Possibility of estimating high-intensity-laser plasma parameters by modelling spectral line profiles in spatially and time-integrated X-ray emission

Abstract

We address an issue of measuring the parameters of an envolving laser-produced plasma commonly observable in high-energy density physics experiments. Available diagnostic equipment does not provide enough temporal, and often spatial, resolution to distinguish the signal coming from the region and timeframe of outmost interest, where deposited energy density reaches its maximum. In this paper, we propose and describe an approach that makes it possible to estimate the plasma parameters existing at the time of the main laser pulse arrival, as well as on later stages of plasma expansion. It is based on the analysis of X-ray spectral line profiles in multicharged ion spectra registered in simple time and spatially integrated mode. As an example, specific calculations were made for Ly_β line of Al XIII and He_β line of Al XII and can be used to diagnose aluminum plasmas with an electron temperature of 400–1000 eV, assuming that expanding plasma was homogeneous at every moment.

1 Introduction

Study of the interaction of high-power laser radiation with matter remains a very important and interesting topic for over half a century. The appearance of lasers with a picosecond, and later with a femtosecond, pulse duration made it possible to increase the laser radiation power by many orders of magnitude and carry out experiments with laser radiation fluxes reaching values of the order of 10²² W/cm² (e.g., [1,2,3]). The purpose of this kind of experiments besides fundamental studies of the equations of high-energy density matter [4, 5] includes the generation of ultra-bright X-ray sources [6,7,8], laser accelerators of charged particles [9,10,11], and inertial confinement fusion [12,13,14].

First, almost every experiment requires measuring the parameters of generated plasma. One of the main diagnostic tools here is X-ray spectroscopy of multiply charged ions, which has proved to be very informative, for plasma generation by both relatively long (nanosecond) and short laser pulses. However, the transition to ultrahigh laser fluxes is accompanied both by the shortening of the laser pulse (up to tens of femtoseconds) and by the improvement of its focusing (up to 1 µm diameter). As a result, a generated plasma object is characterized by extremely high space-time gradients, but an improvement of X-ray spectrometers over the past decades has not been so significant. One can observe X-ray spectra with a temporal resolution not better than 1 ps and with a spatial resolution not better than several microns, which is much more than the space-time gradients inside the objects being studied. Thus, the question arises as to what information the time-integrated emission spectrum can give and how much the measured parameters will reflect the plasma parameters at the end of the heating laser pulse, which is of greatest interest to the researchers. This issue becomes especially important in the case of experiments on heating the so-called mass-limited targets, e.g., nanofoils [15,16,17] or thin wires [18].

It should be noted that the most important parameter characterizing the laser-matter interaction is the density of the region of matter, where the main part of the laser pulse energy is absorbed. For an ideal ultra-short (femtosecond) laser pulse, this density is equal to the density of the unperturbed substance, i.e., for a solid target, it is equal to the density of a solid state. However, if the main laser pulse has a pre-pulse with an intensity that is sufficient for plasma formation, then the main pulse will be absorbed in the plasma region with a critical electron density, and the ion density in this region will be noticeably lower than the solid-state one. Let us note that the transition to relativistic values of the laser intensity increases the value of the critical density up to the so-called relativistic critical density, but it remains significantly lower than the solid state even at laser intensities of the order of 10²² W/cm² (e.g., [19]). If a target is a very thin (nanometer) foil, and the laser pre-pulse has a long duration, then the main laser pulse will interact with an even less dense plasma. Meanwhile, the plasma density in the laser absorption region affects many extremely important parameters, such as energy density, degree of ionization, pressure, and so on. Therefore, determining the density of this region is one of the main diagnostic issues. Moreover, it should be noted that even an answer to a simpler question of whether the absorption region is solid or not is already sufficiently informative since it allows reaching the conclusion on magnitude of the laser contrast, in particular.

In this paper, we discuss a widely known method for determining the plasma density using the widths of the spectral lines of H- and He-like multiply charged ions. We estimate what quantitative information can be retrieved when applying this method to spectra obtained without temporal and spatial resolution.

2 The plasma model and the diagnostic lines selection

Let us assume that the main femtosecond laser pulse interacts with a plasma having an initial electron density n_e,1 and volume V₁. After the end of the heating pulse, the macroscopic motion of the plasma begins (or continues, if the laser pre-pulse has already created the expanding plasma). The plasma does not cease to emit. In the first approximation, one can assume that the expansion is adiabatic, i.e., T_e·V^γ−1 = const, where T_e is the electron temperature and γ is the adiabatic exponent. The adiabatic index for plasma is usually in the range of 1.2–1.67 [20]; therefore, we consider it to be an average value of 1.4. Please note that an adiabatic expansion case was chosen as a general one, but it is reasonably possible that certain experiment expansion model would be different from considered one.

