Study of Formation and Decay of Rare-Gas Excimers by Laser-Induced Fluorescence

The aim of this chapter is to review the experimental and numerical techniques for the estimation of the laser-induced fluorescence (LIF) decay in rare gases using timecorrelated single-photon counting. The advantages of single-photon counting technique are discussed by means of measurement uncertainty analysis. In addition, this chapter provides information concerning the application of this technique to filamentary dielectric barrier discharges (DBD) and radiation trapping of the resonant transitions.


Introduction
Vacuum ultraviolet (VUV) radiation sources are of great interest for many applications such as photochemistry, surface treatment, sterilization, water treatment, photolithography and mercury-free lamps [1][2][3][4]. These applications often use dielectric barrier excimer lamps [5][6][7][8][9][10][11]. Optimization of these VUV sources requires numerical modeling associated with experimental characterization. The determination of the main mechanisms that lead to excimer formation or decay requires accurately determining the reaction rates.
The emission mechanisms of VUV radiations by rare gases are now reasonably understood [12][13][14][15]. Excimers correlated to the first metastable or resonant states are responsible for these emissions. Kinetic models of formation and decay of rare-gas excited states were developed and validated by laser-induced fluorescence (LIF) in the case of argon, krypton, xenon and krypton-xenon mixtures.
Indeed, kinetic studies of rare gases via VUV fluorescence decays, following a brief and selective multiphotonic excitation, recorded by using single-photon counting fluorescence are highly suitable for the determination of reliable kinetic models. Absorption and emission spectroscopies help to complete such studies based on time dependence of VUV fluorescence [12][13][14][15][16]19].
In these experimental studies, VUV fluorescence was monitored by a VUV photomultiplier tube operating in the single-photon counting mode. The emission wavelength was selected by either a VUV monochromator for the recording of the emission spectra or a suitable VUV range passband interferential filter for the excitation spectra and fluorescence decay studies. A multichannel scaler operating with 16,384 channels and a time resolution of 5 ns were used to record the time variation of the VUV luminescence.
Due to the rather weak detected luminous flux, the histograms recorded are characterized by a large number of channels and a low counting rate per bin. The modeling of the fluorescence decay of excited species introduces a temporal function, representing the rate equation model of collisional and radiative decay in a pure gas or a gas mixture. As far as the number densities of excited species in the interaction volume remain low and the laser energy kept low enough to avoid multiphotonic ionization, this function can be fitted by a sum of exponential decays, each one characterized by its own decay frequency associated to a particular single excited state. The number of decay frequencies corresponds to the number of states involved in the formation-decay processes.
In order to determine the temporal evolution law, the experimental histograms must be processed by using the maximum likelihood method, applied to a Poisson distribution which is more suitable than the least squares method for histograms with a low counting rate per channel.
The aim of this chapter is to review the experimental and numerical techniques combined with spectroscopic studies for the estimation of the laser-induced fluorescence (LIF) decay in rare gases. The advantages of single-photon counting techniques are discussed by means of measurement uncertainty analysis. In addition, this chapter provides information concerning the application of these techniques to filamentary dielectric barrier discharges (DBD) and radiation trapping of the resonant transitions.

Mechanisms of VUV emissions by rare gases 2.1. Rare-gas excimers
The emission mechanisms of VUV radiation by rare gases are now reasonably understood. Excimers correlated to the first metastable or resonant states are responsible for these emissions. Figure 1 gives an example of excited molecular states involved in VUV emission of the continua of krypton [17]. The first metastable state and the first resonant one are the precursors to the excimer states. Three-body collisional processes lead to the formation of the high vibrational levels of these excimers. In low-pressure conditions, the transition of these high vibrational levels correlated to the lowest atomic excited states toward the weakly bound ground-state dimers leads to the emission of the first continuum. The second continuum is observed at high pressures, for which collisional relaxation of the high vibrational levels of these excimers is most likely. Thus, the second continuum, arising from radiative transitions of the lowest vibrational levels of the molecular excited states toward the ground state, dominates the spectrum.

