, GVB, Germany).The DBD consists of a driven copper electrode with a diameter of 8 mm and a grounded stainless steel electrode with a diameter of 25 mm that are covered by an aluminium oxide barrier of 1 mm thickness and a single-side polished silicon wafer of 0.8 mm thickness, respectively.The driven electrode is connected to a custom pulsed high voltage (HV) power supply that creates a damped sinusoidal signal at each pulse with a pulse trigger frequency set to 1000 Hz for this work.A similar HV power supply was presented in more detail in [33].The voltage and current measurements are performed using a capacitive voltage divider (P6015A, Tektronix, USA) and a current monitor (Model 2877, Pearson Electronics, USA), respectively, which are read out by a digital oscilloscope (HDO6104A 1 GHz, Teledyne LeCroy, USA).
The molecular emission bands of nitrogen N 2 (C-B) and N + 2 (B-X) transitions are measured using a calibrated ICCD camera (DiCAM-PRO 25-SVGA, PCO, Germany) in combination with two band-pass filters in order to characterise the plasma parameters of the uniform DBD.The maximum and the spectral range of the two band-pass filters transmission, (380 ± 5) nm and (390 ± 5) nm, are chosen to minimise the difference of the measured signals of N 2 (C-B) and N + 2 (B-X) under the considered plasma conditions.In order to synchronise the camera timing with the trigger pulse of the HV power supply and to scan the temporal development of the molecular emission, a digital delay generator (Model DG645, Stanford Research Systems, USA) is used and triggered on the power supply trigger pulse.Both molecular emission transitions used for OES diagnostics consist of several vibrational bands from the same upper electronic level, whose relative intensities depend only on the known Franck-Condon factors and are independent of the excitation process and the plasma conditions.The nitrogen molecular emission in our experiment is assumed to be distributed radially symmetrically in the gap between electrodes and at the lateral surface of the driven electrode.This volumetric photoemission source produces an integrated two-dimensional projection of the radial emission profile at the charge coupled device chip.In order to calculate the radial distribution of emission with the measured data, an inverse Abel transform is performed according to the algorithm published in [34], that is implemented and improved in the PyAbel package [35,36].In general, most inverse Abel transform methods produce mathematically correct negative numbers, but these tend to be physically wrong in many applications.Therefore, a non-negative least square solver was used to obtain non-negative Abel transformed values.This method generates an inverse Abel transformation that reproduces the results of an exact inverse Abel transformation of the input data as closely as possible without producing negative values.
As the uniform atmospheric pressure DBD is assumed to be very stable and reproducible, accumulation and averaging of individual consecutive discharges should lead to correct data.This assumption is confirmed by capturing and comparing several individual images without signal accumulation with an exposure time of 4 ns and without any band-pass filters to ensure a sufficient signal intensity.For the experiments presented in this work, an exposure time of 40 ns with 30 individual on-chip accumulations for one image is selected in order to provide sufficient signal intensity, while at each delay time 128 images are measured and averaged to produce the final image.The pixel edge length of the images is measured additionally using a length standard and amounts to (14.7 ± 0.1) µm.
Emission spectra of the helium-nitrogen plasma are acquired using an echelle spectrometer (ESA4000, LLA Instruments, Germany), with a spectral range between 200 nm and 800 nm and a resolving power of R = λ/∆λ = 13 333, resulting in a spectral resolution of ∆λ = 0.015 nm at 200 nm and ∆λ = 0.06 nm at 800 nm, as described in [16,37].The spectrometer is calibrated using a tungsten ribbon lamp as well as a deuterium lamp as secondary standards, which were calibrated by the Physikalisch-Technische Bundesanstalt (PTB, Berlin, Germany).In this way, the spectral efficiency of the spectrometer (ϵ(λ) in counts photons ) is determined.To absolutely calibrate the ICCD camera, a point-like (diameter 0.6 mm) microwave plasma source is applied and operated at atmospheric pressure with a nitrogen flow of 2 slm as described in detail in [16].Absolute intensities of N 2 (C-B,0-0) and N + 2 (B-X,0-0) emitted by the microwave plasma source are measured using the calibrated spectrometer.At the same time, images of this point-like plasma are recorded using the ICCD camera with the two band-pass filters for the nitrogen transitions N 2 (C-B,0-2) and N + 2 (B-X,0-0) with identical acquisition settings as described above.By comparing the measured absolute intensities of the nitrogen bands of the point-like source with the measured signal of the ICCD camera, calibration factors were calculated to obtain absolute calibrated emission data from the ICCD camera with spatial resolution.
For comparison with the results of the model described in the next section, the line-integrated density of the helium metastable He(2 3 S 1 ) is measured with high temporal resolution in the gas gap between powered and grounded electrodes using TDLAS [38,39].Absorption profiles of He(2 3 S 1 − 2 3 P J , J = 1, 2) at λ = 1083.25 nm and λ = 1083.034nm are measured using a tunable diode laser source (Toptica DFB pro L equipped with an LD-1083-0070-DFB-1 laser diode), a high-pass filter to block light emitted by the plasma with a cut-off wavelength of 1000 nm and an InGaAs detector (DET10N2, Thorlabs) with 5 ns rise time.The line width of the laser (<1 MHz) is considerably smaller than the width of the absorption lines (<10 GHz) mainly due to collisional broadening at atmospheric pressure.In the first step of the experiment, the absorption profile is measured by scanning over two triplet transitions within a range of 80 GHz using a temperature-controlled mode of the laser system.In the next step, the wavelength of the laser is fixed at the edge of the line profile to minimise the saturation effects of the absorption at high metastable densities, and then the time-dependent absorption signal of helium metastables is recorded.By utilising the known line profile, the density of helium metastables is then calculated.The absorption length is assumed to be 10 mm, equal to the diameter of the powered electrode.The detection limit of the system due to the background noise is in a range of 10 10 cm −3 .