The time dependencies of the plasma parameters can be approximated by step functions, i.e., we consider the plasma as stepwise successive stages (or layers) S_j (where j is a stage number) with different density values of ions n_ion,j and electrons n_e,j; the temperatures of the ions T_ion,j and electrons T_e,j and the stage lifetime (the duration of existence) t_j of stage j. Let the lifetime of plasma in the initial state S₁ be equal to t₁. The set of S_j is determined by the condition that the plasma ion density of the j state is α ^j−1 times lower than in the initial stage, that is

$${n_{{\text{ion}},j}}={n_{{\text{ion}},1}} \cdot {\alpha ^{ - (j - 1)}},$$

where α is a constant that characterizes the quality of approximation of real-time dependences by stepwise ones. Then, for the stage S_j, its volume and electron temperature are

$$\begin{aligned} {V_j} & ={V_1} \cdot {\alpha ^{(j - 1)}}, \\ {T_{{\text{e}},j}} & ={T_{{\text{e}},1}} \cdot {\alpha ^{ - \left( {j - 1} \right) \cdot (\gamma - 1)}}, \\ \end{aligned}$$

and assuming that the expansion occurs with a constant speed, the time of existence of stage j (for j > 1) is

$${t_j}={t_1} \cdot {\alpha ^{\left( {j - 1} \right)/m}} - {t_{j - 1}},$$

where m is the dimension of the expansion. Since we are only interested in the relative contributions of different stages to the total spectrum, we can assume that t₁ = 1, V₁ = 1. Below, we consider two cases: m = 1 (planar expansion) and m = 3 (spherical expansion). In both cases, the value of α = 10^0.25 was used, which provides a rather good approximation; the less α the better, but the difference with α = 10^0.125 case is insignificant. The above dependences of the electron temperature, ion density, volume, and existence time of the stage as a function of their numbers n are presented in Fig. 1 (for m = 1).

Fig. 1

Values of electron temperature T_e,j (black solid line), electron density n_e,j (black dashed line), volume V_j (red solid line), and lifetime t_j (red dashed line) of different stages (layers) of expanding and adiabatically cooling plasma as a function of stage number j

Since as density and temperature of the plasma decrease, its luminosity tends to fall (with the exception of a recombining plasma, where the luminescence growth can be caused by a sharp drop in temperature with a slow decrease in density), later stages play an gradually lower role in the formation of observed line widths of time-integrated spectrum. This means that the initial stages “leave their trace” in the final spectrum, from which one can try to evaluate the parameters of these stages. The more the ionization degree of the considered ion is, the higher will be the role of the initial stages. Therefore, let us choose for the further consideration the lines of the ions with a maximum ionization potential. It should be noticed that in this case, the recombination effect mentioned earlier will be rather negligible.

The width of the spectral lines depends both on the ion plasma temperature (due to the Doppler effect) and on its density (due to the Stark effect). Due to the Stark effect, the lines of hydrogen-like ions broaden most strongly, and their widths are proportional to n_q², where n_q is the principal quantum number of the upper level of the corresponding radiative transition. It would be most advantageous to use the most sensitive spectral lines with a large n_q. However, first, line intensities decrease rapidly with the growth of n_q (and there are some difficulties with their experimental registration), and second, these lines can disappear at very high densities due to lowering of the ionization boundary. Therefore, lines with too large values of n_q are not the best choice. A compromise solution may be the second terms of the resonance series of H- and He-like ions, i.e., the Ly_β and He_β lines. It is these lines that are considered in this paper. The results of numerical calculations given below refer to aluminum ions since Al foils are used very often in experiments of high-intensity laser-matter interaction.

The plasma emission spectrum intensity E_j(λ) for each stage S_j was calculated in the stationary approximation by the radiation–collision kinetic code [21]. Then, time-integrated spectrum E(λ) was obtained by simply summing the obtained spectra multiplied by the weight coefficient k_j = t_j·V_j, taking into account the existence time of each stage and the corresponding volume:

$$E(\lambda )=\mathop {\lim }\limits_{{J \to \infty ~}} {E_{{\text{sum}},J}}(\lambda ),$$