Kinetic studies of VUV emissions of rare gases
Kinetic studies using laser-induced fluorescence (LIF) are highly suitable for the determination of kinetic models of formation and decay of rare-gas excimers.
The short-pulsed laser excitation populates only one atomic or molecular state (e.g. only the first resonant or only the first metastable states or the dissociative molecular states correlated to these atomic states). The initial time conditions are known at the beginning of the fluorescence decay, and the analysis of the time-correlated VUV fluorescence is greatly simplified. Non-selective excitation techniques such as electron beam, pulsed discharge or α-particles are not rigorous enough for the determination of kinetic models of the formation and decay of excimers because several processes compete (ionization, electron-ion recombination, electronion formation and destruction, second electron kinetics, etc.) [18,21].
To illustrate the principle of kinetic studies of rare gases, we consider a simple atomic or molecular system of four states displayed in Figure 2. Just after the laser pulse, there is only  This simple model can be used in the case of rare gases. The fluorescence coming from the N 2 molecular state is the first continuum, and the emission of the N 3 molecular state is called the second continuum. The exited state N 1 is the first metastable state.
The radiative decay frequencies α 1 and α 2 are relative to the two radiative states N 2 and N 3 .
The number densities of the different states obey to the following ordinary differential equations: provided that the laser pulse duration is short and the number densities of excited states can be comparatively negligible with respect to the number density of the ground state.
Thus, the initial conditions are The number density N 0 of the ground state is supposed to remain constant. By solving the differential equations and considering the initial conditions, the time evolution of the number densities of the excited states are obtained: Photon Counting -Fundamentals and Applications According to these expressions, the time decay of the VUV fluorescence intensities is given by where V 0 represents the observed volume, Ω is the solid angle of detection, T is the transmission of the optical system of detection, η is the efficiency of the photomultiplier at the fluorescence wavelength λ VUV ¼ c ν and Δt is the time resolution of the detection system. The time acquisition of the fluorescence decay is supposed to be greater than the laser pulse width τ L .
Overall, if the number densities of excited species in the interaction volume V 0 remains weak and the laser energy is low enough to avoid multiphotonic ionization, then collisions between excited species, ion recombination or electron collisions can be neglected. Thus, the simple model developed above can be generalized. The function f(t), which is the rate equation model of collisional and radiative decay in a neutral rare gas, is given by a p e Àb p t þ a 0 (5) where D is the number of excited states involved in the formation-decay processes, b p is the decay frequency of each state and a 0 is the background noise of the detection system.
In pure rare gases, the decay frequency of the p th term with respect to the pressure p is where α p represents the radiative decay frequency, k 2p is the two-body collisional rate constant and k 3p is the three-body collisional rate constant. The sum k 2p N 0þ k 3p N 2 0 is the quenching coefficient Q p . In rare-gas mixtures, the expression of the quenching coefficient is more complex due to the three-body heteronuclear collisional rate constant.
Modeling the experimental decay of the fluorescent intensities at various pressures is the best way to determine the rate equation model of collisional and radiative decay in rare gases. So, a numerical method of computing the decay frequencies b p from the experimental data fitted with a sum of exponential terms is required. The rate constants, α p , k 2p and k 3p , can be determined by processing all the experimental data of the decay frequencies measured in the range of rare-gas pressures. Subsequently, kinetic schemes of formation and decay of rare-gas excimers can be proposed.
3. Single-photon counting method applied to the fluorescence decay recordings 3.1. Single-photon counting As mentioned above, the time function of the fluorescence decay f(t) can be fitted by a sum of exponential terms if the number densities of excited space is very low compared to the number density of the ground state and if the laser excitation is short and selective enough. In order to fulfill these conditions, the laser intensity was considerably reduced to obtain a very low number density of the initially excited species. Thus, the intensity of the fluorescence becomes very weak, and the fluorescence photons are detected as separated pulses using a photomultiplier tube. In our experiments, only a few photoelectrons are detected per laser pulse. The average time intervals between signal pulses are wider than the time resolution of the detection system. In this case, the single-photon counting method using a photomultiplier is very effective and is superior to analog signal measurement in terms of stability and signalto-noise ratio [22].
The single-photon counting method allows the number of photoelectrons detected to be in direct proportion of the fluorescence intensity. The signal pulses are counted by a multichannel scaler during the interval of two successive laser pulses, and the signal measured at each laser pulse period is accumulated at the laser pulse frequency to reproduce the fluorescence waveforms.

