Role of relativistic laser intensity on isochoric heating of metal wire targets

: In a recent experimental campaign, we used laser-accelerated relativistic hot electrons to ensure heating of thin titanium wire targets up to a warm dense matter (WDM) state [EPL 114 , 45002 (2016)]. The WDM temperature proĄles along several hundred microns of the wire were inferred by using spatially resolved X-ray emission spectroscopy looking at the Ti K α characteristic lines. A maximum temperature of ∼ 30 eV was reached. Our study extends this work by discussing the inĆuence of the laser parameters on temperature proĄles and the optimisation of WDM wire-based generation. The depth of wire heating may reach several hundreds of microns and it is proven to be strictly dependent on the laser intensity. At the same time, it is quantitatively demonstrated that the maximum WDM temperature doesn’t appear to be sensitive to the laser intensity and mainly depends on the deposited laser energy considering ranges of 6 × 10 18 Ű6 × 10 20 W/cm 2 and 50Ű200 J.


Introduction
Warm dense matter (WDM) studies [1] are currently receiving increasing attention being important for various Ąelds of science such as astrophysics, physics of plasmas and thermonuclear fusion [2Ű7]. WDM is usually deĄned as matter with density and temperature of about 0.1Ű100 g/cm 3 and 1Ű100 eV, respectively. Over-dense WDM is created in laboratories by compression and heating of matter by shock waves, as has been demonstrated in various experiments on high-power lasers [8Ű11] or even combined with diamond anvils [12,13]. Fast isochoric heating of solid matter may ensure obtaining a near-solid-density WDM. Such experiments can be performed with XFEL [14,15], optical lasers [16Ű19], accelerated particle beams [20Ű25] and laser wakeĄeld [26Ű29]. Isochoric heating of solid-state foils by optical laser pulses is a well-established way for producing WDM [30Ű34], however recently alternative target types have been successfully employed as well [35Ű41].
In our previous work in Ref. [42], thin titanium wire targets were irradiated on their tip by a high-intensity optical laser resulting in the acceleration of electrons up to relativistic energies. These hot electrons (HE) propagate along 100s µm of the wire causing inner-shell ionization of Ti atoms which is then followed by emission of the characteristic X-ray Ti K α line. At the same time, the hot electrons moving in the wire draws a return current of relatively low-energy electrons [43]. Both currents combined may heat matter to a few 10s of eV. Most importantly, heating and ionisation processes occur on a much faster timescale (∼1 ps) than the hydrodynamic expansion of the wire (∼10 ps), thereby keeping the volume constant during heating [41].
In Ref. [42] we focused on a detailed analysis of a single shot mainly using a detailed and advanced simulation tools to interpret the data. Well-established X-ray spectroscopy techniques based on the analysis of Ti K α proĄles [30Ű33] were employed to determine the WDM temperature with spatial resolution along the wire. The advantage of using wire targets is that their prolate shape ensures, the effective collimation of the electron Ćow, thereby providing heating to larger depths. This facilitates high convenience of plasma diagnostics and a clear spatial distinction between the emission of the WDM zone from that of the hot plasma at the laser-irradiated wire tip.
In this paper we present an extended experimental study discussing the dependence of isochoric heating of metal wires on laser parameters (namely, intensity and energy), and their inĆuence on the hot electron distribution and therefore the WDM temperature proĄle.

Experimental setup and K α profile revealing process
The experiment was conducted at the PHELIX Nd:glass (λ = 1064 nm) laser facility at the GSI Helmholtz Centre for Heavy Ion Research in Germany [44,45]. The P-polarized laser pulse was incident on the target at 22.5°, delivering a maximum energy of 210 J in ∼0.4 ps and was focused using a f/3 off-axis parabola to a minimum focal spot of ∼7 µm in diameter. The nominal laser spot contained 40% of the laser energy corresponding to a maximum intensity of about 6 × 10 20 W/cm 2 on target and total power of about 0.5 PW. The use of an ultrafast parametric ampliĄer (uOPA) before the main ampliĄer enabled reaching a main pulse to laser pedestal contrast exceeding 10 10 [46Ű49], thus ensuring the absence of any long-scale length preplasma in front of the solid target. The experimental scheme is shown in Fig. 1. Laser energy, focal spot diameter, and pulse duration were varied in order to get a variety of shots with different parameters as summarised in Table 1. Targets were solid Ti wires of 50 µm in diameter irradiated on their tip. To record WDM emission we employed a focusing spectrometer with spatial resolution (FSSR) [51Ű55] equipped with spherically bent α-quartz crystal with Miller indices 211 (2d = 3.082 Å) with a radius of curvature R = 150 mm. It was designed to record X-ray emission with high spectral resolution to continuously cover a spectral range that extends from 2.45 to 2.8 Å (4.4Ű5.1 keV), including Ti XXII (Ti 21+ ) Ly α , Ti XXI (Ti 20+ ) Heα, and Ti K α1 , K α2 characteristic emission lines. The spectrometer had a spatial resolution along the Z axis of the wire and a Ąeld of view of a few mm with a magniĄcation factor of 0.3. X-ray Ąlms were used as detectors since they provide Ti wire tip and consequent isochoric matter heating by the laser-accelerated hot electron Ćow, resistive heating by return currents, and related processes. Laser plane of incidence (P-polarisation one) coincides with YOZ plane. BSC and FSSR were placed in XOY plane as shown.