Temporally resolved OES diagnostics
Our diagnostics focus on the nitrogen molecular bands, especially the second positive, N 2 (C-B), and the first negative, N + 2 (B-X), system, and allow for determination of the gas temperature, the reduced electric field and the electron density using a combination of OES and numerical simulations of the electron distribution function for different values of the reduced electric field.The second positive system of nitrogen is excited in our helium/nitrogen DBD by electron impact excitation, while the first negative system can also be excited by Penning ionisation of nitrogen molecules due to collisions with helium metastable states.Four main assumptions are made for the discharge conditions in the presented setup: (a) An 'effective' helium metastable state, He met , that is populated by electron impact excitation is considered in the model.The rate coefficient for production of He met is assumed to be the sum of the production rates of the He(2 (1,3) S; 3 These assumptions will be discussed further below.First of all, the gas temperature T g is required as it defines the gas density according to the ideal gas law p = n • k B • T g , where p is the measured gas pressure, n is the total gas number density, k B is the Boltzmann constant.Effective rotational relaxation under atmospheric pressure conditions and thermal equilibrium of five degrees of freedom (three translational and two rotational) of diatomic molecules cause equality of the gas temperature and the rotational temperature in the molecular ground state [22], i.e.T g = T rot .Here, it is further assumed that the rotational distribution of the excited states that are observed using OES is similar to that of the rotational distribution of the ground state.The rotational distribution in the excited state is determined by comparison of calculated and measured spectra.
The rate equations for He met (see below for more details) and the two ground vibrational levels of the monitored excited a Collision cross section used to calculate rate constant from EVDF.b Hemet includes the following states: He(2 (1,3) S; 3 3 S; (2, 3, 4, 5) 1 P; 2 3 P).
nitrogen states (N + 2 (B, 0) and N 2 (C, 0)) give a system of differential equations (equations ( 1)-( 3)).These equations include the effective lifetimes of the above-mentioned species, which are defined in equations ( 4)-( 6) Here, n Hemet , n N + 2 (B,0) and n N2(C,0) are number densities (in cm −3 ) of helium metastables, N + 2 (B, 0) and N 2 (C, 0) states, respectively; k Hemet exc , k are the rate constants of electron impact excitation (in cm 3 s −1 ) of helium metastables, N + 2 (B, 0) and N 2 (C, 0) states, respectively; k Penning is the rate constant of Penning ionisation (in cm 3 s −1 ) of nitrogen molecules by collision with helium metastables [20]; n He , n N2(X) and n e are number densities (in cm −3 ) of ground state helium atoms, ground state nitrogen molecules and electrons, respectively; τ Hemet eff , τ N2(C-B) are the Einstein coefficients (in s −1 ) for spontaneous emission transitions from vibrational level v ′ = 0 to all vibrational levels of the respective lower electronically excited states; k are the rate constants for quenching (in cm 3 s −1 ) of the respective excited states (N + 2 (B, 0), N 2 (C, 0)) by collisions with helium atoms and nitrogen molecules [17,40,41].B1 = 0.41 is the branching factor for production of N + 2 (B, 0) by Penning ionisation of ground state nitrogen molecules [42] in collisions with helium metastables.For all reactions in the model, the parameters are given in table 1.
The rate constants for electron impact excitation k exc are calculated for different reduced electric field strengths E/N using equation (7) kexc Here, m e is the electron mass, C is a constant with C = 1.602 × 10 −12 g cm 2 s −2 eV −1 and E kin denotes the electron kinetic energy in eV.The mechanism of excitation of the two molecular bands N 2 (C-B,0-0) and N + 2 (B-X,0-0) as well as the corresponding cross sections σ exc (E kin ) (in cm 2 ) for the calculation of the excitation rate constants are well known [18].As mentioned, we consider the production of an effective helium metastable state, He met .To calculate the total production rate of He met excitation the excited states He(2 (1,3) S; 3 3 S; (2, 3, 4, 5) 1 P; 2 3 P) [43] are considered.The inclusion of these states for the production of the effective state, He met , is partly motivated by potential mechanisms that can lead to the redistribution of the population of electronically excited states following the initial excitation process.For example, the concept of collisional relaxation from higher electronically excited levels to lower electronically excited levels via the potential curves of the He * * 2 excimer, as described in [45].Several aspects related to this assumption, including the interpretation of density of He met are discussed later.After calculating the rate constants for each helium state and each reduced electric field, the rate constants of the helium states are summed up for each reduced electric field to obtain the rate constant for the production of He met .In order to obtain the electron velocity distribution function f v (E kin ) (EVDF), the program code 'EEDF' by Napartovich et al is used to solve the Boltzmann equation in local approximation for different reduced electric fields and a defined gas mixture [46].The velocity distribution functions represented in the kinetic energy scale are normalised to fulfil equation (8) (see figure 2) Under the assumptions mentioned above, the ratio of electron impact excitation rate constants for the production of He met and N 2 (C, 0) states can be calculated for each recorded delay time t at each radial r and axial z pixel position.All further parameters that are calculated using this ratio are spatially and temporally resolved.Under constant conditions such as gas composition and pressure, this ratio is only influenced by the local electric field strength in the plasma.The electron density can then be determined using the measured absolute intensity of N 2 (C-B,0-0) transition and the electron impact excitation rate constant of the corresponding upper level, calculated for the determined reduced electric field strength.The measured intensities of molecular emission transitions N + 2 (B-X,0-0) and N 2 (C-B,0-0) can be presented in form of equations ( 9) and ( 10) In the frame of above-mentioned assumptions, the differential equations for the density of helium metastables and the intensities of N + 2 (B-X, 0-0) and N 2 (C-B, 0-0) can be given in the form of equations ( 11)-( 13) by inserting equations ( 9) and (10) In a first step, preliminary values for n Hemet were calculated from the experimentally measured intensities I 0-0 using equation ( 12) and used for further calculations.Then, by dividing ( 11) with (13), the system of differential equations can be solved for the ratio of the excitation rate constants using the preliminary values of n Hemet , calculated from the measured intensities of I 0-0 N2(C-B) , and , as all the remaining terms are constants or, in case of electron density, cancel out.
Figure 3(a) shows the simulated data for the reduced electric field at different ratios of excitation rate constant, that are calculated using simulated electron distribution functions (see figure 2) and the cross sections for electron impact excitation available from the literature for the reactions [18,42].Each ratio in the observed range corresponds to a specific reduced electric field strength, therefore it is possible to determine a unique value of E/N.Due to the large amount of data, a fast calculation of E/N from the ratio was ensured by fitting the data with an exponential function (F1, figure 3(a)) and using the fitted equation to calculate the reduced electric field.
The electron density n e can be determined with equation ( 14), the measured absolute intensity I 0-0 N2(C-B) , the density of nitrogen molecules n N2(X) and the rate constant for electron impact excitation of the N 2 (C, 0) state that is calculated from the reduced electric field strength E/N and the electron impact cross section from [18] and fitted with polynomial F2 (see figure 3(b)) To verify the OES plasma characterisation, helium metastable densities are calculated from the results of the model and compared with experimentally measured helium metastable densities using laser absorption spectroscopy.For this, the density of helium metastables is determined by numerical solution of the differential equation equation (11).The electron density is taken from the results of the model and the excitation rate coefficient is determined from F3 (see figure 3  using the reduced electric field obtained from the model with the assumption that Penning ionisation is mainly causing the decay of the metastable density (equation ( 4)).In this case, F3 represents a piecewise cubic interpolation to determine the excitation rate constants.