Photon counter system
The photon counter system is displayed in Figure 3. The fluorescence photons are collected by a photomultiplier tube (PMT). A non-inverting linear amplifier amplifies the negative output pulses of the PMT. The lower pulses are eliminated by the constant fraction discriminator, while the rest are reshaped. The output logic pulse corresponds to the point on the leading edge of the input pulse where the input pulse has risen to a fraction of its maximum amplitude. Thus, the time jitter is reduced, and the time resolution of the circuit system remains the same over a wide dynamic range of pulse amplitudes. The multichannel scaler counts events as a function of time. The laser trigger starts the counter, which segments photon count data into sequential time bins. The width of the bins can be set from a few nanoseconds to several milliseconds. The instrument records the number of photons that arrive in each bin. The multichannel scaler can be programmed to accumulate several records or set to free run. Each record is added to the current accumulator totals.
An example of time-resolved single-photon counting measurement of laser-induced florescence is shown in Figure 4. The experimental florescence is represented by a histogram with the bin number (proportional to the time) on x-axis and the counts on y-axis.
In most cases, the luminous flux detected being rather weak, the histograms recorded are characterized by a large number of bins and a low counting rate per bin width.
The single-photon counting method is effective when the probability of detection of a single photon is greater than the probability of detection of a more than one photon. Therefore, each count corresponds to an individual incident photon, and these incident photons are detected as separate pulses. This is the case if the pulse-to-pulse interval is much greater than the pulse width.

Detection probability
Using a photomultiplier, when the average number of incident photons is n, the probability of observing exactly y counts per time unit is given by the Poisson distribution: where η is the quantum efficiency of the photomultiplier. η is the ratio of the average number of emitted photoelectrons from the photocathode per time unit to the average number of photon incident on the photocathode. For VUV fluorescence detection, a solar-blind photomultiplier was used with a Cs-I photocathode and MgF 2 window. The sensitivity of these photomultipliers is in the range of 110 to 200 nm, and the efficiency reaches typically 15% at maximum.
If the incident number of photons is sufficiently low, the probability of detection of photoelectrons is proportional to the number of incident photons: The detection of photoelectrons is unlikely, but when it occurs, a single photoelectron is detected because the probability of detection of more than a photoelectron vanishes.

LIF setup
The updated experimental setup for LIF experiments shown in Figure 5 is designed to the selective excitation by multiphotonic absorption of the first resonant or metastable states of rare gases (argon, krypton and xenon) and to the recording of the VUV fluorescence decay of rare-gas excimers or exciplexes.
The excitation wavelength range was obtained with the frequency-doubled light of a laser beam coming from a two-stage dye laser amplifier (Sirah Cobra Stretch) pumped by the second or the third harmonic beam of a neodymium YAG laser (Spectra-Physics). A spectral width of 0.002 nm was obtained with a double-grating oscillator. Typically, the dye laser operates in the visible spectral range, and the frequency of the dye laser beam is doubled by a BBO crystal followed by a compensator. The pulse duration of the frequency-doubled beam did not exceed τ L =7nswitha repetition rate of 30 Hz. The UV laser pulse energy is measured with an energy meter (Ophir Nova II) equipped with a pyroelectric energy sensor. The laser energy of the output beam is controlled by a motorized variable laser attenuator (Standard 10 MVAA) having an extremely low output beam deviation and a wide spectral range emission flux variation.
The vertically polarized VUV laser beam entered a cylindrical stainless steel chamber, and it provided excitation along the cell's axis. The luminescence was observed perpendicularly to the laser beam, through a magnesium fluoride porthole, by means of a VUV photomultiplier tube (PMT: EMR-510G 'extremely solar blind') working in a single-photon counting mode. The detection wavelength was selected by either a VUV McPherson monochromator, for the recording of the emission spectra, or an interferential filter, for the excitation spectra and fluorescence decay recordings. The time dependence of the luminescence was monitored by using a multichannel scaler Stanford Research SR430 with 16,384 channels and a time resolution of 5 ns. The laser and the multichannel scaler were triggered by a digital delay-pulse generator (Stanford Research DG 645). An electrometer may also be added to record ionization spectra and to control multiphotonic ionization. The laser beam was on the axis of a plane capacitor where a dc electric field of 40 V cm À1 was applied. The charges were collected with a Keithley 617 electrometer, which could measure electric currents as low as 10 fA. Table 1 displays some examples of configurations of the lasers used in our studies in accordance with the excited state in the case of argon, krypton and xenon (Racah notation is used). Figure 6 shows an experimental histogram recorded at ≈128 nm representative of the fluorescence decay of the second continuum of argon following the excitation of the Ar4s[3/2] 1 state. Typically, the recorded histograms are characterized by a large number of channels or bins (N C = 16,384) and a low bin width (time resolution, R = 5 ns). For that emission wavelength, the quantum efficiency of the photomultiplier is η = 0.15. The photon counting system provides input pulses of width ≈2 ns. This histogram was recorded during 18,000 LASER pulses. This count is the accumulation number (N A ). The total VUV fluorescence acquisition time is

