Electron Temperature Scaling in Laser Interaction with Solids

A precise knowledge of the temperature and number of hot electrons generated in the interaction of short-pulse high-intensity lasers with solids is crucial for harnessing the energy of a laser pulse in applications such as laser-driven ion acceleration or fast ignition. Nevertheless, present scaling laws tend to overestimate the hot electron temperature when compared to experiment and simulations. We present a novel approach that is based on a weighted average of the kinetic energy of an ensemble of electrons. We find that the scaling of electron energy with laser intensity can be derived from a general Lorentz invariant electron distribution ansatz that does not rely on a specific model of energy absorption. The scaling derived is in perfect agreement with simulation results and clearly follows the trend seen in recent experiments, especially at high laser intensities where other scalings fail to describe the simulations accurately.

Figure 1

Comparison of various temperature scalings with selected experimental values (diamonds) and PIC simulations (squares). Simulations include ionizations and collisions, target is a flat foil with thickness $5\lambda$ of copper ions, covered with a $2\lambda$ thick proton layer. The pulse duration of the Gaussian pulse with waist $2\lambda$ (a test simulation with a plane wave at $a_0=100$ yields a similar temperature) was fixed at $100{\omega }_0^{-1}$. To reduce computational demands, the electron density when fully ionized was set to $10n_c$, $40n_c$ or $100n_c$ for intensities with $a_0<8.5$, $8.5\le a_0\le 20$ or $a_0=100$. This choice results in a skin-length $\delta \ge \lambda /2$, a situation corresponding to a realistic solid density case with sufficient preplasma at the front surface.

Figure 2

(a) Trajectories of electrons distributed uniformly at $t_0={\tau }_0=0$, assuming adiabatic ramp up of laser intensity ($a_0=5$). $f_{\tau }$ is constant while $f_t={\gamma }^{-1}$, as indicated by trajectories crossing the red boxes. (b) Electron phase space density in the $z-p_z$ plane at the time the laser maximum reaches the foil front surface. Electrons are emitted into the target in bunches separated by $\lambda /2$. Parameters are the same as in Fig. 1 with $a_0=5$. (c),(d) Limiting cases for the temporal evolution of fields at the position of a virtual particle moving forward with $c$, crossing the critical density surface at $t=0$.

Figure 3

Comparison of proton maximum energy as a function of $a_0$ for ponderomotive electron temperature scaling (8) (solid) and our scaling (11) (dotted). Parameters are the same as for PIC simulations (Fig. 2), assuming a laser absorption of 25% and electron divergence of 40° based on the simulations. Pulse duration $\tau =100{\omega }_0^{-1}$ (black) and $\tau =1000{\omega }_0^{-1}$ (gray), demonstrating the stronger effect of the temperature scaling for long pulses. Dashed lines indicate ${ϵ}_{\max }\propto a_0^2$.