Investigation of the threshold intensity versus gas pressure in the breakdown of helium by 248 nm laser radiation

Abstract

An investigation of the unexpectedly strong dependence of the threshold intensity on the gas pressure in the experimental study on the breakdown of He by short laser wavelength (Turcu et al., in Opt Commun, 134:66–68, 1997) is presented. A modified electron cascade model is applied (Evans and Gamal, in J Phys D Appl Phys, 13:1447–1458, 1980). Computations revealed reasonable agreement between the calculated thresholds and the measured ones. Moreover, the calculated electron energy distribution function and its parameters proved that multiphoton ionization of ground and excited atoms is the main source for the seed electrons, which contributes to the breakdown of helium. The effect of diffusion losses over pressures <1,000 Torr elucidated the origin of the strong dependence of the threshold intensity on the gas pressure. Collisional ionization dominates only at high pressures. No evidence for recombination losses is observed for pressures up to 3,000 Torr.

1 Introduction

The invention of laser devises and its subsequent rapid development to yield very high-power pulses operating at wavelengths ranging from the far infra red to X-ray region of the electromagnetic spectrum lead to the discovery of many new and interesting phenomena during the interaction of laser radiation with matter. One of these phenomena is the electrical breakdown of gases. Although there have been a great deal of researches reported on the breakdown of gases by lasers operated at different wavelengths (see for example [3–12]), the results obtained for shorter wavelengths, however, showed a healthy contradiction concerning the threshold irradiance for breakdown in the nanosecond timescale under laser wavelengths in the UV regime [13, 14]. Generally, the physical models that have been emerged from theory and experiment include processes acting as sources for seed electrons (multiphoton ionization, tunneling ionization and inverse bremsstrahlung) as well as mechanisms responsible for the rapid build up of electrons density (cascade ionization by collision).

The interest of laser induced rare gases plasma is related to the interaction of laser radiation with metal targets during material processing such as drilling and welding. In such processes, the plasma plays an important role in energy conversion between laser and materials. The plasma strongly absorbs and refracts the laser radiation and changes significantly the energy transfer from laser beam to the materials. This shows that rare gases are useful media for material processing; therefore, it is essentially to give a real understanding of the correlation between gas pressure and the physical processes, which contribute to their breakdown mechanism.

Accordingly, in this paper, a detailed numerical study is performed to investigate the threshold intensity as a function of gas pressure when a 248 nm laser source with 18 ns pulse length induced breakdown of helium gas over pressure range 150–3,000 Torr. The laser was focused into the sample gas cell with a f = 9 cm fused silica aspheric singlet lens to produce a ≈ 9 μm diameter focal spot. The effective volume of the breakdown region is taken as a long thin cylindrical shape, which is estimated to be of the order of ~10⁻⁹ cm³. These experimental conditions are deliberately chosen since it showed a strong dependence in the measurements of the threshold intensity as function of gas pressure.

The study based on the modification of an electron cascade model previously developed by Evans and Gamal [2], to investigate the breakdown of simple structure helium atom (a ground state, one excited state and an ionized state) considering an approximate formula for the helium momentum transfer rate v _m which is independent of the electron energy. Moreover, the investigations were devoted to experimental conditions in which there is a combination of pressure, laser pulse length and focal volume, so that there would be no appreciable recombination of electrons and positive ions, nor electron losses by diffusion out of the focal volume. On the contrary, the modified model applied in this analysis takes into account this loss processes and treated the helium momentum transfer rate v _m as a function of the electron energy. Besides, it adopted for the helium atom the assumption given in [5] which considers the group of electronic excitation levels lying above the first excited state to be represented by the meta-stable state at threshold 19.805 eV. This state is actually a composite of four of the lowest excited levels, which are very closely spaced in energy (0.3 eV), and the population of each is strongly mixed by electron collision. This is a reasonable assumption and simplified the calculation. Consequently, in the present work, the helium atom is treated as four levels atom, which in turn enhances the contribution of the two-step ionization processes. The four levels are considered as two exited states having thresholds 19.805 and 21.203 eV, in addition to a ground state and an ionized state at energy of 24.586 eV.

