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Abstract
We report on the spectra of x-rays emitted from dense plasma generated via irradiation of thin stainless steel foils by ultra-relativistic femtosecond laser pulses (intensities ~3 × 1021 W/cm2). Kinetic modelling was used to estimate electron plasma density and temperature, demonstrating Te ~2.1 keV for Ne ~5 × 1022 cm−3 in the hottest emission region. Thus, it is experimentally demonstrated for the first time that the laser pulse of over 1021 W/cm2 intensity is absorbed neither in the solid density plasma nor in a pre-plasma of a common critical density, but in the matter of so called relativistic critical density.
© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Recent investigations of femtosecond laser pulse interaction with matter have focused on the creation of powerful X-ray sources [1–10], gamma-ray radiation sources [11, 12] and high energy ion, electron and neutron beams [13–18]. Such sources are widely applied in laboratory astrophysics, high density energy physics, material science, medicine and bioengineering applications. The characteristics of such sources strongly depend on not only the main plasma parameters (plasma temperature, density, chemical composition) but are also extremely sensitive to the experimental geometry, target material and structure, laser pulse duration and intensity, presence of external sources of radiation or magnetic field, and other experimental conditions.
To optimize and understand the source properties, it is therefore important to monitor as many parameters of the laser-generated plasma as possible. This is challenging due to the small temporal duration (10−13 −10−11 s) and volume (~10 μm3) over which the plasma reaches temperatures of a few keV at near solid density. For example, X-ray laser interferometry [19, 20] is challenging and plasma with solid density cannot be measured. X-ray spectroscopy of multicharged ions [21] is a sophisticated tool for diagnosing such high density, high temperature plasma. X-ray spectroscopy is comparatively simple and, for thin targets, allows direct measurement of the conditions in the dense region of the target. Reaching ultra-high laser intensities requires focusing the laser pulse to focal spot diameters ~2–8 μm, and the resultant plasma therefore has similar spatial scales. Due to this tight focusing, femtosecond laser-produced plasma is very inhomogeneous and plasma parameters vary rapidly along the target, away from focal spot axis, and along the laser beam propagation direction, from the target surface inward. Using relativistic laser intensities significantly modifies the laser plasma interaction. Particularly, the light can propagate into plasma up to densities exceeding the classical critical density . The relativistic critical density depends not only on the laser wavelength, but also on the laser intensity [22, 23]. For a dimensionless laser amplitude a >> 1, it is given by
The effect of ultra-high intensities on the relativistic critical density is therefore determined by a. For example, for a laser intensity of ~1 × 1021 W/cm2 and wavelength λ = 800 nm, the relativistic critical density is , and for laser intensity of ~1 × 1022 W/cm2 the relativistic critical density is , which is more than 30 times higher than the critical density estimated as for moderate laser intensities. The plasma zone near the relativistic critical density would be expected to absorb laser energy more effectively and therefore be heated to the highest temperatures. Reaching ultra-high laser intensities leads to higher temperature plasma generation and more intense hard x-rays emission. Therefore, using medium or high-Z targets allows a more precise measurement of the expected high temperatures and density due to their more complex spectral properties. Modern x-ray spectroscopy instruments are able to observe x-ray emission spectra with spatial resolution no better than 5–10 μm. It means that measured spectra are spatially integrated measurements of x-rays emitted from regions (plasma zones) with significantly different conditions. As was shown in [8–10, 24], although these conditions are continuous functions of spatial coordinates, it is possible to adequately describe the observed spectra by assuming a small number of plasma zones with fixed conditions and combining the resultant spectra from each zone. Note the similar approach was successfully implemented to explain the spectra of buried targets irradiated by short pulse laser [25]In this paper we present, for the first time, experimental evidence supporting the above absorption model by analyzing the K-shell Fe and Cr x-ray spectra emitted from plasma produced via high intensity femtosecond laser pulse irradiation of thin stainless steel solid foils.
2. Results
2.1. Experimental set-up and x-ray spectra observation
The experiments were performed at J-KAREN-P laser facility at the Kansai Photon Science Institute (Japan) [26,27] which uses optical parametric chirped pulse amplification (OPCPA) followed with amplification in Ti:Sapphire to produce a pulse with central wavelength 800 nm and duration τ ~ 40 fs full-width-half-maximum (FWHM). The contrast ratio reached 1012 with a pulse energy up to 10 J. By focusing with an off-axis parabola, the on-target laser intensity reached ~3 × 1021 W/cm2 in a focal spot with diameter 2 μm when target was positioned at focus, and ~5 × 1019 W/cm2 when target was shifted along the laser axis at l = 100 μm from focus position, Fig. 1 [28–30].