Current-voltage characterisation
Figure 4(a) shows the voltage and current waveforms of the plasma during the first positive and first negative half-wave.The total measured current (green dashed line) is a combination of displacement (purple dotted line) and conduction current (red solid line) in the discharge gap.In general, the conduction, or discharge current is the most interesting part with respect to the results discussed later, since it is directly related to the discharge formation.To determine the conduction current, the capacitances C d (capacitance of dielectric) and C cell (total capacitance) as defined in the work of Pipa and Brandenburg have to be estimated [47].C cell was estimated from the Q-V plots as the slope before the discharge ignition, while C d was estimated from the slope of Q max V max plots using a similar method to that shown by Pipa and Brandenburg in [47].With these capacitances, the conduction current j R (t) can be calculated from the measured current i(t) and V(t) using the following equation: Each trigger pulse at a frequency of 1 kHz releases a damped sinusoidal HV pulse sequence with a frequency of about 445 kHz (figure 4(b)).The amplitude decays to zero after ≈ 20 µs (data not shown), while a plasma is ignited only in the first seven half-waves (< 8 µs) under current conditions.The applied voltage has smooth temporal behaviour, but a high frequency modulation can be observed in the first positive halfwave of the conduction current.Phase shifts and frequency changes in conduction current-voltage characteristics prove a strong influence of both the residual charge and a possible change in capacitance on the temporal behaviour of the discharge.In our experiment, one broad smooth current peak can be observed, which is a characteristic of uniform DBDs [5].