high spatial (∼20 µm) and high spectral resolution (λ/δλ ∼ 3000). The emission from the hot plasma corona and from the WDM region were spatially and spectrally distinguished from each other due to the signiĄcant temperature difference and to the target geometry (see Ref. [42]). Spectra showed that, in the spectral region of interest, there was no signiĄcant contribution from bremsstrahlung radiation in comparison to the characteristic emission. All spectral measurements were made in time-integrated mode. The WDM temperature proĄles were retrieved based on analysis of Ti K α X-ray emission spectra recorded with FSSR. A Şbremsstrahlung cannonŤ hard X-ray spectrometer (BSC) was used to estimate the HE energy distribution with Monte-Carlo simulations used to back-calculate the incident bremsstrahlung radiation produced by hot electrons as described in Ref.
[56Ű59]. In our case, the BSC was able to effectively revel the presence of hot electrons of about 0.1Ű5 MeV.
HEs are laser-accelerated in the area close to the target surface, i.e. in the hot plasma volume where laser-matter interaction takes place. Consecutive HE current is complex and there are components which propagate normally to the target surface while a great fraction of HE is accelerated towards the laser direction [60Ű62]. Electric potential induced on the target surface prevents most of the HE from escaping, as it shown in Fig. 1 and discussed in [42]. It is assumed, that heating of deeper-lying areas of the wire are caused by HE, return currents, and other processes whereas ionisation is mainly caused by HE. However, Ąrst few 10s of microns of the wire were heated due to a bulk conductivity process. Therefore, plasma in this region is deeply ionized and its temperature is rather high reaching several hundred eVs. Though such temperature is too high for the ŞneutralŤ K α line to be emitted, the line is also observed there in the temporally integrated spectra. In this case, the neutral K α line is likely generated at the very early stages and comes from the rather cold regions surrounding the hot focal spot. It also should be noted that here we do not consider the initial stage of electron propagation in the target, i.e. from the laser focal spot along the laser beam axis, to the edge of the wire. This process occurs on the scale of wire diameter (∼50 µm), so cannot be distinguished experimentally from the hot plasma zone. We consider and discuss only heating deeper than ∼60 µm where the electrons are expected to move along the wire.
Experimental data processing was performed in a similar way to what described in details in Ref. [30]. Ti K α proĄles were spatially resolved along the wire and, they were initially compared with modelling spectra calculated with the atomic-kinetic code PrismSPECT [63,64]. However, it appeared that SCFLY [65] is more accurate in the atomic-kinetic modelling of X-ray spectra since it includes more atomic levels and conĄguration for low-charged ions with multiple bound electrons (see Ref. [30]); as such only the results of experimental data processing with SCFLY are shown below. Spectra were calculated assuming a 1% 0.5 MeV HE fraction. Spectral line proĄles did not show any signiĄcant dependence on HE energy in a range of 0.5Ű2.5 MeV (which correspond to the range of hot electron energies in the performed experiment). Therefore, the same set of curves can be used to analyse our experimental data from shots with different laser intensities (i.e. regardless of the expected HE energy). All experimental spectra were time-integrated having a contribution from different stages of plasma expansion. Therefore, a two-temperature approach appeared to provide a reasonable description of experimental data [30,66]. WDM temperature was estimated as a sum of relatively low T low bulk and high The position in space where Ti K α total emission reaches its maximum was considered as the laser-target interaction point and was set to be a Ş0 µmŤ position on Z axis. A similar yet simpliĄed method was previously used for revelling temperature proĄles shown in Ref. [42]. The found temperature proĄles are displayed in Fig. 2 and discussed in detail in the next section. Fig. 2. Experimental temperature proĄles of WDM generated inside the titanium retrieved from analysis of its K α emission for the shots listed in Table 1. For ease of comparison error bars are indicated for shot #1 only. ProĄles were retrieved along the axes OZ in Fig. 1; Ş0 µmŤ corresponds to the laser-target interaction point, though the direction is opposed.

