A numerical investigation of laser wavelength dependence on the threshold intensity of spark ignition in molecular hydrogen over a wide pressure range is presented. A modified electron cascade model (Gamal et al., 1993) is applied under the experimental conditions that carried out by Phuoc (2000) to determine the threshold intensity dependence on gas pressure for spark ignition in hydrogen combustion using two laser wavelengths namely; 1064 nm and 532 nm. The model involves the solution of the time dependent Boltzmann equation for the electron energy distribution function (EEDF) and a set of rate equations that describe the change of the formed excited molecules population. The model takes into account most of the physical processes that expected to occur in the interaction region. The results showed good agreement between the calculated thresholds for spark ignition and those measured ones for both wavelengths, where the threshold intensities corresponding to the short wavelength (532 nm) are found to be higher than those calculated for the longer one (1064 nm). This result indicates the depletion of the high density of low energy electrons generated through multi-photon ionization at the short wavelength via electron diffusion and vibrational excitation. The study of the EEDF and its parameters (viz, the temporal evolution of: the electron density, ionization rate electron mean energy, …) revealed the important role played by each physical process to the spark ignition as a function of both laser wavelength and gas pressure. More over the study of the time variation of the EEDF explains the characteristics of the ignited spark at the two wavelengths for the tested pressure values.
Several fundamental analytical methods as well as experimental studies have been initiated to elucidate the interactions associated with the phenomenon of laser spark ignition in a suitable gas. The generated high electron density could continue to absorb energy from the focused laser beam developing eventually what is called laser sustain plasma.
It is of critical importance to determine the laser conditions at which a spark is produced. A knowledge of these conditions is practically important not only for fundamentally understanding the ignition process, but also for the selection of lasers optics windows and beam delivery system for the design of practical laser to be used in the measurement for studying the laser spark ignition.
Although a considerable bodies of study on laser induced breakdown in gases have been reported (see for example [1-8]), the breakdown phenomenon of common combustion gases such as hydrogen has not been available. Because of the importance of such phenomenon, in various applications it attracted the attention of many researchers, in particular the mechanism that convert the laser energy to thrust energy or mechanical energy that could be approached via high power laser produced plasma (LPP) in a gas breakdown process [9-13].
Due to the high efficiency of the molecular hydrogen gas to convert the laser energy into thermal energy, researchers were interested to study this phenomenon in this gas owing to its importance as a thrust gas in rockets or space vehicles. Initially this phenomenon was studied using a CO2 laser radiation [14,15], later on studies are followed to measure the laser threshold intensity for plasma formation in this gas using lasers that operating with wavelengths in the UV, visible and IR regions over a wide range of the gas pressure. Among these studiesthat carried out by [6,16] during their study of antiStokes Raman scattering using two different focal length lenses to focus a 248 nm KrF laser beam into molecular hydrogen gas at a wide pressure range. They observed an optical breakdown at the beam waist of the focused pump beam which is found to be a major limiting factor when one attempts to increase the pump intensity. These measurements are investigated by [
A detailed description of the model is given in [18,19]. Here we summarize only the outlines of the model where the electrons gain energy from the laser field through inverse Bremsstrahlung absorption, while their initial generation proceeds via multi-photon ionization process and grows during collisional ionization of ground state as well as photo-ionization and collisional ionization of the formed excited states.
In our computations the hydrogen molecule is treated as four level molecule which consists of: a ground state, two electronics excited states; the: state at 8.85 eV, and the state at 12.0 eV, and the ionized state. The vibrational and rotational structures of the molecule are considered only for the ground state.
Therefore the model encounters the collisional and radiative processes given as: 1) electron inverse Bremsstrahlung absorption, 2) electron impact excitation of the state, by electrons of energy ε > 8.85 eV, 3) electron impact excitation of the state, by electrons of energy ε > 12.0 eV, 4) electron impact excitation of the vibrationally excited states over the energy range 1.0 - 10.0 eV, 5) electron impact ionization of ground state molecule with electrons having energies ε > 15.43 eV, 6) collisional ionization from the lower excited state by electrons having energies ε > 15.43 - 8.85 eV, 7) collisional ionization from the higher excited state by electrons having energies ε > 15.43 - 12.0 eV, 8) photo-ionization of the lower excited state, 9) photo-ionization of the higher excited state, 10) rotational excitation of the ground state molecule, (this process is treated as an elastic loss mechanism), 11) dissociation of the excited molecules in the state, into two neutral fragments in the electronic ground state, 12) recombination losses and, 13) diffusion of electrons out of the focal volume.
On the basis of these physical processes, the time evolution of the electron energy distribution function described by Boltzmann equation is written as
where represents the number density of electrons with energies in the range between and, is the average oscillatory energy of an electron in the laser field with an electric field amplitude E and angular frequency w, is the momentum transfer collision frequency between an electron and a molecule, and Qv and QR represent the rates of transfer of energy from an electron of energy, to vibrational and rotational levels respectively.
