Dynamics of ultrafast heated radiative plasmas driven by petawatt laser lights

We conducted 1D simulations to investigate the radiation cooling effect. Figure 5 shows the temporal evolution of the bulk electron temperature for the simulations with the radiation cooling. The temperature was measured in the region where the ion density was over 29 $n_{c}$ which corresponds to the half of the initial solid density. We see the degradation of the peak bulk temperature $T_{e}$ for the lower intensity cases. For the case with $a_{0}=2.0$ and $\tau_{l}=3$ ps, $T_{e}$ saturates around keV even though the laser pulse is still irradiated. It is also seen that the life time of 10 keV silver plasma is a few picoseconds. This infers that the plasma internal energy will be converted to the hard X-rays in a few picosecond time scale. We also show the result for $a_{0}=20$ and $\tau_{l}=30$ fs without the radiation loss by the red dashed line. In this case, the temperature decreases only via the adiabatic cooling due to the plasma expansion. We see that the adiabatic cooling is much slower than the radiative cooling. The conversion rate from the laser energy to the hard X-ray emissions, which is discussed later, is about $1\,\%$ for the current laser configurations. When the electron temperature drops to $100\,{\rm eV}$, the emissivity decreases significantly as shown in Fig. 2. Hence, at the time later than $t=4\,{\rm ps}$, the temperature stays at about $100\,{\rm eV}$ in the cases of $a_{0}=3-20$.

In the present research, we have included only non-relativistic bremsstrahlung emissions. Therefore, hot electrons with energies over $100\,{\rm keV}$ do not suffer from the radiation energy loss. Relativistic electrons emit photons with energies over MeV, so called $\gamma$-rays via bremsstrahlung or the radiation damping. The energy conversion rate for $\gamma$-rays is much less than $1\,\%$ [12], so that $\gamma$-rays do not affect the plasma heating.

Figures 6 (a) and (b) show electron energy spectra calculated at the moment when the laser irradiation was over for the cases without radiations (a) and with radiations (b). The spectra have two parts, one for the bulk electrons with energies $E<100\,{\rm keV}$ and the other for hot electrons with energies $E>100\,{\rm keV}$. The peak energy of the bulk electrons is explained by the resistive heating. The Maxwellian distribution $\propto E^{1/2}\,{\rm exp}\left(-E/T_{e}\right)$ with the bulk electron temperature $T_{e}$ calculated from Eq. (1) for $a_{0}=20$ and $\tau_{l}=30$ fs is plotted by the dotted line as a reference. We see that the reference line agrees well with the bulk electron component of $a_{0}=20$. Note here that from Eq. (1), we see the $T_{e}$ slightly increases with the weaker amplitude and longer pulse duration but it is almost same around 10 keV for the same pulse energy cases. The hot electrons are induced in the laser-plasma interaction at the target surface. Their average energy is given by the ponderomotive scaling as $\varepsilon_{p}=[(1+a_{0}^{2}/2)^{1/2}-1]m_{e}c^{2}$. For example, $a_{0}=20$ derives $\varepsilon_{p}\sim 6.7\,{\rm MeV}$ [18]. The hot electron temperature decreases as $a_{0}$ decreases as seen in the spectra in the energy range $E>100\,{\rm keV}$.

The radiation cooling effect on the electron spectra is seen in Fig. 6 (b). Compared with Fig. 6 (a), the peak bulk electron energy of $a_{0}=$ 3.0 and 2.0 shift to the lower energy, while other cases do not, as predicted in the discussion of Fig. 3.

Since the plasma dynamics reflect how the radiations cool down the plasma, it is worthwhile to investigate the dynamics under a situation in which the radiations take the plasma energy significantly. Figure 7 shows the temporal evolution of electron temperature $T_{e}$ (green solid lines) and electron phase $x$-$p_{x}$ plot (dots) around the target observed at the moments when the laser peak reached the solid surface ($t=0\,{\rm fs}$) and at $t=500$ fs. The laser is the long pulse with $a_{0}=2$ and $\tau_{l}=3$ ps. The plots (a) and (b) represent the results with radiations, and (c) and (d) represent those without radiations. Initially, the silver foil is located in the region $x=201-202\,{\rm\mu m}$.

In the interaction region with the laser light ($x<201$ $\mu$m), $T_{e}$ is similar not depending on whether the radiations are taken into account or not. This is because the hot plasma expanding in front of the dense plasma region has a low density, so that the dominant radiation via bremsstrahlung, whose power is proportional to the square of the ion density, is negligible. In the solid region ($x>201.5$ $\mu$m), $T_{e}$ in the case with the radiations stays low around 100 eV while $T_{e}$ without the radiations is hotter exceeding 1 keV. The temperature has a gradient due to the thermal diffusion from the hot region. The electron momentum distribution inside the solid density region shrinks in the vertical direction in (a) and (b) due to the radiation.