Results and discussion
Let us discuss the inĆuence of laser energy and intensity on laser-accelerated HE energy distribution. In general, HE energy spectrum can be roughly described by a Maxwellian distribution with a characteristic temperature T hot [67], which may be estimated according to the laser intensity value using a semi-empirical scaling law. Here we used a modiĄed Beg's law for high intensity range according to Ref. [ Fig. 3(a). Following it, increasing the laser intensity while keeping the laser energy Ąxed leads to increasing T hot and the related HE energy redistribution. In turn, increasing laser energy while keeping laser intensity Ąxed leads to increasing the full energy deposited to HE component while it does not signiĄcantly change T hot . One should therefore expect a more effective target heating despite the same T hot . In order to prove these statements let's compared two different experimental shots (#1 and #2 in Table 1) in which we varied the laser energy while keeping the laser intensity constant. Here the laser energies were different by a factor of four, the focal spot was not changed and the laser intensities were kept constant maintained Ąxed by varying the laser pulse duration: E 2 /E 1 ∼ 4 and I 1 ∼ I 2 , as speciĄed in Table 1. HE temperature and full energy of HE for these shots are shown in Fig. 3(a) and (b), respectively. As expected, the low-energy shot #1 generated almost half as much HE as the high-energy shot #2, while the T hot values for those shots are roughly equal within the measurement's accuracy. This has a certain inĆuence on the temperature proĄle of hot matter, as shown by the green and blue curves for shots #1 and #2, respectively, in Fig. 2. The two proĄles noticeably lie in different temperature regions. Maximum temperatures at 60 µm depth are about 18 and 28 (±1.5) eV for low-energy shot #1 and for the high-energy shot #2, respectively. Both proĄles reveal a linear decrease of WDM temperature up to a maximum distinguishable heating depth of about 400 µm. A similar dynamic is expected for deeper lying layers (depth > 400 µm), although it was not experimentally revealed due to the low signal-to-noise ratio in this region. The experimental results conĄrm our earlier statement regarding the role of laser energy on HE temperature and on their full energy.  Table 1. Experimental values were revealed based on a Şbremsstrahlung cannonŤ hard X-ray spectrometer and are marked with black dots. The black curve corresponds to the modiĄed semi-empirical Bag's law (according to Ref. [50]).
The change in laser intensity is reĆected differently in target heating. This can be seen by comparing the experimental results (temperature proĄles) for the shots #2 and #3 shown in Fig. 2. Here the laser intensities were changed by a factor of two while laser energies remained approximately the same, I 3 /I 2 ∼ 2 and E 3 ∼ E 2 . We see that the temperature proĄles are nearly coincident in the whole observed region. This is also in agreement with HE parameters revealed from BSC. HE electron temperatures and full energy are equal within the measurement error bars, as indicated in Fig. 3(a) and (b). This illustrates that the increase of laser intensity by a factor of two is not enough to produce a signiĄcant change of energy deposited by HE, i.e. to produce an increase of WDM temperature and depth of heating, at least for the considered range of intensities. To provide a basic interpretation and qualitative description of discussed experimental results we implement a simple physical approach which considers interaction of HE of different energies with matter. We made Monte-Carlo GEANT4 [68,69] simulations consider heating of a Ti solid-state laterally inĄnite material block by HE of different energies injected normal to the target surface. Though, initially in experiment HE start to propagate with some angular distribution along a laser axis, electric potential induced on the target surface prevents HE from escaping and redirects them to move along the wire [70]. Therefore, considering the HE propagation and matter heating many hundreds of microns deep into the target we simulate the process by injecting HE without initial divergence. Figure 4(a) shows the energy deposited in solid Ti by Hs of different energies as a function of depth. The higher the HE energy, the lower its collisional frequency [71]. It is clearly seen that the depth corresponding to peak energy deposition increases with HE energy, while the temperature proĄles become smoother. Figure 4(a) shows that HEs of energy < 1 MeV must be the main heating source of matter up to ∼250 µm depth (orange area in Fig. 4). A deeper lying target region marked in yellow corresponds to target regions mainly heated by HEs with energies > 1 MeV. Beg's law suggests that the greater the laser intensity, the larger the T hot value. Despite the fact that low energy electrons always dominate in the electron energy spectrum, the number of relatively high energy HEs increases for higher values of T hot , as demonstrated in Fig. 4(b), which shows the dependence on depth of energy deposited by HE Ćuxes having a Maxwellian distribution for different T hot (the total electron energy was set to be the same for all curves). It is clear that the temperature curves for T hot ≥ 1 MeV don't undergo dramatic changes for relatively small increase of T hot (i.e. of laser intensity). Calculated total deposited energy in solid-state Ti by a hot electron current having a single-temperature Maxwell distribution energy spectrum. Total