Following to this assumption, the processes which are involved in the model can be summarized as: (i) electron inverse bremsstrahlung absorption, (ii) electron impact excitation, (iii) electron impact ionization of the ground state atoms, (iv) photo-ionization of the ground state atoms, (v) photo-ionization of the excited state atoms, (vi) collisional ionization of the excited state atoms, (vii) electron diffusion out of the focal region and finally (viii) electron-ion recombination processes (three-body recombination). For more realistic results, the rate coefficients and cross sections of these processes are considered as a function of the electron energy.

In the present work, direct multiphoton ionization of ground state atoms is very limited compared to those of the excited ones. Moreover, ionic dimmer formation, followed by electron recombination which leads to the formation of excited atoms, is ignored for the experimental conditions under investigation [9]. In contrast to previous electron cascade models [4, 6, 15] in which steady-state calculations of the electron energy distribution function were performed disregarding the effect of laser pulse shape, in this model the electron energy distribution function is assumed to vary with time and thus more realistically represents the conditions found in practice. Accordingly, the model solves numerically the time-dependent Boltzmann equation for the electron energy distribution function (EEDF) and a set of rate equations describing the rate of change of the formed excited states population.

Taken these considerations into account, a computer program was run to investigate the following relations: (i) the threshold intensity as a function of gas pressure, (ii) the effect of loss processes on the variation in threshold intensity with gas pressure, (iii) the EEDF at the peak and end of the laser pulse, (iv) the time evolution of electrons and excited states density, (v) variation in the excitation rate, ionization rate and electron mean energy and with time, and (vi) the time evolution of the EEDF during the laser pulse. The last four relations are calculated at three selected values of the experimentally tested gas pressures that represent different regimes.

2 Theoretical formulation

2.1 Basic equations

A detailed description of the theoretical approach used to describe the electron cascade model is given in [2]. Here, we summarize only the outlines of the model. The time-dependant Boltzmann equation for the EEDF which represents the energy gained from the laser field by electrons is given by,

$$\frac{\partial n(\varepsilon ,t)}{\partial t} = \frac{2\varepsilon }{{3m\nu_{m} (\varepsilon )}}\nabla^{2} n + \frac{1}{3}\varepsilon_{0} \nu_{m} (\varepsilon )\frac{\partial n}{\partial \varepsilon } + \frac{2}{3}\varepsilon_{0} \nu_{m} (\varepsilon )\frac{{\partial^{2} n}}{{\partial \varepsilon^{2} }} + {\text{Inelastic}}\;{\text{collision}}\;{\text{term}}$$

(1)

where ε ₀ = e ² E ²/2 mω ² is the average oscillatory energy of an electron in the laser field with electric field E and angular frequency ω, e and m are the electronic charge and mass, ν _m (ε) is the momentum transfer collision frequency and n(ε) is the electronic density at energy range ε and ε + dε. In this equation, the first term on the right-hand side represents the rate of electrons loss from the radiated volume due to diffusion. This term is responsible for the focal volume dependence of the breakdown threshold and is influenced by the frequency of the incident radiation [16]. Here, we shall follow the custom to rewriting this by setting ∇² n = −Λ² n, where Λ is the diffusion length [4]. The second term on the right-hand side represents the electron energy gain; the third term expresses the diffusion of electrons along the energy axis. This energy diffusion term has been obtained from classical arguments based on expressions for the mean energy gain per collision for various types of interactions given by Pert [17], which is analogous to the term involving energy diffusion appeared in the energy gain equation derived by Zel’dovich and Raizer [15] as a consequence of using a quantum–mechanical treatment. The inelastic collisional terms include generation of electrons with energy ɛ through ionization of ground and excited atoms as well as atomic excitation by electron impact. It also comprises the loss terms due to electron diffusion and electron recombination.

The model is applied corresponding to the considered experimental conditions given in [1] for helium gas.

