Isochoric heating of solid-density plasmas beyond keV temperature by fast thermal diffusion with relativistic picosecond laser light

The interaction of relativistic short-pulse lasers with matter produces fast electrons with over megaampere currents, which supposedly heats a solid target isochorically and forms a hot dense plasma. In a picosecond timescale, however, thermal diffusion from hot preformed plasma turns out to be the dominant process of isochoric heating. We describe a heating process, fast thermal diffusion, launched from the preformed plasma heated resistively by the fast electron current. We demonstrate the fast thermal diffusion in the keV range in a solid density plasma by a series of one-dimensional particle-in-cell simulations. A theoretical model of the fast thermal diffusion is developed and we derive the diffusion speed as a function of the laser amplitude and target density. Under continuous laser irradiation, the diffusion front propagates at a constant speed in uniform plasma. Our model can provide a guideline for fast isochoric heating using future kilojoule petawatt lasers.

Figure 1

(a) Temporal evolution of the bulk electron temperature in the solid target ($n_{\mathrm{i}}=50, Z=10$) during the irradiation of the peak intensity ($a_0=1$) and trajectory of the diffusion front (orange circles). The position $x=0$ corresponds to the initial boundary position between the preplasma and the solid. The time $t=0$ is the time at which the laser peak reaches the position of the initial preplasma surface. (b) Time-space diagram of the bulk electron temperature $T_{\mathrm{e}}$ with contour lines. The orange circles show the trajectory of the diffusion front, as defined in (a). Each solid line is a contour line that represents $T_{\mathrm{e}}=$ 0.1, 0.2, 0.5, 1, 2, 5, and 10 keV.

Figure 2

(a) Initial electron density profile for each case. The scale lengths are 0.32, 0.80, 1.60, and 2.56 $\mu \mathrm{m}$. The maximum densities of the preplasma for the cases with the scale length of 2.56 $\mu \mathrm{m}$ are 50 and 500 $n_{\mathrm{c}}$. (b) Trajectory of thermal diffusion front in each case. The color of each line corresponds to that shown in (a).

Figure 3

(a) Temporal evolution of electron phase plot $x-p_{\mathrm{z}}$ for $a_0=1, n_{\mathrm{i}}=50, Z=10$, and the preplasma scale length 2.56 $\mu \mathrm{m}$. The arrows indicate the positions of the diffusion front at times $t=$ 0.83 ps (pink), 1.67 ps (yellow), 3.33 ps (green), and 5.0 ps (blue). (b) Trajectory of the fast thermal diffusion front in the time-space diagram. The orange line represents the result of the PIC simulation shown in (a). It is as same as the orange circles in Fig. 1 and the orange line in Fig. 2. The gray line represents the numerical solution of Eq. (2) with the factor $f=0.1$. The arrows indicate the observation times in (a).

Figure 4

Fast thermal diffusion speed $v_{\mathrm{heat}}$ vs $a_0/{\lambda }_{\mathrm{L},\mu \mathrm{m}}$ ($\propto I^{1/2}$). The target parameters were fixed ($n_{\mathrm{i}}=50, Z=10$). Black dots and white circles represent the simulation results for ${\lambda }_{\mathrm{L}}=$ 1.0 and 0.5 $\mu \mathrm{m}$, respectively. The gray shaded area and striped area are $v_{\mathrm{heat}}$ obtained from Eq. (5) with the factor $f=0.065$ and the absorption rate ${\chi }_{\mathrm{ab}}=0.1$–0.17 for ${\lambda }_{\mathrm{L}}=$ 1.0 $\mu \mathrm{m}$ and ${\chi }_{\mathrm{ab}}=0.13$–0.22 for ${\lambda }_{\mathrm{L}}=$ 0.5 $\mu \mathrm{m}$, respectively.

Figure 5

Fast thermal diffusion speed $v_{\mathrm{heat}}$ vs target electron density $n_{\mathrm{e}}$ when the laser normalized amplitude and the laser wavelength are fixed ($a_0=2$ and ${\lambda }_{\mathrm{L}}=1.0$). The black square, circle, and triangle dots represent the simulation results of $Z=5$, 10, and 20, respectively. The gray shaded area is the result of the theoretical model with the absorption rate ${\chi }_{\mathrm{ab}}=0.17$–0.2.

Figure 6

2D heat map of electron temperature at $t=1000\tau$ for ${\varphi }_{\mathrm{L}}\approx 50\mu \mathrm{m}$ (large), and ${\varphi }_{\mathrm{L}}\approx 7\mu \mathrm{m}$ (small). The white line is the contour of electron temperature of the inflection point at the center, $Y=0$. The parameters for the two-dimensional simulation are as follows: A 1.65 ps (500 ${\tau }_{\mathrm{L}}$) flat-top laser pulse (${\lambda }_{\mathrm{L}}=1\mu \mathrm{m}$) is injected from the left boundary with the peak intensity of $a_0=$ 1.4. The pulse has a Gaussian rising edge with the time width of 2.31 ps (700 ${\tau }_{\mathrm{L}}$). The laser focused spot size (FWHM) was set to ${\varphi }_{\mathrm{L}}\approx$ 50 $\mu \mathrm{m}$ (large) and 7 $\mu \mathrm{m}$ (small). The laser was irradiated perpendicular to the target surface. A 30-$\mu \mathrm{m}$-thick aluminum with the ion density of 59.8 $n_{\mathrm{c}}$ (2.7 g/cc) is placed in a $60\times 120\mu \mathrm{m}$ simulation box with 20 $\mu \mathrm{m}$ of vacuum in front of the target. Note here that we start from the partially ionized aluminum ($Z=1$). Each cell includes 2 and 26 superparticles for ions and electrons, respectively, in the plasma region. We used a numerical resolution of $\mathrm{\Delta }x=\mathrm{\Delta }y=1/50\mu \mathrm{m}$ and $\mathrm{\Delta }t={\tau }_{\mathrm{L}}/50$, where ${\tau }_{\mathrm{L}}$ is the laser period.