In this equation, the first term on the right-hand side represents the rate of electrons loss from the radiated volume due to diffusion. Here, we shall follow the custom to rewriting this by setting, where Λ is the characteristic diffusion length. This length characterizes the distance over which a particle should diffuse in order to be lost from the plasma. Following standard optical focal theory, the minimum focal volume is assumed to be cylindrical [
Empirical formulae for collisional cross sections are obtained using curve-fitting technique for the most recent experimental data published in literature by [
The momentum transfer cross section was obtained as a function of the electron energy, using a least-squares fit for experimentally measured values given by [
In general, vibrational losses in hydrogen represent an important energy sink for electrons which cover a wide range of electron energies (1 eV < ε < 11 eV). Here, these losses are considered as being due to elastic collisions for electron energies ε < 3 eV, and as being due to inelastic collisions for electron energies in the range 3 eV < ε < 11 eV. Corresponding expressions for cross sections based on curve-fitting of experimental data given by [24,25] for the low and high energy ranges respectively are:
The obtained formulae for cross sections of the two excited electronic states considered here are represented by [
The expression for the ionization cross section was derived from experimental measurements carried out by [
Owing to lack of experimental data, rate coefficients for collisional ionization of the two electronic excited states are obtained using an analytical formula given by [
where k = 1, 2 indicate the two excited states and Ɛki is the energy difference between each of the excited states and the ionization limit, and respectively.
The photo-ionization coefficients of the excited states are estimated using a formula given by [
where s is the photo-absorption cross section of a molecule, u is the frequency of the laser light and K is the number of photons absorbed by an excited molecule leading to its ionization.
As a function of the electron energy, the cross section for rotational excitations of the ground state molecule can be represented by the following expression, which is based on the experimental results of [23,25]:
The expression of the dissociation cross section is obtained from the experimental data of [
The three body recombination rate constant as a function of the electron energy is taking from a relation given by [
In this work the rate of diffusion losses is defined as:
Here, the characteristic diffusion length Λ is considered to be energy independent. Adopting the experimenttal conditions considered in this analysis [
Equation (1) is solved numerically using a step-by-step integration method. The energy step-length Dɛ is chosen so that the complete energy distribution could be represented by about 25 equally spaced steps. This covers and exceeds the first ionization energy threshold of the molecular hydrogen (15.43 eV). The derivatives and are evaluated using the finite difference technique. The inelastic collision terms (ionization and excitation as well as loss processes) could have been included as difference terms in Equation (1), but there are good reasons for treating them separately [
The modified electron cascade model is applied to investigate the experimental measurements that carried out by [
son the measured thresholds of [
Reasonable agreement is obtained between the calculated and measured values for the two laser wavelengths with the thresholds obtained for the short wavelength 532 nm, lies above those obtained for the longer one 1064 nm. This result gives an evidence for the validity of the model to investigate the experimental measurements. This in turn encouraged us to carry out calculation of the EEDF to study the physical processes (responsible for spark ignition) dependence on the gas pressure for the two laser wavelengths.
From this figure it is noticed that the values of the calculated EEDF at the peak of the pulse are lower than those obtained at its end. Also a decrease of its values at 1.0 eV is observed only at higher pressures for the two considered wavelengths. This decrease reflects the effect of vibrational excitation with electrons having energies < 3.0 eV. Moreover the reduction of the EEDF at the low energy region indicates the high rate of inelastic collisional processes which acts to deplete electrons with high energies.
To study the physical process responsible for the source of the electron growth during the early stages of the laser pulse,
torr (curve 1), 760 torr (curve 2) and 3000 torr (curve 3).
It is shown from this figure that for the longer wavelength
via collisional processes. Diffusion losses are effective only at the lower gas pressure region. At the shorter wavelength (λ = 532 nm), however, (
It is noticed here that the exhaustion of the excited molecules is more pronounced at the higher pressure value (curve 3) for the shorter wavelength (532 nm). At the longer wavelength (
In
the shorter wavelength increasing the gas pressure results in a slight decreases of the ionization rate. This gives an evidence for the high competition of the diffusion and vibrational losses at this wavelength.
To assure this result,
To find out the characteristics of the formed spark under the tested experimental conditions Figures 8-10 show the contour representation of the time evolution of the electron density corresponding to explicit electron energy range for the three values of gas pressures at the wavelengths (a) 1064 nm and (b) 532 nm. These figures examined precisely the time evolution and generation of the formed plasma as a function of both gas pressure and laser wavelength. It is clear from these figures that at low pressure (
increases the plasma generation extends over a longer period during the descending part of the laser pulse. This indicates the important role played by collisional processes which are more effectives at high pressures. Moreover the size reduction of the formed plasma at the shorter wavelength shown in Figures 8, 9 and 10(b) gives an evidence for the high rate of energetic electron loss out of the focal volume through diffusion.
In the present work a modified electron cascade model is applied to investigate the experimental measurements that carried out by Phuoc (2000) [
assumed physical processes included into the model. Moreover the calculation of the EEDF as well as the time evolution of the electron density, electron mean energy and ionization and excitation rates, revealed the exact role played by low energy electrons through vibrational excitation and its competition with diffusion losses at the low pressure region for the wavelength 1064 nm and over the whole pressure range for the shorter wavelength 532 nm. The study of the time variation of the EEDF clarifies the exact period for spark ignition as well as its characteristics (size, electron density and temperature) as a function of both gas pressure and laser wavelength.