2.2 Cross sections and rate coefficients

The cross sections and rate coefficients of helium gas for the kinetic reactions considered in this analysis are taken from experimental data available in literature as follows:

2.2.1 Momentum transfer cross section

The cross section of the momentum transfer (σ _m) due to electrons collision with helium atoms has been obtained from the results given in [18] such as:

$${\sigma}_{\text {m}} = 6. 8 7 4\times 10^{ - 1 6} { \exp }( - \varepsilon / 2 4. 5 8 6)$$

(2)

where ε is the electron energy expressed in eV. The rate coefficient of this process is written as:

$$\upsilon_{\text{m}} (\varepsilon ) = {\text{N}}_{0} p\sigma_{\text{m}} (\varepsilon )[ 2\varepsilon \left( {\text{eV}} \right)/{\text{m}}]^{ 1/ 2}$$

where N₀ is Loschmedt number (3.56 × 10¹⁶) and p is the gas pressure in Torr [19].

2.2.2 Collisional excitation rate coefficients

The rate coefficients of collisional excitation for the two considered excited states are taken from [5]. The formula for the excited states 3s and 3p at threshold energies 19.805 and 21.203 eV, respectively, is given by:

$$\begin{gathered} \upsilon_{\text{ex1}} (\varepsilon ) = 0\quad \varepsilon < 1 9. 80 5\;{\text{eV}} \hfill \\ \upsilon_{\text{ex1}} (\varepsilon ) = \{ [{\text{A}} + {\text{B}}(\varepsilon - \varepsilon_{\text{ex1}} )]\upsilon_{\text{m}} (\varepsilon )\} \quad \varepsilon > 1 9. 80 5\; {\text{eV}} \hfill \\ \end{gathered}$$

(3)

$$\begin{gathered} \upsilon_{\text{ex2}} (\varepsilon ) = 0\quad \varepsilon < 2 1. 20 3 \;{\text{eV}} \hfill \\ \upsilon_{\text{ex2}} (\varepsilon ) = \{ [{\text{B}} + {\text{A}}(\varepsilon - \varepsilon_{\text{ex2}} )]\upsilon_{\text{m}} (\varepsilon )\} \quad \varepsilon > 2 1. 20 3\; {\text{eV}} \hfill \\ \end{gathered}$$

(4)

where A = 7.2 × 10⁻³, B = 7.5 × 10⁻², ε_ex1 and ε_ex2 are the energies of the lower and upper excited levels.

2.2.3 The ground state collisional ionization rate coefficient

Using the empirical formulae that based on the experimental measurements given by [5], the ionization rate coefficient is written as:

$$\begin{gathered} \upsilon_{\text{i}} (\varepsilon ) = 0\quad \varepsilon < 2 4. 5 8 6 {\text{ eV}} \hfill \\ \upsilon_{\text{i}} (\varepsilon ) = [C\upsilon_{\text{m}} (\varepsilon ) \, (\varepsilon {-}\varepsilon_{\text{i}} )]\quad \varepsilon > 2 4. 5 8 6\;{\text{eV}} \hfill \\ \end{gathered}$$

(5)

where C is a constant and equals 6.5 × 10⁻³, ε_i is the ionization energy.

2.2.4 Two-step collisional ionization rate coefficient

Due to the lack of experimental data, we adopt the same assumption for the rate of this process given in [2], which implies the same functional form as the ionization rate of the ground state atoms as follows:

$$\upsilon_{\text{exi 1}} (\varepsilon ) = [C_{ 1} \upsilon_{\text{m}} (\varepsilon ) \, (\varepsilon {-}\varepsilon_{\text{i}} + \varepsilon_{\text{ex1}} )]\quad \varepsilon > 2 4. 5 8 6 { }{-}\varepsilon_{\text{ex1}} \;{\text{eV}}$$

(6)

$$\upsilon_{\text{exi 2}} (\varepsilon ) = [C_{ 2} \upsilon_{\text{m}} (\varepsilon ) \, (\varepsilon {-}\varepsilon_{\text{i}} + \varepsilon_{\text{ex2}} )]\quad \varepsilon > 2 4. 5 8 6{-}\varepsilon_{\text{ex2}} \;{\text{eV}}$$

(7)

In these relations C ₁ = C ₂ = 6.5 × 10⁻¹.