Experimental histograms
The number of photoelectrons detected is about N PE = 10,000, and the maximum of counts per channel is N PEmax = 15. In average, during this experiment, the probability of counting one photoelectron during a bin width is less than 1.5 Â 10 À5 and reaches 3.5 Â 10 À4 at maximum. The pulse-to-pulse interval is more than 15 μs that is larger than the photoelectron pulse width. Thus, single-photon counting conditions are fully met.
To understand the mechanisms of formation and decay of rare-gas excimers, the histograms recorded at different pressures must be analyzed to determinate the frequency decay rates and also the lifetimes of the radiative states and the quenching rates.

Methods for fitting multiexponential histograms of the fluorescence decay
The luminescence decay curves follow a multiple exponential form after short laser excitation. Two methods for the curve fitting of multi-exponentials to experimental data have been compared. The chosen method needs to be accurate if the experimental signal contains exponential components with similar decay rates or both fast and slow decay rates.

Maximum likelihood method derived from a Poisson distribution
The luminescence decay curves f(t) follow multiple exponential form: ft ðÞ¼ X p¼D p¼1 a p e Àb p t þ a 0 : The fitting method allows to obtain the amplitudes a p and the decay frequencies b p . The number D of exponential terms depends on the number of states involved in the process of decay. For the sake of concision and clarity, let θ ¼ a 0 ; a 1 ; …; a D ; b 1 ; …; b D ðÞ be the 2D +1 dimension vector of unknown parameters of the function f(t).
For fitting multi-exponential decay curves to experimental data, many methods of data analysis can be proposed to minimize the least squares deviation. The estimator to be minimized is where N C is the number of channels (bins) of the histogram, y i is the number of counts for the i th channel observed at time t i and σ i is the standard deviation of the statistical distribution. Here, Taking into account the weak probability p(y i ) of detection of y i counts in the ith channel at time t i given by Poisson's law another estimator is more relevant: the likelihood function defined as the product of the probabilities: The objective of fitting method is to determine the θ parameter that maximizes the likelihood function.
Gradient vanishing ∂L ∂θ ¼ 0 leads to the non-linear systems of D equations: The function f(t) can be specified introducing the resolution R of the system of detection (e.g. the bin width of the histograms) and the channel number i: Thus, a new set of equations is obtained: These equations are solved numerically using the Newton-Raphson method. The iterative root-finding procedures of the Newton-Raphson method need an initial estimate θ 0 graphically obtained. Then, an improved estimate θ u is produced after u iterations. The iterations go on until the relative variations of the estimate θ become negligible.
Linear equations are obtained with a first-order Taylor development of the non-linear equations about the estimate θ u . Thus, the system of linear equations is solved with a classical Gauss algorithm.