Fig. 1 X-ray spectra measurements. (a) Schematic of experimental setup; (b) Typical x-ray spectra recorded on the x-ray CCD when target was positioned at focus, and when target was shifted from focus position along the laser axis by l = 100 μm.
For measuring the x-ray radiation emitted from 5 μm thick stainless steel foils (AISI 304: Fe 72%, Cr 18%, Ni 10%), a Focusing Spectrometer with Spatial Resolution (FSSR) [31–33] with resolving power λ/δλ ~3000 was implemented. The instrument was equipped with a spherically bent mica crystal with lattice spacing 2d = 19.94 Å and a radius of curvature R = 150 mm. The crystal was aligned to operate at the m = 8 order of reflection to record K-shell emission of Fe ions (lines Heα, Kα and Lyα) in the 1.67 – 2.1 Å wavelength range. Simultaneously we measured K-shell emission from He-like Cr ions and neutral Cr Kα at the m = 7 order of reflection in the 1.93 – 2.28 Å wavelength range. X-ray radiation from m = 1 – 5 orders of reflection was strongly attenuated by a 100 μm thickness Mylar (C10H8O4) film installed in front of crystal to protect it from contamination.
The spectrometer viewed the laser irradiated front side of the target surface at an angle ~82°. X-ray spectra were detected with an Andor DX-440 X-ray CCD with pixel size of 13.5 μm, which was protected against exposure to visible light by two layers of 1 μm thick polypropylene coated with 0.2 μm Al.
The high laser intensity interaction provided sufficient signal that the spectral measurements could be taken in a single laser shot. However, to improve the signal-to-noise ratio, the measured spectra were averaged by three shots made at the same laser pulse parameters. Typical stainless steel x-ray spectra are shown in Fig. 1(b). The x-ray spectra obtained for a target in and out of focus are notably different. At low intensity (top) there are lines corresponding to transitions in neutral atoms of Fe and Cr (Fe Kα λ = 1.94 Å, Cr Kα = 2.29 Å), but at high intensity (bottom) transitions in the vicinity of resonance lines of He-like Fe and Cr ions dominate the spectrum.
2.1. Experimental results for the low laser intensity case and modeling approach
The plasma conditions can then be inferred by comparison of experimentally measured x-ray spectra with numerical modeling. There are various codes [24, 34–36] dedicated to modeling x-ray spectra emitted from various plasmas (magnetic confinement plasma, laser plasma, vacuum spark, Z- and X-pinches, plasma generated under the influence of power x-ray lasers). These codes typically operate in the framework of the collisional-radiation model [37–39], including the effect of radiation transport. Here, we have used the open access code FLYCHK [40] with the NIST atomic database [41], which is commonly used for X-ray spectra modeling.
We first consider the x-ray emission from plasma generated when the target is located at a defocused position with a laser intensity on target of I ~5 × 1019 W/cm2 (see Fig. 2(b)). This x-ray spectrum mostly contains emission lines corresponding to transitions in neutral atoms of iron and chrome. Existence of bright Fe Kα and Cr Kα lines suggest a bulk plasma electron temperature Te ~10 – 30 eV. At the same time, the presence of Kα lines can only be explained when including a fraction of hot electrons in the calculation.
Fig. 2 Comparison of theoretical and measured spectra for a target shifted from focus position along laser axis by 100 μm. Theoretical spectra modeling by FLYCHK for different plasma temperature values: (a) Te = 10 eV; (b) Te = 50 eV; (c) Te = 130 eV; (d) Te = 170 eV; in all cases the electron density is Ne = 1023 cm–3 and plasma thickness d ~1 μm. We include 1% hot electrons with temperature 10 keV.