2.2.5 Multiphoton ionization rate coefficient

The rate coefficient of multiphoton ionization of the ground and excited states is calculated using a relation given by [20], such as

$$A = \frac{{\sigma^{k} }}{{\upsilon^{2k - 1} \left( {k - 1} \right)!h^{k} }}$$

(8)

where σ is the photo-absorption cross section (10⁻¹⁶ cm²), ν is the laser frequency, h is Planck’s constant and k is the number of photons absorbed by an atom to be ionized.

2.2.6 Electron diffusion

Diffusion losses take place during the laser pulse with high probability at the low gas pressure values. The rate coefficient of this process (free electrons diffusion) is given by

$$\nu_{\text{D}} = 2\varepsilon /( 3 {\text{m}}\varLambda^{ 2} \nu_{\text{m}} (\varepsilon ))$$

(9)

where Λ = b/2.405 is the diffusion length of electrons in a cylindrical focal volume of radius b and length l ≫ b.

2.2.7 Electron-ion recombination

The electron-ion recombination is considered to be three-body process. The rate coefficient is estimated from a formula given in [15] such as:

$$R = 8. 7 5\times 10^{ - 2 4} n(\varepsilon )/\varepsilon^{ 9/ 2 } \left( {{\text{cm}}^{ 3} /{\text{s}}} \right)$$

(10)

where n is the electron density and ε is the electron energy.

2.3 Method of calculation

Equation (1) is solved numerically using step-by-step integration method. The step length in energy Δε was chosen, so that the complete energy distribution could be represented by 30 equally spaced points. This covers and exceeds the first ionization energy threshold of helium at 24.586 eV. The derivatives ∂n/∂ε and ∂²n/∂ε² are solved using the finite difference technique.

The ionization and excitation as well as loss processes could be included as difference terms in the basic differential Eq. (1), but there are good reasons for treating them separately [2]. The breakdown criteria used in this analysis are taken as attainment of degree of fractional ionization δ = 0.1 % of the neutral gas atoms present in the focal volume. Adopting the experimentally considered pulse shape, the temporal variation in the laser intensity in the focal volume is assumed to have a Gaussian distribution in the form

$$I(t) = I({\text{o}})A(e^{{\frac{{ - (t - \tau )^{2} }}{{4\tau^{2} }}}} - B)$$

(11)

where A and B are constants having the values 4.5208 and 0.7788, respectively, I(o) is the laser peak intensity and 2τ is the laser pulse width (FWHM). To avoid the discontinuity of the EEDF, the calculations started with one electron having a Gaussian energy distribution with a mean energy of 4 eV. In addition, the depletion of the ground state as well as the excited states due to ionization and excitation is also taken into account.

A computer program was undertaken including the Boltzmann equation, a set of rate equations describing the change in excited states population together with the relations that represent the rate coefficient of each physical process considered in this model.

3 Results and discussion

A computer analysis is performed to investigate the threshold intensity dependence on gas pressure in the breakdown of He by KrF laser radiation under the experimental condition given in [1]. In this experiment, He over a pressure range 100–3,000 Torr is between 4 × 10¹¹ and 2 × 10¹³ W/cm² cm². The laser was focused into the sample gas cell with a f = 9 cm fused silica aspheric singlet lens to produce a ≈ 9 μm diameter focal spot. To account for the strong dependence of the threshold intensity on the gas pressure observed experimentally, computations are carried out to obtain first a relationship between the threshold intensity and gas pressure when both diffusion and recombination losses are taken into account. Figure 1 represents a comparison between the measured thresholds [1] (scattered solid circles i) and the calculated ones (curve ii). It can be seen from this figure that a reasonable agreement is obtained between the calculated thresholds and the experimentally measured ones over the whole range of the tested gas pressure, both show a strong dependence on gas pressure. This result assures the validity of the model in investigating the considered experimental measurements. This decrease is attributed to the effect of diffusion losses over the pressure range, since at this short laser wavelength (high photon energy hν ~ 5 eV) one anticipate that ionization and breakdown are governed mainly by photo-ionization mechanism for which the threshold intensity is almost independent on the gas pressure. To confirm the origin of this slope, the calculations are repeated in the absence of diffusion losses. These are represented by curve (iii) on the same figure. The obvious deviation between the calculated threshold intensity and the experimentally measured ones over the low-pressure range (100–1,000 Torr) shown by this curve indicates the significant contribution of this process to the breakdown of helium at this experimental condition as well as it interprets the origin of the observed slope.