where

$${E_{{\text{sum}},J}}(\lambda )=\sum\limits_{1}^{J} {{k_j}{E_j}(\lambda )}.$$

The profile of the spectral line A with the central wavelength λ₀ was determined by the spectrum E(λ) in the vicinity of λ₀; the contributions of the bremsstrahlung and photorecombination radiation were subtracted from it. We denote by W_H(J,A) the total width of line A at a relative height of H (for, e.g., H = 0.5 corresponds to full-width at half-maximum, FWHM) calculated using the function E_sum,J(λ). Then, the width of the line observed in the absence of time resolution will be determined by the value W_H(A) = W_H(∞,A), and the line width corresponding to the first stage of plasma evolution (the initial stage) will be equal to W_H(1,A). We also denote their ratio by R_H(A) = W_H(A)/W_H(1,A). The value of W_H(A) could be measured in an experiment, and some values of R_H(A) were estimated in this study; therefore, it makes possible to determine the line width and then the plasma density at the initial stage of plasma generation.

3 Modelling results

Let us first consider the calculations results for the Ly_β line of the Al XIII. The plasma parameters corresponding to the S_j states for the case of plane expansion (m = 1) were determined in accordance with the discussed model and are shown in Fig. 2. The initial value of the ion density n_e,1 was taken of 6 × 10²² cm⁻³ (solid-state aluminum) and the initial electron temperature T_e,1 was of 1 keV. This figure shows the calculated k_j·E_j(λ) profiles of the Ly_β spectral line, as well as the timely and spatially integral profile E_sum,J(λ) (the grey-filled area in the figure; note that it was multiplied by 0.4 for better visual comparison; J = 6 in this case). Figure 3 shows the time- and spatial-integrated intensities E_sum,J(λ) for different J, i.e., this figure shows the process of plasma stages evolution.

Fig. 2

Spectral profiles k_j·E_j (λ) (color lines) of the Ly_β line calculated for different plasma stages S_j (for j = 1–6); the sum of all these stages E_sum,6 (λ) multiplied by 0.4 for better visual comparison (the grey-filled area in the figure) and also widths of the observed lines W_0.5 in legend

Fig. 3

a Sums of the spectral profiles of first J stages of plasma evolution (for J = 1–8) of the Ly_β line, E_sum,J (λ) and b spectral profiles for the same sums, but normalized by the condition E_sum,J (λ) = 1

As can be seen in Fig. 2, the maxima of the spectral intensities of the lines corresponding to steps S₃ and S₄ and are higher than for the initial stages of the plasma S₁ and S₂ [E₄(λ₀) > E₃(λ₀) > E₁(λ₀) > E₂(λ₀)]. In addition, the width of the integral line, taken at FWHM, differs from the FWHM of the initial state S₁ more than two times, but is relatively close to the FWHM of the layers S₃ and S₄ [W_0.5(1, Ly_β) > W_0.5(Ly_β) ~ W_0.5(3, Ly_β)]. From Fig. 3, it can be seen that the late stages of expansion give an essential contribution to the central regions of the line profile (for ex., the most central region stop changing noticeably only starting from J > 5), while the line profile of peripheral region is already formed during first stages.

This can be explained by the fact that the full width of each subsequent spectral line decreases with the evolution of the plasma, but a relative increase in values of the weight coefficients k_j partially compensates it for few initial stages. In other words, since each subsequent stage exists longer than the previous one and has a larger volume (t_j+1 > t_j, V_j+1 > V_j), the stages with j = 1–6 have a noticeable effect on the central part of the resulting spectral line profile.

Different stages make a different contribution to the formation of the final profile of the spectral line, with the influence of earlier stages of expansion being more noticeable at the peripheral part of the integral profile, and the influence of the later stages, on the contrary, in its central region.

Figure 4 shows the ratio of the full widths of the total profiles E(λ) to the full widths of the profiles E₁(λ) corresponding to the initial states, i.e., R_H(Ly_β) as a function of relative height H. Calculations were carried out for different values of the initial plasma parameters. The widths of the profiles were determined at different relative heights (1/10, 1/4, 1/e, ½, and 3/4 of the maximum). As can be seen, the widths of the lines by 0.1 of relative height are least likely to change (the lower the initial density, the less this change). This is due to the fact that the cooling and expanding plasma contributes less and less to the periphery region of the total spectral line.

Fig. 4

Values of the R_H(Ly_β) parameter as a function of H for one-dimensional adiabatic plasma expansion for different initial ion density values n_ion,1 and electron temperature T_e,1

This means that in order to diagnose the initial state of the plasma, it is preferable to use its full width at the lowest possible relative height. Unfortunately, with this approach, we confront with a limitation caused by the presence in the observed spectrum of both noises of a different nature and the nearby satellite structures that do not allow us as accurate measurement of the width of the spectral line at the desired low relative height.