The dependency of modelled x-ray spectra of Fe and Cr ions on plasma electron temperature is shown in Fig. 2. Spectra from stainless steel foils were calculated as a superposition of modelled spectra for iron and chrome components. Note that, in Fig. 2. and beyond we use the following labels: red curve - modelled spectrum of Fe contribution, blue curve - modelled spectrum of Cr contribution, gray curve - measured x-ray spectrum. From Fig. 2 one can see that increasing the electron temperature led to a significant decrease of the relative intensities of the Cr Kα (λ = 2.08 Å) and Fe Kβ line (λ = 1.76 Å). Further temperature increase resulted in appearance of new spectral lines on the left wings of Fe and Cr Kα that were not observed experimentally.
Kinetic calculations obtained for the inferred temperatures showed an averaged ion charge of ~1 – 3. Iron has initial solid density Ni,solid = 8.4 × 1022 cm−3, and therefore the electron density in such plasma is estimated to be Ne,solid ~(1 – 3) × 1023 cm−3. To proceed with the comparison with experimental spectra of iron and chrome we therefore assume values of bulk electron temperature and density of Те = 10( ± 3) eV and Ne = 1( ± 0.2) × 1023 cm–3 (see Fig. 2(a)).
2.2. Experimental results for ultra-intense laser intensity case and comparison with theoretical modelling
As was shown on Fig. 1(b), x-ray spectrum obtained for the case of relativistic laser intensity reaching I ~3 × 1021 W/cm2 on target consisted of many different lines including cold K-shell transitions and transitions in deeply ionized (H-, He-like) ions. Unlike at low intensity, it is not possible to describe such a complicated spectrum using one bulk plasma temperature. We therefore used the approach previously developed in [8–10, 42]. Following this model, the plasma volume was divided into four zones, each of which has notably different parameters and is responsible for the emission of different spectral lines.
In this model, Zone I (see Fig. 3.) corresponds to the plasma directly heated by the laser. It is therefore limited in the transverse direction by the focal spot size, and extends inward into the target up to the relativistic critical density. Zone I will be described by the highest temperature Te1 and an electron density Ne1. Zone II and Zone III are located further inside the target and have overcritical density, approaching solid density (Ncr,rel < Ne2 ~Ne3 < Ne,solid). These regions are heated due to energy transport processes from Zone I. Zone II is close to Zone I with a lower electron temperature than Zone I, but higher than Zone III (Te3 < Te2 < Te1). Finally, Zone IV corresponds to heated area of solid target (Ne4 = Ne,solid) which is located even deeper into the target and at significant transverse distance from the laser axis in the near-surface layer. This zone is mostly heated by energetic electrons generated in the laser plasma interaction (“hot” electrons) or intense x-ray emitted from the first plasma zone (Zone I). Zone IV has a lower temperature than Zone III (Te4 < Te3).
Let us now consider the x-ray spectral contribution from Zone I. This zone is the hottest plasma area, and therefore responsible for the resonance lines of H- and He-like Fe and Cr ions. To provide a good match between the theoretical and experimental spectra we varied the values Te1, Ne1 and plasma thickness – d1.
We stress that our modeling shows that spectrum observed in experiment could not be described by any plasma conditions except the case when we choose Ne1 = Ncr rel. Note that the modelled spectrum was also not consistent when using optically thin plasma (d → 0). We included plasma self-absorption effect using value di as a plasma thickness for each zones in the direction of observation (see Fig. 3). Here d2, d3 and d4 are limited to d ≤ 1.5 – 2 μm, as they cannot be larger than the target thickness.
Assuming d1 ~2 μm the best match between experimental and theoretical spectra was reached when Ne1 = 5 × 1022 cm−3 = Ncr,rel estimated for relativistic laser intensity I ~3 × 1021 W/cm2 using Eq. (1). Importantly, the theoretical spectrum was quite sensitive to the plasma temperature. When Te is 1500 – 1800 eV the relative intensities of Cr Heα 1,2 could not describe experimental spectrum. If Te is 2400 eV or higher we observe an increase in Fe Heα1 that is not matched to the measured spectrum (see Fig. 4). Thus, by optimizing the line fitting the electron temperature was found to be Te1 = 2100( ± 40) eV.
X-ray radiation emitted from Zone II is primarily due to satellite transitions in Li-, Be- and B-like ions of iron and chromium. The existence of such ions is possible in a cooler plasma with a temperature Te ~300 – 1200 eV. Simulation of the X-ray spectrum of the iron component shows that the generation of satellites on the right wing of the Fe-Heα 2 line depends strongly on the plasma temperature (see Fig. 5).