Fig. 1

Comparison between experimental (i) and calculated threshold intensities as function of the gas pressure obtained for, all processes (ii), no diffusion (iii) and no recombination (iv)

On the other hand, the effect of recombination losses on the threshold intensity is verified by repeating the calculations when this process is ignored (curve iv). The calculated thresholds are found to be in consistency with its values that obtained in its presence as shown by curves (ii) and (iv). This gives concrete evidence that this process has a negligible contribution to the breakdown phenomenon even at the higher pressure region. This may be attributed to the fact that the recombination coefficient R of helium is very much smaller than for other noble gases [21]. It is not surprising therefore that there is no indication for any increase in the threshold intensity at the high-pressure values. From this result, it is clear that there is a significant correlation between gas pressure, threshold intensity and the physical processes which contribute to the breakdown mechanism. Consequently, it is worthwhile studying deeply the effect of gas pressure on the physical mechanisms which contribute to this phenomenon. In doing so, secondly calculation of the electron energy distribution function (EEDF) and its parameters are carried out at three selected values of the gas pressure, namely 150, 825 and 2,100 Torr, at laser intensities 1.65 × 10¹³, 5.48 × 10¹² and 2.76 × 10¹² W/cm², respectively. (Hereafter, these will be corresponding to curves (1), (2) and (3), respectively, on the figures). These values are deliberately chosen since they represent different pressure regimes.

In Fig. 2, the EEDF is plotted for the three values of the gas pressure at the peak (A) and end (B) of the laser pulse. From Fig. 2A, it is clear that the values of the EEDF’s are almost coincide over the energy range from 1.0 to 20.0 eV as shown by curves (1), (2) and (3) where a peak is observed at 2.0 eV followed by a gradual decrease up to 20.0 eV (first excitation threshold). Beyond this energy, a sharp decrease is observed with their tails directed toward an energy region above the ionization threshold for the higher pressure value curve (3). While for curves (1) and (2), the tails are directed toward the high-energy region. This behavior can be explained due to the fact that free electrons gain energy from the laser field through elastic collision process; this allow them to accumulate enough energy exceeding the excitation threshold of the gas atom; eventually, they lose this energy in excitation. This process in turn results in high population of low-energy electrons (due to ionization of the formed excited atoms by photo and collisional processes), which can repeat the same process. Accordingly, increasing the gas pressure leads to an increase in the inelastic collision rate. This in turn increases the population of excited atoms and hence the free electrons density. This result is observed clearly in Fig. 2B where the values of the EEDF are increasing with the increase in the gas pressure over the energy range from 1.0 to 20.0 eV. Immediately after this energy, the EEDF showed a steep decrease with their tails directed toward a lower energy region (from 25.0 eV down to 22.0 eV). It is also noticed here that as the gas pressure increases, the tail is more directed below the ionization threshold.

Fig. 2

Electron energy distribution function at the peak (A) and end (B) of laser pulse plotted for pressures (1) p = 150 Torr (dash line), (2) p = 825 Torr (solid line), (3) p = 2,100 Torr (dash dot line)

Figure 3 represents the time evolution of electron density during the laser pulse at the three considered cases as represented by curves 1 (150 Torr), 2 (825 Torr) and 3 (2,100 Torr). It can be seen from this figure that the curves start with one electron followed by a fast increase during the first nanosecond in a descending manner where curve (1) lying above curves (2) and (3). Immediately after the peak, a reverse behavior is noticed, where the rate of electrons growth represented by curve (3) (the higher pressure) exceeds those represented by curves (2) and (1). Moreover, the rate of electrons growth suffers a leveling off behavior at the lower pressure (curve 1). This behavior assures that electrons generate with high rate at the low-pressure regime through multiphoton ionization process, and hence, they can easily overcome the losses due to electron diffusion. While at the high pressures, the slow electron growth rate gives an evidence for the domination of the collisional ionization processes. This behavior is confirmed by the fast electron growth rate observed during the descending part of the laser pulse. The low value of the electron density shown during the early stages of the laser pulse indicates the effect of diffusion losses as a process which acts to retard the electron growth rate, in particular over the low-pressure regime.