OES characterisation of DBD plasma
As an example, a measured emission spectrum of the source is presented in figure 5.With these spectra, the gas temperature is determined on the basis of the rotational distribution of molecular nitrogen.Using an in-house software, the emission spectra of nitrogen molecules is simulated with the same spectral resolution of the applied echelle spectrometer for various temperatures.The N 2 (C-B,0-0) rotational band at λ = 337.1 nm is the most intense feature in the emission spectrum of the helium/nitrogen DBD and typically used for determination of the gas temperature [16].Due to residual impurities of hydrogen (n H2 ⩽ 0.1 ppm) in the applied helium or residual water from the chamber walls, a low intensity emission of NH(A-X) is also observed in the measured spectra.The emission band NH(A-X,0-0) at λ = 335.8nm partially overlaps the rotational structure of N 2 (C-B, 0-0) and can influence the determination of rotational temperature.Therefore, we use the N 2 (C-B, 0-1) emission band (λ = 357.7 nm) for this purpose.Spectra for the N 2 (C-B, 0-1) emission band are calculated at different rotational temperatures.For all calculated spectra, an interval of 355.34 nm to 355.48 nm is chosen where the mean intensity value is determined and normalised to the maximum intensity at 357.7 nm.This relative intensity versus rotational temperature for the simulated spectra is fitted with a polynomial function.The same calculation procedure is applied to the measured spectra and the relative mean intensity is determined in the same spectral interval.The rotational  temperature is then evaluated using the relative mean intensity values of the measured spectra and the polynomial function.The gas temperature of T g = (317.3± 3.2) K in the discharge is determined using 10 spectra measured with the spectrometer without temporal resolution.
Time-resolved OES using the camera and two band-pass filters is used in order to analyse the plasma parameters with temporal and spatial resolution.The images of the uniform helium DBD with an admixture of 0.45% nitrogen are measured with an exposure time of 40 ns and the temporal discharge development is scanned with a time step of 40 ns using the delay generator.As reported by Goldberg et al [11], a solution of the rate equations ( 11)-( 13) can result in non-physical outcome at the negative gradient of measured intensities.The reason for this is the derivative term in the equations that is negative at falling intensities.If the absolute value of this term is larger than the constant term, the left side will be negative, resulting in negative excitation rate constants.Therefore, for the analysis, we focus only on intervals of rising intensity during the first two current pulses of the uniform DBD.
As mentioned, because of the insufficient sensitivity of available detectors and the required accumulation of measured images to reach a sufficient signal, a very important aspect of the applied diagnostic method is the reproducibility of spatial and temporal distributions of the plasma properties.This was verified without band-pass filters by a reduction of the exposure time down to 4 ns where accumulation of multiple images was disabled, as well as analysis of the current voltage characteristics.The signal of multiple images acquired at the same delay time varies in the range of the noise, while the distribution of plasma emission does not show larger variations that indicate stochastic behaviour.Based on these measurements, we can conclude that the helium DBD with 0.45% nitrogen is diffuse with reproducible intensity distribution and the chosen exposure time of 40 ns with a collection of 30 images gives the possibility to receive reliable plasma conditions with sufficient temporal and spatial resolution to apply the collisionalradiative model.
Background corrected images of the discharge captured through (380 ± 5) nm (N 2 (C-B)) and (390 ± 5) nm (N + 2 (B-X)) band-pass filters at different delay times and as recorded by the camera without any modification besides cropping to the region of interest and adding the outline of the driven electrode are presented in figure 6.The delays of the images presented in this figure are chosen to present the variation in the spatial emission characteristics during the discharge ignition with ≈ 170 ns occurring at the start of the first positive half-wave and ≈ 250 ns occurring close to the halfmaximum of the current in the first positive half-wave.At the moment of ignition, the plasma is confined in the gas gap with a width of 0.8 mm between the powered and grounded electrode.With increased time and electric current, the plasma  spreads along the electrode surface, taking a form similar to a surface discharge.A similar temporal behaviour is repeated also in the first negative half-wave of the applied HV.
The DBD is characterised with spatial and temporal resolution during rising intensities, as explained later, at the rising edge of two current pulses, the first positive and the first negative pulse (see figure 4).Briefly, the following steps are required for each image in order to calculate plasma properties: • Determination and subtraction of the background signal.
• Extraction of the region of interest and centring of the radial symmetry axis at the central pixel.• Signal averaging of the left and right halves of the discharge to account for the assumption of a radially symmetric intensity distribution for inverse Abel transform.• Determination of the radial distribution of the emission intensity using an inverse Abel transform of measured images (shown in figure 7).• Intensity calibration considering the pixel volume of (3.16 ± 0.07) × 10 −9 cm 3 , calculated from the pixel edge length of (14. Figure 7 depicts the radial distribution of the intensity of the respective nitrogen molecular emission bands at 170 ns and 250 ns after the HV trigger pulse.As mentioned, the inverse Abel transform calculates the radial distribution of two-dimensional projections of any measured axisymmetric object.Here it must be noted, that in the region of the driven electrode shown with a contour line in figure 7 and the following figures, no light can be emitted, because the electrode is a solid body and located there.The data that is calculated there is set to zero to avoid misleading information and results for subsequent calculations.Further, by comparing the radial distribution of N 2 (C-B) in figure 7 at 250 ns with the recorded image in figure 6, no emission appears at a radius smaller than 3.5 mm, while there is emission distributed across the gap.This light is not emitted from the centre of the gap, but from an annular region with an inner radius of approximately 4 mm.
Figure 8 shows the phase resolved emission of I 0-0 N2(C-B) in two spatial areas R 1 and R 2 that correspond to the regions marked in figure 7. The data in figure 8 is averaged across the radial dimension r to allow an analysis of the time development of the emission of I 0-0 N2(C-B) .R 1 is located at the lateral surface, spanning the total height of the observed region, while R 2 is located in the inner area of the gap.
Initially, the ignition occurs in the gap (R 2 ) during each half-wave (beginning around 0.17 µs and 0.81 µs, respectively), resulting in a conventional volume discharge.Subsequently, the plasma propagates outwards to the lateral surface (R 1 ), forming a ring at the edges and then expanding both upwards along the surface and downwards towards the grounded electrode.This spatial propagation on the lateral surface resembles the behaviour of a surface discharge.The discharge dynamics exhibit characteristics of both volume and surface discharges, occurring at different points in time within the development of the discharge.In the first positive halfwave, the intensity of the volume discharge in the gap peaks during the first observed image (0.17 µs).However, the intensity during the second half-wave has two local maxima: at the first to second image (0.81 µs to 0.85 µs) and at the second to last image (0.97 µs).For the lateral surface discharge, the maxima occur during the final image of the selected sequences for both, positive and negative, half-waves.Overall, these results demonstrate that the discharge originates in the gap and subsequently expands to the lateral surface.
To focus the analysis on the rising edges of the current, the images under investigation are limited accordingly.From this data, the two selected sequences as marked in figure 8 provide consistently rising intensities throughout most of the observed regions, with only minor exceptions such as the rapid collapse of the gap discharge.These selected images are then subjected to the collisional-radiative model.
Using the spatial distributions of emission, the reduced electric field and the electron density are determined in the frame of the collisional-radiative model (see figure 9).In figures 7 and 9, two extreme operating conditions of the OES plasma characterisation are presented, demonstrating the range of the diagnostic and some typical results.At first, the analysis of plasma conditions with low signal intensity, especially outside the gap at 170 ns can be discussed: figure 7 shows a very small measured intensity in the considered area (z > 1.5 mm, |r| > 5 mm), though the measured CCD signal is reliable and reproducible after a detailed analysis of the underlying data.Using the measured OES signal, a quasi homogeneous reduced electric field and the electron density outside the gap at the moment of ignition are determined (see figure 9, delay of 170 ns).Here, electron densities at the lowest end of the applied scale amount to < 10 7 cm −3 and are not visible on the chosen colour scale and range.
Plasma parameters (reduced electric field and electron density) in the gas gap at a delay of 250 ns cannot be reliably determined, but for the region outside the gap, the plasma parameters can be derived (see figure 9).The reason for this effect is the low emission intensity of the N 2 (C-B) band inside the gap and the comparably high intensity in the annular region (see figure 7, delay 250 ns).Under such circumstances, the radial distribution is challenging to determine, because the small signal from the inner region sums up with the large signal of the intense ring.This leads to a radial distribution with non-positive values in the centre, that may be mathematically correct, but not physically meaningful.The temporal development of the discharge in the first positive half-wave shows a reduction of the electric field in the gap, leading to a reduction of high energy electrons that could excite N 2 (C-B) emission.At the same time, the annular region with |r| > 4 mm shows a significant increase of electron density of approximately one magnitude during the sequence.
The spatial distributions of reduced electric field and electron density obtained at six consecutive delay times during the first negative half-wave of the measured current are presented in figure 10.Similar to the first positive half-wave of applied voltage, the characteristics of the measurements and related plasma properties are apparent, i.e. an annular intensity distribution with small intensity at the centre at later time steps.The propagation of the plasma along the driven electrode after a short ignition phase in the gas gap can also be observed in this phase.The duration of the discharge inside  the gas gap is limited to ≈ 80 ns, similarly to the first positive half-wave, presumably because of a fast charge accumulation at the dielectric surface and therefore a decreased electric field in the gap.After this phase, the plasma propagates along the lateral surface of the driven electrode out of the observed region.This phase coincides with the rise of the applied voltage, which, similarly to the first positive phase of the applied voltage, results in an electron density more than one order of magnitude higher than in the plasma located in the electrode gap.
Helium metastables are one of the most important excited species in helium DBDs under the presented conditions.The determination of the helium metastable densities in this study is interesting for two reasons, first, for the validation of the applied OES diagnostic and the collisional-radiative model and second, for the exploration of possible applications of the studied discharge.To validate the quantitative diagnostic method, we calculate helium metastable densities in the frame of the collisional-radiative model using the determined plasma parameters and compare these values with those obtained for the He(2 3 S 1 ) state using TDLAS.This comparison is performed in the gas gap between driven and grounded electrodes, because other locations are not accessible for absorption spectroscopy.
The helium metastable densities are calculated in each pixel by solving the differential equation equation ( 11) using the determined plasma parameters, F3 (see figure 3) and equation equation (14).Figure 11 shows the spatial distributions of the helium metastable densities at the maximum of the positive (t = 250 ns) and the negative (t = 1010 ns) half-wave of the measured current.As shown, helium metastables are not only produced inside the gas gap between the electrodes, but also on the lateral surface of the driven electrode outside the gap.The maximum of the spatially resolved metastable density in the first positive half-wave is about one order of magnitude higher compared to the first negative half-wave.Furthermore, during the first half-wave, there is a considerable density in the gap, while the maximum is located in the annular region around the gap.In contrast, for the first negative half-wave, there is barely a measurable density in the gap, and the maximum is shifted upwards to the lateral surface of the electrode.