Numerical simulations
For a simulated data derived from experimental fluorescence decays of the second continuum of krypton in krypton-xenon mixtures [17]. The exact value of the fluorescence decay is Different data were simulated using a Monte Carlo method with a number of total accumulated counts varying from from 500 to 500 000. The simulated data were fluctuated by a Poisson distribution.
An example of simulated data is shown in Figure 7. Typically, the histograms recorded are characterized by a large number of N C channels and a low counting rate per R = 5 ns width channel. This allows to measure with the same recording, the simultaneous measurement of both slow and fast decay rates with the same resolution and without time integration.
The decay frequency values (b 1 , b 2 , b 3 ) resulting from a least squares fit or a likelihood fit were compared to exact values. Estimation of uncertainties was performed through a statistical analysis (Type A evaluation) [25]. The standard deviation of the first convergent values was combined with the standard deviation of estimation of the decay frequencies obtained by processing the numerical method with a series of Monte Carlo-simulated histograms.
The mean values and standard deviations of the parameter estimations are summarized in Figure 8.    Least squares method gives accurate results for high counting rates, but the method does not converge for low counting rates when the number count per channel is weak and many zeros are recorded per channel. As already demonstrated in the case of time-correlated singlephoton counting (TCSPC) fluorescence decay analysis [24], the maximum likelihood method gives stable results over the whole count range, even for total counts less than 1000, where the least squares analysis delivers unreasonable values or does not converge.
For low counting rates, the maximum likelihood method gives excellent parameter estimations for multi-exponential fits of fluorescence decay curves. This method is ideally suitable for estimating the decay frequencies when the fluorescence decays are recorded with a singlephoton counting system.

Application to the kinetic studies of rare gases
The kinetic studies following the selective excitation of the first metastable or resonant atomic states and of g or u molecular dissociative states correlated to these atomic states are highly relevant. Indeed, in operating conditions of applications, excimer production in discharges is obtained by populating relay states of higher configuration than the first metastable and resonant states, but the latter are quickly populated because the radiative and collisional decay frequencies of these higher states are fast. Finally, the A1 u as well as the B0 + g molecular states are at the origin of VUV emissions. Molecular transitions can also intervene in these cascades, and the dissociative g states can serve as intermediates in populating the first metastable or resonant atomic states.
Laser-selective excitation of the first metastable state or the g parity molecular states can only be achieved with absorption of two photons, while selective excitation of the first resonant state or the u parity molecular states is done by absorption of three photons. The single-photon counting method adapted to laser-induced fluorescence kinetic studies of rare gases ensures to accurate determinations of the rate constants k p for each decay frequency b p involved in the VUV fluorescence decay. The values of the rate constants, k 1p , k 2p and k 3p , are determined using the least squares method by processing all the experimental data of the decay frequencies measured in a range of rare-gas pressures. The combined standard uncertainties of the constant rates were estimated according to the ISO Guide to the Expression of Uncertainty in Measurement [25]. Full details about the method used for the estimation of uncertainties can be found in Ref. [15].

Example of results from analyses of VUV emissions of xenon excimers in pure xenon
A comparative kinetic study of the first and second continua following the excitation of either the dissociative g molecular states correlated to Xe6s[3/2] 1 state or the Xe6s[3/2] 2 state by twophoton absorption has been performed by our team for the first time in a wide range of pressure up to 600 Torr.
Two or three frequency decay rates, depending on the initial state populated, characterize the time evolution of the second continuum of xenon. The pressure dependence (Figure 9) of these decay frequencies allows to determine the formation and decay mechanisms of the xenon excimers.
The first decay frequency b 1 is attributed to the decay of the metastable state by collisions with two ground-state xenon atoms: The second decay time b 2 is unambiguously identified as the lifetime of the lowest vibrational levels of the state 1 u 3=2 ½ 2 (or the O À u 3=2 ½ 2 state): The third decay frequency, only observed when the resonant state is initially populated, corresponds to the decay of the resonant state by collisions with two ground-state xenon atoms: The kinetic study of formation and decay of xenon excimer combined with the spectroscopic studies allows establishing a kinetic scheme describing all the radiative and collisional mechanisms involved in the VUV emission of xenon [16] (Figure 10).
Three-body rate constants relative to the formation of excimers and radiative lifetime of Xe 2 A1 u 3=2 ½ 2 ÀÁ lowv are consistent with those reported in the literature [26][27][28] and are more accurate.
Analysis of the amplitudes of the fluorescence decays allowed the determination of the binary collision rate constant for vibrational relaxation and energy transfer arising from the high vibrational levels of the xenon excimer correlated to the resonant state toward the metastable.