Fig. 3

Time evolution of the electron density at pressures; (1) p = 150 Torr (dash line), (2) p = 825 Torr (solid line) and (3) p = 2,100 Torr (dash dot line)

To study the role played by the formed excited atoms in the breakdown of helium, Fig. 4 illustrates the time evolution of the excited atoms growth for the three pressure values as shown by curves (1), (2), and (3). It is clear from these curves that the rate of growth of the excited atoms showed similar behavior for curves (1) and (2), where they start with low values, and then, a fast increase followed by a gradual one is continued up to the end of the laser pulse. As the gas pressure increases (curve 3), different behavior is observed, where the growth of the excited atoms showed a noticeable oscillation during the first half of the laser pulse followed by a fast increase up to its end. These oscillations envisage the competition between the processes which act to populate (electronic excitation) and depopulate (photo and collisional ionization) the excited states. Beyond the peak, the rate of growth of the excited atoms increases in an ascending way with the gas pressure. This indicates the role of inelastic collisional processes, which are responsible to populate the excited states.

Fig. 4

Time evolution of the excited states density at pressures; (1) p = 150 Torr (dash line), (2) p = 825 Torr (solid line) and (3) p = 2,100 Torr (dash dot line)

To confirm this result, an investigation of the significance of the time evolution of the calculated excitation rate per electron at the three values of the gas pressure is demonstrated by curves (1), (2) and (3) in Fig. 5. It is shown from this figure that during the early stages of the laser pulse, the excitation rate is more effective at the low pressure as indicated by the peak observed in curve (1). For the higher pressures, in contrary, a noticeable drop over the same time interval as showed by curves (2) and (3). Beyond this time, the excitation rate follows almost the Gaussian pulse shape but with very low values. The behavior shown by these curves indicates the high collisional excitation rate during the first few nanoseconds at the low-pressure regime. This opposes the drop of the excitation rate shown at the higher pressures, which explains the low growth rate of the excited states population shown by curves (2) and (3) in Fig. 4.

Fig. 5

Temporal variation in the excitation rate plotted for pressures; (1) p = 150 Torr (dash line), (2) p = 825 Torr (solid line) and (3) p = 2,100 Torr (dash dot line)

To correlate between the depletion rate of the excited atoms and the growth rate of the electron density as a function of gas pressure in Fig. 6, the time evolution of the calculated ionization rate per electron is plotted for the three pressure values (curves 1, 2 and 3). It is clear from this figure that the ionization rate starts with high value at 1.0 ns followed by a fast drop reaching a very low value at 4 ns. Afterward, the ionization rate follows the same Gaussian pulse shape with its peak coinciding with the laser pulse. It is also shown that the peak value is slightly increasing with the increase in the gas pressure. The high value of ionization rate shown during the early stages of the laser pulse indicates the important role played by multiphoton ionization process, whereas this Gaussian shape behavior assures the domination of collisional processes in populating the excited states, which can immediately ionized by multiphoton and collisional processes during the period 4.0 ns up to the end of the laser pulse.

Fig. 6

Temporal variation in the ionization rate plotted for pressures; (1) p = 150 Torr (dash line), (2) p = 825 Torr (solid line) and (3) p = 2,100 Torr (dash dot line)

From the above results, it can be seen that collisional processes are mainly dependent on the electron energy; therefore, it is advisable to study the effect of gas pressure on the variation in the electron mean energy with time. Figure 7 presents this relation for the three gas pressures as shown by curves (1), (2) and (3). It is clear from this figure that the rate of growth of the electron mean energy is faster for the low gas pressure on the period from 1.0 to 4.0 ns, while its value is reduced as the gas pressure increase. Above this time, the electron mean energy shows almost leveling off behavior up to the end of the laser pulse with the three curves (1), (2) and (3) corresponding to values 7.3, 6.7 and 6.3 eV, respectively. This behavior confirms that photo-ionization processes govern the breakdown of helium over the low-pressure regime, while collisional ionization is mostly prevailing at the high pressures, which is indicated from the lower values of the electron mean energy due to electron energy loss through inelastic collisional processes. This result is consistent with the electron density growth rate shown in Fig. 3 (Table 1).

