Dynamic model of target charging by short laser pulse interactions

A model providing an accurate estimate of the charge accumulation on the surface of a metallic target irradiated by a high-intensity laser pulse of fs–ps duration is proposed. The model is confirmed by detailed comparisons with specially designed experiments. Such a model is useful for understanding the electromagnetic pulse emission and the quasistatic magnetic field generation in laser-plasma interaction experiments.

Figure 1

Scheme of target charging by laser and the key interaction parameters.

Figure 2

Maxwell-Juttner distribution function and the potential barrier $\mathrm{\Phi }$.

Figure 3

Density profiles for ions (full line) and electrons (dashed line) near the target edge.

Figure 4

(top) Three-dimensional geometry of the potential numerical reconstruction and (bottom) the map of potential for $R_h/{\lambda }_{\mathrm{Dh}}=50$, negative values for $x/{\lambda }_{Dh}>0$ and positive values for $x/{\lambda }_{Dh}<0$.

Figure 5

Comparison of the potential ${\widehat{\varphi }}_{\mathrm{th}}$ from the analytical calculation or from the numerical reconstruction.

Figure 6

(bottom) Comparison of the minimum and the maximum of the thermal potential and (top) the minimum position of the thermal potential. Data from the analytical calculation or from the numerical reconstruction.

Figure 7

Scheme of four successive charged disks at a time $t$, which were born at four arbitrary times $t_1<t_2<t_3<t_4<t$.

Figure 8

Two examples of the total potential $\mathrm{\Phi }={\varphi }_{\mathrm{th}}+{\varphi }_E$, and the effective potential barrier $\mathrm{\Delta }\mathrm{\Phi }$.

Figure 9

Experimental measurements of the charge (symbols) and the model predictions (curves) for scans on the laser energy and on the laser pulse duration: $E_{\mathrm{las}}=20$ to 80 mJ, $t_{\mathrm{las}}=30$ fs to 5 ps, ${\lambda }_{\mathrm{las}}=800$ nm, $r_{\mathrm{las}}=6\mu \text{m}$, and ${\eta }_{\mathrm{las}}=40\%$.

Figure 10

Ejection regime limits for $E_{\mathrm{las}}=80$ mJ, ${\lambda }_{\mathrm{las}}=800$ nm, $r_{\mathrm{las}}=6\mu \text{m}$, and ${\eta }_{\mathrm{las}}=40\%$ for the aluminum (blue), tantalum (red), and copper (green) targets. The regime limits are close for all materials as shown by the three set of curves.

Figure 11

Temporal evolution of the cloud radius, $R_h$ (a), the total number of hot electrons, $N_h$ (b), and the ejection current, $J_h$ (c) in the steady-state regime (dashed lines) and the comparison with their approximated values in this regime (solid lines). $t_{\mathrm{las}}=10$ ps, $E_{\mathrm{las}}=80$ mJ, ${\lambda }_{\mathrm{las}}=800$ nm, $r_{\mathrm{las}}=6\mu \text{m}$, and ${\eta }_{\mathrm{las}}=40\%$.

Figure 12

Temperature and charge evolution in the full ejection regime. The charge is already at 80% of its maximum at the time $t_{ee}$. $E_{\mathrm{las}}=0.8$ J, $t_{\mathrm{las}}=5$ fs, ${\lambda }_{\mathrm{las}}=800$ nm, $r_{\mathrm{las}}=6\mu \text{m}$, and ${\eta }_{\mathrm{las}}=40\%$.

Figure 13

Comparison between $Q_{\mathrm{tot}}$ and the final charge obtained in the full ejection regime: $E_{\mathrm{las}}=0.8$ J, ${\lambda }_{\mathrm{las}}=800$ nm, $r_{\mathrm{las}}=6\mu \text{m}$, and ${\eta }_{\mathrm{las}}=40\%$.

Figure 14

Time evolution of $N_h/(R_h-r_{\mathrm{las}})$ for three materials. The electron surface charge is material independent except a slight deviation in the thermal regime. $E_{\mathrm{las}}=0.8$ J, ${\lambda }_{\mathrm{las}}=800$ nm, $r_{\mathrm{las}}=6\mu \text{m}$, and ${\eta }_{\mathrm{las}}=40\%$.

Figure 15

Effect of the target size on the EMP signal observed on different laser facilities. Dots are experimental measurements for the charge (a) and for the EMP electric field [(b) and (c)]. Lines are charge model predictions. (a) Top curve and dots are from ECLIPSE facility [6] $(E_{\mathrm{las}}=80$ mJ, $t_{\mathrm{las}}=50$ fs, ${\lambda }_{\mathrm{las}}=800$ nm, $r_{\mathrm{las}}=7.5\mu \text{m}$, and ${\eta }_{\mathrm{las}}=40\%)$. Bottom curve of panel (a) and the curve of panel (b) are from Ref. [19] $(E_{\mathrm{las}}=150$ mJ, $t_{\mathrm{las}}=150$ ps, ${\lambda }_{\mathrm{las}}=1064$ nm, $r_{\mathrm{las}}=150\mu \text{m}$, and ${\eta }_{\mathrm{las}}=40\%)$. (c) Data from the TITAN facility [18] $(E_{\mathrm{las}}=200$ J, $t_{\mathrm{las}}=20$ ps, ${\lambda }_{\mathrm{las}}=1050$ nm, $r_{\mathrm{las}}=10\mu \text{m}$, and ${\eta }_{\mathrm{las}}=40\%)$.

Figure 16

Experimental measurements of the charge and the model predictions for a scans on the laser energy and on the laser pulse duration with a Teflon target. Stars and squares are two identical targets. Lines are the model prediction. ${\lambda }_{\mathrm{las}}=800$ nm, $r_{\mathrm{las}}=6\mu \text{m}$, and ${\eta }_{\mathrm{las}}=40\%$. (a) Laser energy scan $E_{\mathrm{las}}=30$ to 90 mJ at $t_{\mathrm{las}}=30$ fs. (b) Laser pulse duration scan $t_{\mathrm{las}}=30$ fs to 5 ps at $E_{\mathrm{las}}=80$ mJ.

Figure 17

Amplitude of the EMP electric field at 50 cm of the target from various laser facilities versus the laser energy. Blue zone (left) shows the EMP induced by the target charging. Red zone (right) shows the EMP damped by the neutralization.

Figure 18

Modified diffusion model of electron beam penetration in a target: $R_m$ is the maximum range, ${\chi }_D$ the penetration depth, ${\chi }_E$ the maximum energy dissipation depth, and $r_B$ the backscattering range. Figure extracted from Ref. [21] with permission.

Figure 19

Notation of the distances and a scheme of the potentials among the target, the holder, and the ground at equilibrium.

Figure 20

Time dependence of the neutralization charge through the holder with a perfect conductor target and with a Teflon target. Laser parameters: $E_{\mathrm{las}}=80$ mJ, $t_{\mathrm{las}}=50$ fs, and $r_{\mathrm{las}}=4$ $\mu \text{m}$. Data obtained from numerical simulations with the setup described in Ref. [6].

Figure 21

Schematic process of the equation solver.

Figure 22

(a) Final charge obtained for various values of $dt$ and $d\varepsilon$. The converged result equals 3.40 nC. (b) Temporal evolution of the current for different resolutions: $E_{\mathrm{las}}=0.08$ J, $r_{\mathrm{las}}=6$ um, ${\lambda }_{\mathrm{las}}=800$ nm, $t_{\mathrm{las}}=0.5$ um, and ${\eta }_{\mathrm{las}}=40\%$.