Figures 8 (a), (b), (d), and (e) show the temporal evolution of ion densities $n_{i}$ (green lines) normalized by $n_{c}$ and electron energy densities (PPa) (purple lines) observed at the same time in the same simulations of Fig. 7. The broken lines indicate the initial foil region. (a)-(c) are the plots for the simulations with radiations, and (d)-(f) are those for the simulations without radiations.

At $t=0$, the foil surface is pushed forward by the hot electrons’ pressure. The compressed ion density $n_{i}$ at the foil surface in (a) is about 4 times higher than the initial density. The increase of $n_{i}$ is significantly higher than the case without radiations. Due to the radiation cooling, the electron pressure inside the solid region decreases. Hence, the bulk plasma can be compressed more easily with the radiation cooling effect. A strong radiation happens in the compressed region, which results a dent in the temperature profile at $x=201.4\,{\rm\mu m}$ in Fig. 7 (b).

Figures 8 (c) and (f) display ion phase $x-p_{x}$ diagram and ion density distribution at $t=500\,{\rm fs}$. The ion momentum is normalized by $M_{i}c$ where $M_{i}$ is the ion mass. It can be seen that the collisional shock is formed because of the pressure gradient. Since the upstream pressures in both with and without radiations are similar, the speed of the compressed surface $u_{p}$ is similar, $u_{p}\sim 0.0005c$ with radiations and $0.0007c$ without radiations as indicated by the dotted lines in (c) and (f). The acoustic speed at $T_{e}=$1 keV is $C_{s}=(T_{e}Z_{i}/M_{i})^{1/2}\simeq 0.00055c$ with $Z_{i}=30$ for the case with radiations and $C_{s}\simeq 0.00063$ with $Z_{i}=40$ for the case without radiations where $Z_{i}$ is the average ion charge state in each simulation. The shock speed $u_{0}$ is explained well by $u_{0}=u_{p}\rho_{1}/(\rho_{1}-\rho_{0})$ [19] where $\rho_{0}$ and $\rho_{1}$ are uncompressed and compressed ion densities, respectively, through the shock surface. In the case with radiations, $u_{0}\sim 0.00067c$ which is 3 times slower than the case without radiations. And thus the target with radiation takes 3 times longer time to expand into vacuum than the case without radiation. We also confirmed the momentum and energy conservations through the shock surface [19]. These results show that the radiation cooling affects the plasma dynamics by reducing the shock speed in the dense plasma.

A hot metal plasma with keV electron temperature has a potential to be a hard X-ray source. We calculated the conversion efficiency from the laser light to hard X-ray photons with energies of 18-75 ${\rm keV}$. The photons with this energy range could be a backlight source for the high density plasma experiments [10]. Figure 9 (a) shows the conversion efficiency of Ag (circles) and Al (triangles) foils with thickness of 1 $\mu$m. The emitted X-ray energy was obtained by integrating the X-ray emissivity over photon frequency and solid angle in each PIC cell and summing the energy during the simulations. It is found that the conversion efficiency of Ag is nearly two-orders of magnitude higher than that of Al. The bulk electron temperatures achieved by the laser irradiation are almost same around 10 keV in both materials. At such high temperature, the Al plasma is fully ionized and the Ag plasma is almost fully ionized to the charge state of 45.

Below we explain the high X-ray yields in the Ag target. The bremsstrahlung is the dominant radiation process and its emission power density $E_{ff}$ (W/cm³) depends on the charge state $Z_{i}$ of the plasma as $E_{ff}=1.69\times 10^{-32}n_{e}T_{e}^{1/2}\sum_{Z_{i}}\left(Z_{i}^{2}n_{i}(Z_{i})\right)$ where $n_{e}$ is electron number density (1/cm³), $T_{e}$ is electron temperature (eV), and $n_{i}(Z_{i})$ is ion number density (1/cm³) with charge state $Z_{i}$ [20]. If we assume the charge neutral state, $n_{e}=\sum_{Z_{i}}\left(Z_{i}n_{i}(Z_{i})\right)$. Since the ratio of the atomic number of Ag and Al is $47/13$, the radiation power of bremsstrahlung from the Ag plasma is $\sim(47/13)^{3}=47.11$ times higher than that from the Al plasma. In addition to bremsstrahlung, strong emissions through the radiative recombination process are expected in the Ag plasma. The power of the radiative recombination emission of Ag is close to that of bremsstrahlung, so that the Ag plasma has about two-orders of magnitude higher emission than the Al plasma.