Fig. 7

The electron mean energy as a function of time represented at; (1) p = 150 Torr (dash line), (2) p = 825 Torr (solid line) and (3) p = 2,100 Torr (dash dot line)

Table 1 Calculated values of photo-ionization coefficient determined for the two electronic excited states and ground state of helium with the corresponding photon energy, minimum number of photons absorbed by the atom to reach the ionization threshold and the residual energy carried by the emitted electron

For a deeper understanding, the EEDF at different time intervals is calculated and plotted for the three pressures as shown in Fig. 8 (i) 150 Torr, (ii) 825 Torr and (iii) 2,100 Torr.

Fig. 8

Time variation in the electron energy distribution function during the laser pulse calculated at; (i) p = 150 Torr (dash line), (ii) p = 825 Torr (solid line) and (iii) p = 2,100 Torr (dash dot line)

From this figure, it is observed that the overall behavior of the EEDF is almost the same for the three pressures showing a cut-off at about 20 eV (the excitation limit). Moreover, it is evident that the shape of the EEDF is time dependent. The peaks appearing at about 2 eV for the three pressures at time intervals covering the laser pulse duration are probably due to those electrons that overcome diffusion losses and reach higher energy values up to the excitation threshold (20 eV). On the other hand, for the low-pressure value (Fig. 8 i), the rate of growth of the EEDF is higher during the early stages of the laser pulse. This indicates the important role played by the multiphoton absorption process in ionizing the formed excited states. The fast decay of the EEDF at 2 ns immediately after the peak (2 eV) refers to the effective contribution of inelastic collision processes. This result gives an evidence for the high excitation rate during this time interval which is consistent with the results shown in Fig. 5.

This is not the case at the high pressure (Fig. 8 iii) where the EEDF showed a slower growth rate at 2 ns which assures that ionization occurs mainly via collisional processes during this time interval. At the intermediate pressure, however (Fig. 8 ii), diffusion losses is found to be less pronounced. This is shown from the higher values of the EEDF at 2 ns. It is also noticed from this figure that the growth of the EEDF follows the temporal variation in laser intensity. Moreover, as the gas pressure increases, the tail of the EEDF ends with lower values and their tail is directed toward the low-energy region. This behavior indicates the negligible contribution of recombination losses to the breakdown of helium under the investigated experimental conditions.

4 Conclusion

Applying the modified electron cascade model, a study is preformed to investigate the strong threshold intensity dependence on gas pressure observed in the measurements that carried out by Turcu et al. [1], in their experiment which is undertaken to analyze the breakdown of helium by short laser wavelength (248 nm). The calculated threshold intensities showed reasonable agreement with the experimentally measured ones over the tested gas pressure range when diffusion and recombination losses are taken into account. This result confirms the validity of the model. Moreover, the inclusion of these processes into the model could explain the strong dependence of the threshold intensity on the gas pressure. This was confirmed by studying the separate contribution of each process to the threshold intensity. Diffusion losses are found to be effective over pressures less than 1,000 Torr, while no evidence is observed for the effect of recombination.

The study of the EEDF and its parameters revealed that multiphoton ionization of the ground and formed excited states is responsible for the production of high electron density during the early stages of the laser pulse over the low-pressure regime. This process is offset by diffusion losses which act as a snick for the low-energy electrons. This in turn retards the ionization growth rate and hence explains the high threshold intensity observed at this pressure region.

The experimental condition applied in this model clarified an important fact that at high photon energies, collisional processes are still dominant during the interaction of high-power laser with gas target. This result is extracted from the obtained constant value of the mean energy retained by electrons during a long time interval of the laser pulse. Increasing the gas pressure leads to a decrease in the mean energy. This behavior is confirmed from the study of the temporal evolution of the electron density as a function of gas pressure where it was found that during the descending part of the laser pulse, the electron density increases as the gas pressure increases.