Comparison of the model with TDLAS data
In order to compare the spatially resolved helium metastable densities from the OES and collisional-radiative model with the line integrated He(2 3 S 1 ) densities measured by TDLAS, the average values from the spatially resolved measurements are calculated in the cylindrical absorption volume of the laser beam with a diameter of 1 mm through the centre of the gap as follows.A forward Abel transform of the radially resolved helium metastable densities is performed to obtain line integrated helium metastable densities that are comparable to densities measured through the gap using TDLAS.These line integrated values are then normalised to the absorption path length of 10 mm and the circular region of interest at the same position where the laser beam is passing through is extracted.In the end, the average of the extracted values represents a good comparison to the values measured with the TDLAS.These values, which are presented as red circles in figure 12, are compared with the He(2 3 S 1 ) densities measured Comparison of helium metastable He(2 3 S 1 ) densities measured using laser absorption spectroscopy gas gap between DBD electrodes (TDLAS, blue solid line) with densities calculated using the measured plasma parameters in the frame of the collisional-radiative model (CRM, see text).At a delay of ≈160 ns no plasma emission was observed.At this delay, the lowest TDLAS densities of ≈ 10 10 cm −3 may not be reliable, but are presented to show the rising edge of the helium metastable density curve.Confidence intervals of measured and calculated values are determined using error propagation as discussed later.
using TDLAS (solid blue line in figure 12).Confidence intervals of measured and calculated values, presented in figure 12, are calculated using error propagation.While the agreement between the densities inferred from the model and those measured by TDLAS is generally good, it is important to emphasise that these densities are not exactly equivalent.The TDLAS approach measures the He(2 3 S 1 ) directly, whereas the OES model is dependent on assumptions regarding the production and consumption of an effective metastable level, He met .As a result, an exact agreement between the densities derived from both approaches is not necessarily to be expected.The good agreement between the densities inferred by the OES model and those measured by TDLAS may provide an indication that, under our conditions, the density of He met is a reasonable proxy for the density of the He(2 3 S 1 ) state.
Aside from the comparison between helium metastable densities measured by TDLAS and those inferred from the OES data, the temporal variation of the metastable densities in the gap demonstrates some interesting behaviour.Specifically, the density of metastables within the gap is much higher around the positive current peak (≈ 0.2 µs to 0.6 µs) than during the negative current peak (≈ 1 µs to 1.4 µs).This is consistent with the observations in figures 10 and 11 that the majority of electrons and helium metastables are produced at the sides of the electrode close to the time of the negative current peak (≈ 1 µs) and not within the gap itself.