Other rare gases and mixtures
Studies of VUV emission using two-or three-photon absorption laser-induced fluorescence (TALIF) technique were undertaken in argon and krypton. The decays of the first resonant and the first metastable state of argon Ar4s[3/2] 1 and Ar4s[3/2] 2 or Kr5s[3/2] 1 and Kr5s[3/2] 2 were explained with respect to the kinetic model already proposed for xenon [15,20].
The decay of the first resonant state leads to very efficient population of the metastable via the molecular state (B0 u + ) highv and dissociative states correlated with the metastable state. Thus, at high pressure, emission of the second continuum comes from radiative de-excitation of the excimer states (A1 u ) lowv toward the molecular ground state. The study of the temporal behavior of the second continuum of argon shows the formation of excimers only by three-body collisions. The three-body rate constant of formation was measured for the first time under selective excitation of the first resonant state of argon and the first resonant and metastable state of krypton ( Table 2). All these studies allow the clarification of the mechanisms of VUV emission involved in the dielectric barrier discharge.
More complex kinetic schemes were proposed in Kr-Xe mixtures [17]. In mixtures, instead of the continuum of krypton, when a small amount of xenon was added, the first and second continua of xenon were observed, even though Kr5s[3/2] 1 was initially excited, proving the Photon Counting -Fundamentals and Applications efficiency of the energy transfer. From the temporal analysis in pure krypton, the three-body rate constants for Kr5s[3/2] 1 and Kr5s[3/2] 2 and the lifetime of the excimer Kr 2 A1 u have been measured again. In the mixtures, we clearly showed the occurrence of an energy transfer from Kr5s[3/2] 1 to the Xe[5d(7/2)3] level, and the two-body collision rate constant has been estimated. A weak coupling could also intervene between these two states.
When the Xe5s[3/2] 1 or Xe6s[3/2] 2 were initially excited, the formation of the heteronuclear KrXe * exciplexes was clearly shown [18]. The decay frequency of heteronuclear excimers correlated to the xenon metastable state obeys different scaling laws depending on the xenon pressure. The roles of heteronuclear and homonuclear excimers in the formation of VUV emissions of the gas mixture are interpreted.

.1. Monofilamentary dielectric barrier discharge
When performed in a neutral gas, kinetic analysis, following a brief and selective excitation of an identified state, is very reliable and proves to be a powerful tool. But, this analysis cannot be extrapolated with the same gas, subjected to discharge operating conditions, nor can it be considered as a mere juxtaposition of the kinetics of several distinct states. Nevertheless, if, on the one hand, the discharge excitation phase is quite short and, on the other one, time origin is perfectly defined, then an in situ kinetic analysis of the discharge can be achieved with the single-photon counting technique. A monofilamentary dielectric barrier discharge (MF-DBD) meets these requirements and can thus be a good candidate for such studies. Figure 11 shows a schematic diagram of our experimental setup meant to produce spatially stable MF-DBDs. These micro-discharges were achieved in a glass cell, between two identical in-house-made electrodes in a classical double barrier discharge configuration. Each electrode was a 4 mm-diameter cylindrical aluminum rod covered with 0.5 mm-thick alumina, thus ensuring a monofilamentary micro-discharge (MF-DBD) [29]. The adjustable interelectrode distance d was set to 2 mm. A tuneable sinusoidal high voltage, ucell(t) ≈ usup(t), was applied to the electrodes by means of a step-up ferrite transformer driven by a power amplifier. The current, icell(t) ≈ isup(t), flowing through the cell, consisted of a fast-rising pulse (pulsed current) superimposed on a displacement current, was obtained by measuring the voltage drop across a 50 Ω series-grounded resistance. Electric signals were recorded with a 10 GHz digital sampling 1 GHz oscilloscope (TDS7104). For each micro-discharge, the instant of origin Rare gas k 31 (10 À32 cm 6 s À1 ) k 31 (10 À32 cm 6 s À1 ) τ 1u (ns) for photon counting was defined on the leading edge of the fast cell current pulse, using a constant fraction discriminator.
High-speed photography of the micro-discharge was performed with a 3 ns gated intensified charge-coupled device (ICCD) 512 Â 512 array camera (PI-MAX Princeton Instruments) covering the visible to near-infrared spectrum (360-920 nm). Snapshots of the global visible luminescence were taken at regular intervals of 4 ns with an exposure time of 3 ns by accumulating 2000 micro-discharges on the positive half cycle. For each snapshot of the ICCD camera, triggering was performed on the leading edge of the preceding negative current pulse and with a constant delay of about half the signal period. During the high-speed photography of the discharge, the statistical fluctuations on the time lag between two successive current pulses for 2000 events corresponded to a pulse-to-pulse jitter (external jitter) of 3 ns. So, the time position of each snapshot was known with an uncertainty of about 3 ns. The cell was filled with laboratory-grade pure argon, xenon or krypton. Before filling up at the required pressure, the cell was evacuated down to at least 10 À7 Torr. During filling up, the gas was drifted through an aluminum-zirconium getter pump for further purification.
The VUV emissions were recorded with the same device used for LIF experiments and processed with the same numerical techniques ( Figure 12).