Note here that the dependence of the conversion efficiency on the laser intensity in Fig.,9 (a) is weak in the intensity regime $>10^{19}\,{\rm W/cm^{2}}$, since the achievable electron temperature via the resistive heating is not much changed when the total laser energy is same. In the case of laser intensity below $10^{19}$ W/cm², the efficiency decreases to a half of other cases. This is due to the radiation cooling that occurs in the picosecond time scale during the laser plasma interaction. Namely, the electron temperature cannot reaches keV-level in the dense plasma region with radiations as seen in Figs. 6 and 7, and thus over-keV X-ray emissions are suppressed. Figure 9 (b) shows emission energy spectra in the simulations with the different laser amplitudes. Here, the spectra for photon energy $h\nu_{i}$ were calculated as

where $\rm i$ and $\rm j$ denote indices of the photon energy group and the spatial grid, respectively. $\eta_{\rm i}$ is emissivity of photon energy of $h\nu_{\rm i}$ where $h$ and $\nu$ are Planck constant and photon frequency, respectively, and $\Delta\nu_{\rm i}=\nu_{\rm i}-\nu_{\rm i-1}$. The time integration is taken over the total simulation time $T_{\rm sim}$. Figure 9 (b) indicates that the photons with energies above $1\,{\rm keV}$ from bremsstrahlung and recombination processes mainly contribute to the cooling.

We see that the emission energy over 3 keV decreases significantly in the case of $a_{0}=2$ due to the radiation cooling as indicated with an arrow in Fig. 9 (b). This result is consistent with the lower conversion efficiency seen in Fig. 9 (a). The emissivity of bremsstrahlung is proportional to $T_{e}^{-1/2}\exp(-h\nu/T_{e})$ [21], and therefore, the peak of the emission energy appears around the energy comparable to the electron temperature.

We conducted 2D PIC simulations to see the multi-dimensional aspect of the radiation cooling effect. A solid silver foil was placed at the region of $x=21-22$ ${\rm\mu m}$ initially. The initial charge state of the silver ion is 1. We use the laser parameter of $a_{0}=2$ and $\tau_{l}=3$ ps, which shows the significant radiation cooling during the laser irradiation in the 1D PIC simulation. The laser light is focused with a spot diameter $3\,{\rm\mu m}$ normally at the center ($y=10\,{\rm\mu m}$). The laser electric field oscillates in the $y$-direction, namely, the laser is p-polarized.

Figure 10 shows the electron temperature distribution with (a), (b) and without (c), (d) radiations. Figures (a) and (c) show the temperature distribution at the moment when the peak of laser arrived at the foil surface. In front of the target $x<21\,{\rm\mu m}$, the pre-plasma is heated to $T_{e}>10\,{\rm keV}$. The electron temperature in (a) and (c) have almost identical distribution in the pre-plasma region. Since the emissivity is proportional to the plasma density, e.g., bremsstrahlung depends on the square of the plasma density as discussed in Sec. 3.3, the radiation cooling effect is small in the low density region.

Electron temperature inside the foil increases along the incident laser axis via the resistive heating, and reaches $4\,{\rm keV}$ with radiations and $8\,{\rm keV}$ without radiations. The heat propagates from the interaction surface via the thermal diffusion. Overall, as expected from the 1D calculation results, the temperature of the case with radiation is lower than that without radiation. These results are consistent quantitatively with 1D results as shown in Fig. 7. (b) and (d) show the temporal evolution of the electron temperature distribution in the $y$ direction. Here, the temperature is averaged inside the foil. In the case with radiation, the peak temperature was reduced to $4\,{\rm keV}$. On the other hand, in the case without radiation, the temperature continued increasing during the laser irradiation. Comparing (a) and (c), we see that the target expansion front slows down with the radiation cooling. This is due to that plasma expansion speed is given approximately by the ion acoustic velocity $C_{s}$ which is proportional to $T_{e}^{1/2}$. Since the expanding plasma density is low, the radiation cooling is not significant, and consequently, the temperature at the rear side was higher than that inside of the solid density foil.

Figure 11 shows spectra of the areal density of the radiation energy $E_{\rm tot}$ during the laser plasma interaction period $T_{\rm LPI}$ calculated as

where $\eta$ is the emissivity. The spatial integral is taken over the laser spot region inside the foil $S$, i.e., $x=21-22\,{\rm\mu m}$ and $y=8.5-11.5\,{\rm\mu m}$. The solid line represents the sum of the emissivity from the plasma in the 2D simulation with the radiation cooling. The dashed line shows the result when the cooling effect is neglected during the interaction. Below the photon energy of $4\,{\rm keV}$ which mainly comes from plasma at temperature below 1 keV, the two lines are similar. Above the photon energy of $4\,{\rm keV}$, the emission energy without radiation cooling is about 2 times higher. The difference was due to the difference in the temperature distribution seen in Figs. 10 (b) and (d), namely, the higher temperature in (b) results larger hard X-ray emissions. This means that, without considering the radiation cooling effect, the emission energy will be overestimated especially for higher energy X-rays.