Assumptions used in the collisional-radiative model
As was mentioned before, plasma parameters are determined from the collisional-radiative model under the following assumptions: (a) An 'effective' helium metastable state, He met , that is populated by electron impact excitation is considered in the model.The rate coefficient for production of He met is assumed to be the sum of the production rates of the He(2 (1,3) S; 3 Firstly, some important factors related to the assumption of an effective helium metastable state are addressed.One area of importance is the relationship between the density of the effective metastable and those of individual excited states, in particular, that of the He(2 3 S 1 ) measured by TDLAS.For this, the production and consumption processes of various excited states need to be considered.
In general, different electronic states of helium can be excited by electron impact in the uniform DBD.The largest electron impact excitation rates under atmospheric pressure discharge conditions can be attributed to the He(2 (1,3) S; 3 3 S; (2, 3, 4, 5) 1 P; 2 3 P) states.For the reduced electric field values and gas compositions considered in this study, the rate constants for electron impact excitation of the He(2 3 S), He(2 1 S), He(2 3 P), He(2 1 P) states are similar and together represent the majority of the total electron impact excitation rate.In addition to electron impact excitation, each state can, in principle, be populated by collisional relaxation of excited helium atoms with higher excitation energies during collisions with ground state He atoms.Radiative decay, collisions with electrons and the formation of various He ions can also play a role in the overall kinetics of these excited states.A detailed overview of the importance of several of these processes in high pressure He discharges can be found in [48], for example.
The electronic relaxation of excited helium atoms after collisions with ground state He atoms can occur via formation and dissociation of the highly unstable collisional complex He * * 2 , i.e. a diatomic excimer with energy equal to or higher than the dissociation limit [45,49,50].
In the first two reactions, where a stabilising collision does not occur, there are two possibilities.In the first, the level of excitation remains unchanged via a direct process, or through electronic transfer.In the second, a lower energy He state, He * ↓ , can be formed in an electronic relaxation process.
In the third reaction, a stabilising collision occurs leading to a stable helium excimer, He * 2 , as for reaction 17 below.The total cross section, and the cross section for electronic transfer interactions between (He(2 1 S, 2 3 S)) and ground state He atoms has been determined by [51].From this it is known that these collisions occur at high rates, particularly at atmospheric pressure.However, the precise outcomes of such collisions depend on the specific helium excited state taking part in the collision process, and the potential energy curves of He * * 2 related to the interaction.These have been studied and discussed by a number of authors [50,52].In principle, electronic relaxation from higher levels can occur due to the approaching or crossing of different potential curves [53].From the potential energy curves given in [52], electronic relaxation from the He(2 1 S), He(2 3 P), He(2 1 P) levels towards lower levels appear to be possible.
A subset of the potential reaction pathways have been studied in [54].For example, collisional relaxation from the He(2 1 P) level to the He(2 1 S) was proposed from the potential energy curves of [52], and the work of Payne et al [55].However, this reaction is not supported by the data of Lawler et al [54], which favours the formation of stable He * * 2 under the conditions studied.Lawler et al [54] also found that, at least up until pressures of 300 Torr, there was no evidence of collisional relaxation from He(2 3 P) to He(2 3 S) [54].Rather, under those conditions, radiative decay from the He(2 3 P) to the He(2 3 S) (A = 1.02 × 10 7 s −1 , [56]) level was the most significant process.Beyond this, a detailed set of rate constants for the collisional relaxation of different excited states and the corresponding product channels is, to our knowledge, not available, which makes a quantitative understanding of the electronic distribution of He excited levels after the initial excitation process challenging to determine.With respect to other radiative transitions of importance, the He(2 1 P) decays quickly to the ground state with A = 1.8 × 10 9 s −1 [56].
While these considerations do not give an absolutely complete picture of the excited state kinetics in He, several points of relevance to the OES model presented here can be concluded.If collisional relaxation of all the higher levels to He(2 3 S) would be perfect, the density of He met in the model could be considered equal to the density of He(2 3 S).When this is not the case, and other channels, such as the formation of He * 2 , or radiative processes which do not in the end contribute towards the formation of He(2 3 S) are significant, then the density of He met inferred from the OES is more difficult to interpret, and represents a convolution of the densities of the states whose electron impact cross sections are used to define the production rate i.e.He(2 (1,3) S; 3 3 S; (2, 3, 4, 5) 1 P; 2 3 P).As mentioned earlier, the good comparison between the He met densities inferred from the OES model and those of the He(2 3 S 1 ) state measured by TDLAS provides an indication that the density of He met is a good representation of the density of the He(2 3 S 1 ) under the conditions used in this study.
The second assumption, that Penning ionisation is the main loss channel for the effective metastable state will now be considered.In the presence of molecular nitrogen, the formation of helium excimers according to reaction (17) [57] competes with reaction (18), the Penning ionisation of molecular nitrogen [20,42] for the loss of He met He met + 2He → He * 2 + He ( 17) In case of the He(2 3 S) state, the rate constant of the process in reaction ( 17) is known in the literature and amounts to 1.5 × 10 −34 cm 6 s −1 [57].Under the presented plasma conditions, the frequency of Penning ionisation (8 × 10 6 s −1 , using the rate constant for He(2 3 S), see above) is about two orders of magnitude higher than the frequency of excimer formation (9.4 × 10 4 s −1 ).
To confirm the assumption of negligible participation of metastable N 2 (A) and the ground state of ion N + 2 (X) for excitation of N 2 (C, 0) and N + 2 (B, 0) states, respectively, a comparison of the reaction pathways is performed.For the production of N + 2 (B, 0), there are three possible production pathways, direct electron impact excitation, stepwise excitation via N + 2 (X) and Penning ionisation.The rate constants are calculated assuming the following plasma parameters: a reduced electric field of 40 Td (see figure 10), electron and molecular nitrogen ion density of 10 10 cm −3 and a helium metastable density of 10 12 cm −3 (see figure 11) according to our DBD conditions.With these assumptions, the rates for reactions ( 19)-( 21) amount to 10 12 cm −3 s −1 , 5 × 10 16 cm −3 s −1 and 3 × 10 18 cm −3 s −1 , respectively.
The trigger frequency of the HV power supply amounts to 1000 Hz in our experiment, while we characterise the plasma conditions during the first microsecond of the HV pulse sequence and the total discharge time amounts approximately 8 µs.With an extended collisional radiative model including the stepwise excitation of N 2 (C) via N 2 (A) [16] and the determined plasma parameters, the density of the nitrogen metastable N 2 (A) was estimated to increase during the discharge time up to a maximum of ≈ 1 × 10 12 cm −3 , but should not accumulate to the next HV trigger.The electron impact excitation rates of N 2 (C, 0) amount to 4 × 10 14 cm −3 s −1 and 6 × 10 17 cm −3 s −1 for reactions ( 22) and ( 23), respectively Based on all of these measurements and calculations, we can conclude that the aforementioned assumptions of the presented study are reliable under the current conditions.