Fig. 5 Emitted spectrum of iron plasma component calculated at fixed plasma parameters: Ne = 5 × 1023 cm−3, d = 2 μm and for different plasma temperatures.
As seen in Fig. 5., an increase in the plasma temperature leads to the generation of satellites due to transitions in increasingly ionized ions, and there is no temperature value at which transitions to Li- and B-like ions would be simultaneously sufficiently intense. Therefore, for an adequate description of the observed spectrum we separated Zones II and III with practically the same electron density, but with different electron temperatures (see Table 1.).
Table 1. The plasma parameters used in the model calculations for zones 1–4.
X-ray emission from Zone IV corresponds to transitions in neutral atoms of iron and chrome. Existence of Fe Kα and Cr Kα spectral lines can be described by one set of plasma parameters, as previously shown for the low intensity case. We note that 0.1% of hot electrons with Thot = 10 keV is already sufficient for good agreement with the experimental spectrum and further increase of the hot electron fraction did not affect the modeling in Zone IV.
As was predicted, hot electrons are generated in the laser plasma interaction in Zone I and then penetrate into the peripheral target area, leading to the generation of neutral Kα and Kβ lines from Zone IV. We therefore modelled how the hot electron fraction influenced our emitted spectra from Zone I; the results are presented in Fig. 6.
Fig. 6 Dependence of x-ray spectrum emitted from stainless steel foils on hot electron fraction with temperature Thot = 10 keV. Theoretical spectra were calculated for the following plasma parameters: Te = 2000 eV, Ne = 5 × 1022 cm −3, d = 2 μm.
Figure 6 shows that the spectra depend slightly on hot electron fraction. However, a ten times larger hot electron fraction results in different intensities relation between the resonance and intercombination Fe Heα lines. A choice of 1% hot electrons with temperature Thot = 10 keV gave an improved agreement between experimental and modeling spectra compared to 0.1%. Therefore a 1% hot electron fraction was used for modelling of all zones, based on its influence on emission from Zone I.
Fig. 7. shows a modelled spectrum combining the X-ray emission spectra of all 4 zones. The plasma parameters used in the model calculations are listed in Table 1.
Fig. 7 Comparison of theoretical calculation obtained in the assumption of x-ray emission from four different plasma zones, and experimentally measured spectrum. (a) Typical x-ray spectrum, observed at J-KAREN-P laser facility when the intensity on target reached 3 × 1021 W/cm2; the target was stainless steel foil with thickness of 5 μm. Red curve shows the modeled spectrum obtained as a superposition of iron and chrome plasma component spectra emitted from 4 plasma zones. (b) Different plasma zones contribution in observed stainless steel x-ray spectrum calculated for plasma parameters shown in Table 1.
Generally speaking, not only spatial distribution but temporal gradients of the plasma parameters should be considered in the model of high-temperature plasma. However, a femtosecond duration of the heating laser pulse and the time of hard X-ray emission shorter than 1 ps make it possible to use more simple static approach. A good agreement between the experimental and calculated spectra shows that such approach is good enough not only for qualitative but quantitative description for X-ray emission of femtosecond laser produced plasma.
3. Conclusion
A good agreement between the experimental and calculated spectra allows us to conclude that the adequate modelling of the entire plasma region requires at least 4 zones. A key conclusion is that the zone with highest electron temperature and ionization does not correspond to the classical critical electron plasma density Ne,cr. but a much higher electron density. This is inferred from kinetic modeling, which shows that at relativistic laser intensities, a plasma region with near relativistic critical density becomes the most significant, which under the conditions of the present experiment is 30 times higher than the classical value. It is this region that is responsible for the generation of ions with maximum ionization stages, and due to the high density the ionization process itself occurs much faster than in the classical case of plasma produced by laser with non-relativistic intensities.
Funding
Russian Science Foundation (RSF) (17-72-20272); Japanese Ministry of Education, Culture, Sports, Science and Technology (Supplemental budget, C-PhoST); Japan Society for the Promotion of Science (JSPS) (KAKENHI JP 26707031, 26247100, 16K05506); Japan Science and Technology Agency (PRESTO JPMJPR16P9 6813804).
Acknowledgments
We thank the J-KAREN-P technical teams at Kansai Photon Science Institute for their support during the experiments.
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