We note that a rather weak correlation is observed between the values of R_H(Ly_β) and n_ion,1. The situation is different for the temperature: it can be seen that the lower the initial temperature of the plasma, the bigger R_H(Ly_β), i.e., the smaller the difference between the FWHW of the profiles E₁(λ) and E(λ). This can be clearly seen in Fig. 4, and has a simple explanation: at a low initial temperature, when T_e is much smaller than the excitation energy of the Ly_β line, a further decrease in temperature during the expansion process leads to an exponential decrease of this line luminosity. This means that even with a small value of J ~ 1–2, the relation E(λ) ~ E_sum,J(λ) will be satisfied.

Figure 5 shows a comparison of the calculation results for different cases of the expansion models: three-dimensional (m = 3), one-dimensional (m = 1), and zero-dimensional (in latter case, there is no plasma expansion and the plasma layer with the initial parameters n_ion,1 and T_e,1 = 500 eV is considered). This picture shows a correlation between W_0.5(Ly_β) and n_ion,1 demonstrating that the one-dimensional and three-dimensional models of expansion give us rather close results. Therefore, the choice of the plasma expansion model corresponding to a particular experiment can give some refinements into the observed values of the full widths of the spectral lines, but in the first approximation, they do not depend on the expansion type. Since the current importance of experiments on heating of ultrathin foils with thickness less than the diameter of the laser spot, we will discuss below only the one-dimensional expansion model, which is still adequate for such experiments.

Fig. 5

Values of the R_0.5(Ly_β) parameter as a function of the initial value of the ion density n_ion,1 for cases of a three-dimensional (red line), one-dimensional (blue line), and zero-dimensional (orange line) plasma expansion during adiabatic cooling with the initial temperature T_e,1 = 500 eV

Regardless of the relative height H at which the value of W_H is determined, the width of the observed profile E(λ) differs from the width of the profile E₁(λ). This difference [i.e., the value of the parameter R_H(Ly_β)] depends mainly on the initial temperature T_e,1 and, to a much lesser extent, on the initial ion density n_ion,1. This can be clearly seen in Fig. 6. If the initial temperature of the plasma is known (e.g., it can be determined from the relative intensities of the dielectronic satellites [22]), then the data of this figure can be used to measure the initial ion density of plasma. It should be noticed that dielectric satellites of radiative transitions 2p^{2 1}D₂–1s2p ¹P₁ and 1s2p^{2 2}D–1s²2p ²P of the Ly_α and He_α lines, respectively, are most convenient for measuring the initial temperature. These satellites are both the most intense of all satellite lines and the most distant from the corresponding resonance lines so that even in case of solid-state density plasma, they do not overlap the main lines (e.g., [23]).

Fig. 6

Values of the R_0.5(Ly_β) (a) and R_0.5(He_β) (b) as a function of the initial plasma temperature for different initial ion density values for a one-dimensional expansion of aluminum plasma

To measure plasma density, we need to divide the measured line width Ly_β by the value of the parameter R_H(Ly_β), thereby determining the value of its width W_H(1,Ly_β) at the initial instant of time, and then determine the plasma density by the standard method (e.g., [24]). If the Ly_β line intensity is too low to measure the width of its profile (which likely indicates a relatively low plasma temperature), we can use the He_β line of the Al XII ion instead (see Fig. 6).

It is possible to calculate the dependence of the widths of both spectral lines Ly_β and He_β on the initial density and plasma temperature if we use the Holtsmark formula for the distribution of the ion field in the calculation of the Stark widths and take into account the electronic shock broadening and the thermal Doppler broadening. A reader may prefer to use other than Holtsmark formula to calculate Stark broadening (from a point of getting more accurate calculation or satisfying specific experimental parameters); in this case, one can follow the steps of described technique. The results of the calculations are shown in Fig. 7 for the one-dimensional expansion case m = 1 and H = 0.5. If both lines are observed in an experiment, then the simultaneous measurement of their widths makes it possible to determine both plasma parameters, n_ion,1 and T_e,1. It also should be noticed that the distribution of ion microfields in a dense laser plasma may noticeably differ from Holsmark and lead to other values of the spectral lines widths. However, it can be expected that the value of a ratio R_H will be less sensitive to different distribution functions than the FWHM values will. This means that to obtain a smaller measurement error, perhaps, it is preferable to use Fig. 6, rather than Fig. 7.