Electrical waveforms and discharge development
The development of a single micro-discharge is clearly shown by the successive snapshots given in Figure 13a, together with the cell current depicted in Figure 13b on an expanded time scale. The rhombuses correspond to the first six snapshots given in Figure 13a.    characteristic peak corresponding to the arrival of this avalanche at the anode. The positive streamer (cathode-directed streamer), which is, in fact, an ionization wave due to secondary avalanches, propagates toward and reaches the cathode at about t = 16 ns, forming a conducting channel between the anode and the cathode. The diameter of the luminous filament is 0.26 mm (the maximum current density is about 320 A cm À2 ). This value is comparable with the current density proposed by Kogelschatz [30].
Operating at 10 kHz and at low pressures, between 25 and 150 Torr (Figure 14a), the xenon MF-DBD emitted, within 118-300 nm, both the first and second xenon continua centered, respectively, at 152 and 172 nm, the first continua being more significant up to 150 Torr. Above 300 Torr (Figure 14b), only the 172 nm emission was present. No significant emission was present above 180 nm in the VUV range.

Temporal decays of the observed VUV emissions
After a time t = t 0 , always less than 80 ns, the luminescence decays of both the first and the second continua of xenon were fairly well described, over the whole pressure domain, by a sum of exponential terms. When we included the first 80 ns in the time range of our data processing, the histograms were not correctly represented by exponential terms: the maximum likelihood computation used for the estimation of the time constants did not converge. This feature is consistent with the current flowing through the cell, showing that the currentlyisnearlychokedafter80ns( Figure 15).
For all the working xenon pressures, the luminescence decays of the first continuum were well described, after t 0 , by only two exponential terms. The first decay frequency, b 1 , of this emission is attributed to the decay of the Xe[3/2] 1 resonant state through collisions with two ground-state xenon atoms, leading to the formation of Xe 2 [B0 u + (3/2) 1 ] highν excimers with k 31 = (16.7 AE 2.7)10 À32 cm 6 s À1 and α 11 = (1.6 AE 0.6) Â 10 6 s À1 . α 11 is the inverse of the apparent lifetime of the Xe[3/2] 1 due to radiation trapping of the resonant photons in the cell. Its value depends on the cell's configuration.   [18,37,38] or theoretical works [39],where like us two-body collisions of the resonant state are not observed as well.
After the initial active phase lasting at most 80 ns, the luminescence decays of the second continuum were fairly described, over the whole pressure domain (100 to 550 Torr), by only two exponential terms with decay frequencies b 2 and b 3 . The decay frequency, b 3 , of the second continua is attributed to the radiative decay of the Xe 2 [A1 u (3/2) 2 ] lowν state itself, resulting from vibrational relaxation on the Xe 2 [A1 u (3/2) 2 ] highν excimers, through very fast binary collisions with a ground-state xenon atom. This reaction is too fast to be observed by our photondetecting device. The radiative lifetime of the Xe 2 [A1 u (3/2) 2 ] lowν excimers is τ 1u = (96 AE 6) ns. This value is concordant with literature values [16,26,27,37,38,41,42].