electron energies were set to be the same for all cases. The target was considered as a solid-state, laterally inĄnite material block. The effect of the conĄnement of hot electron Ćow due to the wire geometry was not considered. When increasing the laser intensity even further (whilst keeping the laser energy on target constant), one can expect changes in the WDM temperature proĄle and an effective heating of the target to deeper lying layers. This is demonstrated by comparing the green and blue curves for T hot = 1 and 2 MeV in Fig. 4(b). The energy deposition in the region 0Ű300 µm is roughly the same, while the difference is more pronounced in the region 300Ű1000 µm. The experimental temperature proĄles for shots #2 and #4 conĄrm these expectations. The laser intensity here was increased by a factor of 100 while the energy was constant, I 4 /I 2 ∼ 100 and E 4 ∼ E 2 . The T hot values for shots #4 and #2 were measured to be 2.1 (±0.4) and 0.4 (±0.2) MeV, respectively, see Fig. 3(a), which is in general agreement with Beg's law (while the estimated HE full energy changes by a factor of 1.5), see Fig. 3(b). The WDM temperature proĄles for these shots are equal within the measurement's accuracy up to depths of ∼300 µm along the wire, see Fig. 2. This illustrates that laser pulse intensity does not signiĄcantly affect the WDM temperature proĄle on this spatial range. At the same time, the heating depth for high intensity shot #4 is signiĄcantly higher than that of shot #2 (see the range of ∼300Ű550 µm in Fig. 2) and in particular the WDM temperatures noticeably differ at 400 µm depth, 14 and 10 eV for shot #4 and #2, respectively. This conĄrms that the bigger the laser intensity, the higher the heating depth and the smoother the proĄle in general.
The approach we have used to explain the temperature proĄles dependence on laser intensity and energy considerably simpliĄes the design of future experiments. It provides a good qualitative description of dynamics and of the observed dependence of WDM temperature proĄles on laser parameters. However, it does not include the description of target edges and of associated effects such as the presence of induced surface currents and self-generated electromagnetic Ąelds, inĆuence on electron current, time evolution of these processes etc. In particular, we neglect the collimating effect due to the Ąnite geometry of the wire. Such collective effects would change the penetration of HE in matter, their stopping ranges, and ultimately their energy deposition. However, we believe that the observations we made in our paper concerning the relative changes as a function of laser energy and intensity will remain valid. In any case, developing complete numerical models describing HE generation, penetration, and heating of matter is of great interest and the experimental results presented in our work may be used to benchmark such models. These effects were indeed taken into account at least partially in Ref. [42].
It is worth mentioning that following the discussed experimental data and simpliĄed physical approach, it may be expected that a further increase of the laser intensity at a constant energy would lead to a signiĄcant increase of the heating depth, while the maximum WDM temperature would change slightly. Our observations also suggest that it can be possible to tune the laser pulse parameters in order to obtain a speciĄc ŞdesiredŤ WDM temperature proĄle. Coating the wire tip with high-Z material may lead to the simultaneous decrease of T hot and increase of HE number, which may increase the WDM temperature in the Ąrst few hundreds µm of the target. However, despite being in practice a very cost-intensive option, the most reliable way of increasing the wire WDM temperature is by enhancing the deposited laser energy.
The discrepancy between the maximum temperature measured in present and previous work (see. Ref. [42]) should also be mentioned. This is due to the fact that we used different spectroscopic codes for analysing spectra which provided more accurate description of experimental data. Most likely, the previous temperature values from [42] were overestimated by a factor ∼2. However, although quantitatively important, the presented data show the same qualitative trends and therefore this does not affect the conclusion of our paper.

Conclusions
In this article, we experimentally studied the inĆuence of laser pulse parameters on temperature proĄles of WDM generated in a titanium wire heated by a laser-accelerated hot-electron current. Laser intensity and energy were varied in ranges of 6×10 18 Ű6×10 20 W/cm 2 and 50Ű200 J, respectively. The increase in laser energy (while keeping intensity constant) leads to an increase in the total HE energy and to the subsequent rise of WDM temperature in the Ąrst few hundred microns of the target. Beyond laser intensities of about 1×10 19 W/cm 2 (which corresponds to ∼1 MeV characteristic temperature of hot electron distribution), the maximum WDM temperature becomes weakly sensitive to the laser intensity and mainly depends on the deposited energy. On the other hand, an increase of laser intensity at constant laser energy leads to heating of deeper target layers. These dependencies are explained with basic physical principles and simpliĄed numerical simulation of HE Ćows and WDM heating to show the inĆuence of laser parameters on the WDM temperature proĄle. Our experimental data may also be used to construct and verify more complex models of WDM heating by laser-accelerated HEs in further studies.