Fig. 7

FWHM of the Ly_β line of the Al XIII ions W_0.5(Ly_β) (a) and He_β line of the Al XII ion W_0.5(He_β) (b) as a function of the initial ion density n_ion,1 for different initial temperatures T_e,1

To do this, it is necessary to use a simple procedure. First, it is necessary to get from Fig. 7 two sets of points {n_ion,1,T_e,1}(Ly_β) and {n_ion,1,T_e,1}(He_β) for the measured values W_0.5(Ly_β) and W_0.5(He_β) for which the calculated values of the widths coincide with the corresponding experimental values, and then to determine their intersection, which gives the desired values of n_ion,1 and T_e,1.

4 Discussion

Several certain limitations in the proposed diagnostic approach are considered. The most serious limitation is the model not accounting for the recombination channel of excitation of the spectral lines. This channel can be easily taken into account using time-dependent kinetic equations. However, in this case, it would be necessary to disclaim the universality of the technique, since the time in kinetic equations would have an absolute value and would differ from the relative time of j stage existence (t_j) used above. This would require specifying both the initial size of the plasma and the velocity of its expansion. As a result, calculated line widths would depend on too many free parameters, and their diagnostic use would, in fact, become impossible.

On the other hand, it is possible to easily obtain a negligible contribution of recombination effects of excited levels under experimental conditions. As noted above, for this purpose, it is necessary to consider the spectral lines of ions with ionization potentials much higher than the expected plasma temperature. For this purpose, it may be necessary to add specific tracer elements with sufficiently large atomic numbers into the laser target.

In the plasma expansion model, we described, it is assumed that plasma expansion does not occur at the stage of its heating. In other words, we supposed that the duration of the main laser pulse is much less than the characteristic gas-dynamic time. This limitation is significant in the case of plasma heating by long (nanosecond) laser pulses, which was not considered in this paper. Other assumptions for plasma are (a) plasma remains homogeneous during the expansion, (b) its expansion is assumed to be linear (but it was checked that 1D and 3D expansion models give us close results) and adiabatic, (c) plasma remains optically thin, (d) noticeable population of highly ionized ions, (e) plasma produced by interaction of matter with ultra-short laser pulse, and (f) a potential spectrometer records the emission from the entire volume V_j at each stage of expansion. If some experimental parameters do not satisfy the mentioned assumptions (e.g., expansion of plasma cannot be considered as adiabatic one), then the recalculation of E(λ) is required following the described technique. All certain experimental features also must be taken into account. Let us mention that satellites were taken into account in our calculations, and as it turned out, they do not interfere with the observation of the resonance lines profiles.

In our calculations, we assume that at all the stages of plasma evolution, its ionization state corresponds to the electron temperature. In this case, the main channel for populating the excited states of ions with a charge Z is the electron impact to low-lying states of this ion, and the processes of recombination of ions with a charge (Z + 1) practically do not contribute to the population. However, if at the initial moment the plasma is too strongly ionized, i.e., if a number of ions N_(Z+1) with a charge, (Z + 1) is larger than a number of ions with a charge Z, then during plasma expansion and its temperature decreasing, the recombination channel becomes the main channel for excitation of spectral lines. To avoid this, it is necessary to use ions with N_Z ≫ N_(Z+1) at the initial time for diagnostic. To satisfy this condition, as calculations of the ionization balance show, it is sufficient that T_e at the initial moment does not exceed ~ 0.1·I_Z, where I_Z is the ionization potential of the Z ion.

The usage of optically thin plasma in our calculations is our main limitation. Nevertheless, at least two experimental setups can be proposed.

• (1) One can use a target which is a mixture of two different substances with close atomic numbers; for example, Mg or Si with a small amount of Al as a tracer element. The percentage of Al in the target depends on the expected size of a plasma (on the thickness of the target and the diameter of the laser focal spot). The number of electrons in such a plasma would not much differ from the number of electrons in a pure Al plasma (less than 10% as expected).

• (2) On the other hand, we can use a non-uniform target. In this case, the central region (0.1 µm or less) should be made of Al (and the rest can be made from a different material). Laser axis comes through Al region. When expanding, we would have something like a plasma aluminum “needle” surrounded by a cloud of another material plasma. Therefore, Al spectral diagnostic results would correspond to the central plasma areas, i.e., we can determine the density at the very center of the focal spot.

5 Conclusion

Taking into account the limitations stated above, computer modelling results described in this article make it possible to retrieve the plasma parameters at most extreme initial stage of short pico- and femtosecond intense laser pulse interaction with solid matter and, particularly, to control actual conditions in case of thin-foil or other types of mass-limited targets, using the spectra of multiply charged ions registered without a time resolution.