Kinetic scheme
The kinetic scheme given in Figure 16 summarizes the main reactions occurring in the xenon MF-DBD during the post-discharge phase. Only the reactions (R-6), (R-7), (R-8) and (R-11) were directly identified in our kinetic study. Reactions (R-3), (R-4), (R-9) and (R-10) were indirectly identified but are too fast to be measured by our detecting device. Finally, reactions (R-1) and (R-2) were not observed in the kinetic study, but they were identified in our emission spectra at low pressures.
At early stages, electrons and ions, as well as higher excited states, are present in the discharge. Electrons likely excite the lowest resonant and metastable states, which are also indirectly populated via fast cascades issuing from these highly excited states. All these reactions occur during the first 80 ns, while the current is extinguished within 30 ns. Finally, these reactions lead to the formation of Xe 2 [A1 u (3/2) 2 ] highν and then Xe 2 [A1u(3/2) 2 ] lowν which emits, nearly exclusively, the narrow energetic second continuum.

Radiation trapping in rare gases
Resonant states play an important role in excimer formations and are involved in emission of UV sources based on rare gases. Atoms excited in a resonant state will decay to the ground level by emission of resonant photons themselves having a good probability of being absorbed by surrounding ground-state atoms and then reemitted. Imprisonment of resonance radiation or radiative trapping is caused by multiple emission, absorption and re-emission of photons before they leave the cell [28,40].
The escape factor g which characterizes this process is defined as the mean number of absorption-reemission processes. It depends on both the gas used and the geometrical characteristics of the experimental device. The measured apparent lifetime τ a is expressed as a function of the natural lifetime τ n : τ a ¼ g τ n .

Photon Counting -Fundamentals and Applications
Each time a photon is reabsorbed, it can be lost by collisional quenching of the resonant atomic state. So, resonance radiation trapping allows collisional decay to prevail over radiative decay, permitting excimer formation from resonant states.

Experimental results in pure xenon
The selective excitation of Xe[3/2] 1 was achieved by three-photon excitation using a tunable laser. The luminescence of the first resonance line and the first continuum of xenon was observed through an interferential filter centered at 145 nm (FWMH = 17.5 nm). Luminescence was recorded by means of a VUV photomultiplier set in the photon counting mode.
When the pressure is less than 15 Torr, the temporal luminescence decay is well described by a single exponential term. The variation law of the decay frequency (b 1 ) is parabolic b 1 ¼ α 11 þ k 31 N 2 Xe ÀÁ .
This time constant is attributed to the Xe[3/2] 1 state, and the apparent lifetime is 3.79 μs. The natural lifetime is equal to 3.98 ns [26]. The escape factor g = 952: a photon undergoes 952 absorption-re-emission processes, on average, until it leaves the cell in our experimental conditions. Constant k 31 expresses excimer formation: A second exponential term appears only for pressures greater than 15 Torr. The best fit of b 2 gives b 2 ¼ k 32 N 2
The absence of a constant term suggests an excited state with a long lifetime. This time constant was attributed to the Xe[3/2] 2 metastable state. This state is created during the vibrational relaxation of the O þ u state.

Experimental results in xenon-krypton mixtures
This study was performed in the same excitation and detection conditions as for pure xenon. This study shows that the foreign gas (krypton) at high concentration modifies trapping phenomena and that van der Waals broadening by krypton should be taken into account. There exists no energy transfer between the two rare gases when the foreign gas (krypton) is lighter than the excited one (xenon).
On the contrary, efficient energy transfers occur from a lighter rare gas toward a heavier one, like in Kr-Xe mixtures following excitation of Kr.

Conclusion
Kinetic studies of rare-gas VUV fluorescence decays under short and selective multiphotonic excitation conditions using single-photon counting fluorescence are highly suitable for the determination of reliable kinetic models. The use of the maximum likelihood method offers the possibility to lower excitation energies and to determinate fast and slow decay frequencies in a same scan. This allows us to determine the main mechanisms involved in the formation and decay of rare-gas excimers.
For high pressures, emission of the second continuum comes from radiative de-excitation of the excimer states correlated to the first metastable state toward the molecular ground state. The study of the temporal behavior of the second continuum of pure rare gas shows the formation of excimers only by three-body collisions. The three-body rate constant of formation of excimers was measured for the first time under selective excitation. These studies allowed us to clarify the mechanisms of VUV emission involved in the dielectric barrier discharge. Resonant states do not seem to contribute to the production of excimers in a dielectric barrier microdischarge. The resonant state contributes to the formation of excimer through the transfer toward the metastable state at high pressures. The second continuum essentially comes from radiative transition of the low vibrational levels of the excimer correlated with the metastable state.