Uncertainty of the applied diagnostics
The uncertainty of the presented OES diagnostics with temporal and spatial resolution under atmospheric pressure conditions depends not only on the calibration uncertainty of the applied measurement system, but also on the reliability and uncertainty of the simulation procedures, including the used rate constants and cross sections.We assume that at atmospheric pressure in helium, the effective lifetime of the N 2 (X, C) states is long enough to ensure equilibrium between five degrees of freedom (two rotational and three translational) of the diatomic nitrogen.The temperature value of 317.3 K from the determination of the rotational temperature of N 2 (C-B,0-1) shows an statistical uncertainty of 1%, expressed as the standard error of the mean.The systematic uncertainty is currently difficult to estimate in this case, as this relies on the uncertainty of the simulated emission spectra.The uncertainty of the radial emission distributions calculated from the inverse Abel transform is mainly influenced by the measured data and its deviation from cylindrical symmetry, and to a smaller extent by the selected parameters and algorithm used to perform the inverse Abel inversion.The selected method adds baseline noise filtering due to the nonnegative optimization procedure and suppresses very highfrequency oscillations, leading to a slightly smoothed distribution.But in general, the uncertainty in the inverse Abel inversion method could be considered negligible to the other error sources.
As mentioned in [16], we currently cannot determine the reliability and uncertainty of the electron distribution functions calculated using the Boltzmann solver [46].The most important issue is the reliability of the electron impact cross sections applied within this program.Comparison of the electron velocity distribution functions calculated using 'EEDF' by Dyatko et al [46] and BOLSIG+ by Hagelaar and Pitchford [58] shows only small differences.
The uncertainty of measurement and calibration procedures makes the most important contribution to the confidence interval of the determined plasma parameters.The calibration of the secondary standards (at the PTB, Germany), namely the deuterium lamp (7% at 200 nm − 400 nm) and the tungsten-ribbon lamp (4% at 350 nm, 2.3% at 380 nm, 1.8% at 600 nm and 2.3% at 780 nm) yields an uncertainty of the spectrometer efficiency of 8% over the whole spectral range.The standard deviation of the intensity measurement, using the calibrated spectrometer and the plasma parameters, is determined using quadratic error propagation.The uncertainty of the ICCD camera calibration with the point-like microwave plasma source and the calibrated spectrometer, amounts to 2% measuring N 2 (C-B,0-0) and 5% by determination the intensity of N + 2 (B-X,0-0).The uncertainty of collisional quenching of N 2 (C-B,0-0) and N + 2 (B-X,0-0) amounts to 2% and 10% at the used experimental conditions [17,40,41].The uncertainty of the cross sections of excitation of N 2 (C, 0) and He(2 (1,3) S; 3 3 S; (2, 3, 4, 5) 1 P; 2 3 P) by electron impact of N 2 (X) and He(1 1 S 0 ) are 14% and 30% respectively [18,43].Using quadratic error propagation and data presented in figure 3(a) we determine confidence interval by determination of reduced electric field and electron density to 33% and 23% respectively.

Density of helium metastables
The density of helium metastables in the gas gap between the electrodes is used for validation of the applied diagnostic methods.For this, the metastable densities, calculated using the measured plasma parameters and averaged over a defined volume, are compared with densities measured by TDLAS (see figure 12).The spatially resolved metastable density distributions (figure 11) show a large variation in the densities, that can not be detected using TDLAS, where the averaging happens over the line of sight as well as within the diameter of the laser beam.Therefore, the spatially resolved measurements had to be reduced in dimensions in order to be comparable with the TDLAS measurements.The averaging method presented here is based on the forward Abel transform and reconstructs the line of sight averaged densities, which are observed by the laser, by averaging across the corresponding volume.Therefore, the same region is accounted for in both methods.The experimental and theoretical data presented in figure 12 show a good agreement with overlapping confidence intervals.These results act as an indirect validation of the plasma parameters measured using quantitative OES diagnostics with temporal and spatial resolution.
The standard deviation of helium metastable calculation from the estimated plasma parameters amounts to 38% and is estimated using error propagation of the uncertainty of the collisional-radiative model.Because of the saturation of the absorption signal in the time interval of 0.16 µs to 0.5 µs during the TDLAS measurements, the time dependent metastable density had to be measured at the edge of the determined absorption line profile.The line profile was measured at a constant time (≈ 0.8 µs) after the maximum density to avoid saturation effects and assumed to be time independent.After all, the relative error of TDLAS measurement procedure amounts to about 30%.
The effective production of helium metastables in the surface discharge forming at the lateral edge of the electrode surface may be of interest for different applications.Calculations based on the OES and collisional-radiative model show that the densities of helium metastable in the region of this surface discharge are about one order of magnitude higher than in the gas gap, which is formed by the primary discharge ignition (see figure 11).Possible applications of discharges in such a configuration could relate to the production of extreme UV radiation from helium excimers (λ = 80 nm) or the production of high metastable densities used in ion sources for mass spectrometry.

Conclusion
In this work, an atmospheric pressure DBD operated in helium with 0.45% nitrogen admixture is characterised with temporal and spatial resolution using optical emission spectroscopy, Abel inversion of measured CCD images, numerical simulation and a collisional-radiative model, all combined to a new diagnostic method for the estimation of plasma parameters.The reduced electric field and the electron density are determined with an uncertainty of 33% and 23%, respectively.Comparing the electron densities in the volume and the surface discharges, which are occurring during different timeperiods of the driving voltage and at different locations within the system, the electron density measured in the surface discharge outside the gas gap is about one order of magnitude higher than in the volume discharge in the gap.The new OES diagnostic is successfully validated by the comparison of helium metastable densities in the gas gap calculated using OES-measured plasma parameters with those densities of metastables measured using laser absorption.In each observed half-wave, the studied DBD first ignites in the gas gap of 0.8 mm between the driven and grounded electrodes as a volume DBD.Afterwards, it propagates (slides) along the lateral surface of the driven electrode outside the gas gap as a surface (or sliding) discharge without contact to a grounded electrode, while the volume discharge extinguishes rapidly.The surface discharge is a particularly effective source of helium metastables, with a maximum measured density several times higher than in the gas gap, and, at the same time, a considerably larger volume.In principle, this means that the presented discharge could be applied in different applications, where a high density of helium metastables is favourable.For example, a novel, flat extreme UV light source (60 nm-110 nm [59]) based on this discharge type could be designed, where the discharge is ignited at the edge of a large, planar surface and then propagates across this surface creating high densities of helium metastables without the need for a second electrode for the surface discharge.Further possible applications are surface treatment and mass spectrometry, where the helium metastables can be used as a soft ionisation source.
'PlasNOW-Plasma generated Nitric Oxide in Wound healing' (Project number 430219886) and the SFB 1316 (Project number 327886311) project A5.The authors would like to thank Ahmad Abou Haileh for partial support in conducting the experiments.

Figure 1 .
Figure 1.Schematics of the experimental setups for the characterisation of uniform dielectric barrier discharge in helium/nitrogen mixture.

Figure 2 .
Figure 2. Electron velocity distribution functions (EVDF) simulated in kinetic energy scale obtained by the solution of the Boltzmann equation in a helium plasma with an admixture of 0.45% nitrogen.EVDFs are simulated at different reduced electric field strengths determined in Townsend (1 Td = 10 −17 V cm 2 ). (b))

Figure 3 .
Figure 3. (a) Reduced electric field plotted against the ratio of electron impact excitation rate constants (×) of helium metastable and N 2 (C, 0) state in a helium plasma with an admixture of 0.45% nitrogen.(b) Calculated electron impact excitation rate constants of N 2 (C, 0) (×) and He metastables (•) under the assumption of fast relaxation of electronic excitation.The solid (a), dotted (b) and dashed (b) lines represent a fit (F1, F2) or interpolation of calculated values (F3), respectively.

Figure 4 .
Figure 4. Current-voltage characteristics of the helium DBD with an admixture of 0.45% nitrogen.(a) Detailed view of the first positive and first negative half-wave including total measured, conduction and displacement current.Black arrows indicate the trigger times of the images that are used for plasma characterisation with spatial and temporal resolution later in the manuscript.(b) Overview of total measured current and voltage with extended timescale until 8 µs.

Figure 5 .
Figure 5. Emission spectrum of the DBD in helium with an admixture of 0.45% nitrogen measured in UV spectral range.Positions of several electronic transitions and intense vibrational bands are shown.

Figure 6 .
Figure 6. Background corrected images of the helium DBD with 0.45% admixture of nitrogen for N 2 (C-B) ((380 ± 5) nm) and N + 2 (B-X) ((390 ± 5) nm) with an exposure time of 40 ns.The delay time of the trigger signal is depicted in the images.Reflections of the plasma emission at the surface of the silicon wafer are cropped.The red contour line shows the outline of the cylindrical electrode.

Figure 7 .
Figure 7. Radial distribution of N 2 (C-B, 0-0) and N + 2 (B-X, 0-0) nitrogen molecular emission bands of the helium DBD with an admixture of 0.45% nitrogen at 170 ns (top) and 250 ns (bottom) after the HV trigger pulse.Temporal resolution amounts to 40 ns.The black contour line shows the outline of the electrode.The dashed rectangles marked with R 1 and R 2 show regions that are analysed in detail in the next figure.
7 ± 0.1) µm.• Correction for collisional quenching by dividing the intensities I with A 0 • τ eff , where A 0 and τ eff are the Einstein coefficient and the effective lifetime of the respective transition.• Determination of the reduced electric field in each pixel volume.• Calculation of the rate constant for electron impact excitation of N 2 (C, 0) using the reduced electric field and the polynomial F2 (see figure 3(b)).• Finally, determination of the electron density in each pixel volume.

Figure 8 .
Figure 8. Phase resolved development of I 0-0 N2(C-B) in two regions R 1 (top, at the lateral surace) and R 2 (middle, in the gap) as shown in figure 7 with the current voltage characteristic (bottom) for comparison.The time step and the exposure time for the images amount to 40 ns each.The vertical dashed lines indicate the start and end of the investigated time intervals during the first positive and the first negative half-waves.

Figure 9 .
Figure 9. Reduced electric field (left column) and electron density (right column) determined in the helium DBD with an admixture of 0.45% nitrogen at 170 ns (top) and 250 ns (bottom).Temporal resolution amounts to 40 ns.The black contour line shows the outline of the electrode.

Figure 10 .
Figure 10.Reduced electric field and electron density determined in the helium DBD with an admixture of 0.45% nitrogen in the first negative half-wave of the HV.The delay to the trigger pulse of the HV power supply is depicted.The temporal resolution amounts 40 ns.The black contour line shows the outline of the electrode.

Figure 11 .
Figure 11.Spatial distribution of the helium metastable densities calculated using the OES measurements and collisional-radiative model at two delay values.

Figure 12 .
Figure 12.Comparison of helium metastable He(2 3 S 1 ) densities measured using laser absorption spectroscopy gas gap between DBD electrodes (TDLAS, blue solid line) with densities calculated using the measured plasma parameters in the frame of the collisional-radiative model (CRM, see text).At a delay of ≈160 ns no plasma emission was observed.At this delay, the lowest TDLAS densities of ≈ 10 10 cm −3 may not be reliable, but are presented to show the rising edge of the helium metastable density curve.Confidence intervals of measured and calculated values are determined using error propagation as discussed later.

Table 1 .
Overview of all reactions and their corresponding rate constants with standard deviation considered in the model.
The following reactions show three possible reaction pathways following the collision of helium excited states with helium ground state atoms.Here, He * represents an excited state of He above the lowest excited state, He(2 3 S 1 ), and He * ↓ represents an excited state with lower energy than He *