Superluminal sheath-field expansion and fast-electron-beam divergence measurements in laser-solid interactions

We show that including a sufficient description of the target’s rear surface significantly affects the interpretation of a wide range of laser-solid experiments. A simple Debye sheath model will be shown to be adequate. From this the sheath field responsible for ion acceleration has been shown to expand at superluminal speeds, leading to very large ion-emission regions on the target’s rear surface; a new explanation for the dynamics of the ion-accelerating sheath field accounts for this observation and demonstrates the inaccuracy of measuring the angular divergence of the injected electron beam, crucial to fast ignition, from the lateral extent of the ion emission. However, it is shown that on careful probing the sheath field can provide unique insight into details of the fast electron’s distribution function. The relative merits of probing other physical quantities has been examined. The width of the background temperature spot overestimates the divergence by a factor of 2 unless electron recirculation is prevented.

Figure 1

The two types of targets considered in this paper.

Figure 2

The peak sheath field (top) and potential (bottom) against the fractional number density of fast electrons injected in 1D simulations ($n_f/n_{0f}$) various times into the simulation.

Figure 3

The sheath field 100 and 500 fs after the beam enters the vacuum (top). The potential of the sheath after 500 fs (bottom).

Figure 4

Fast-electron number density just inside and sheath field and potential just outside the target after 100 fs (top). Profiles outside the target constructed from current inside after 285 fs with (R) and without (NR) refluxing (bottom). The peak values are as follows: $n_f^{\text{peak}}=2.0\times {10}^{27}$ ${\mathrm{m}}^{-3}$, $E_y^{\text{peak}}=8.1\times {10}^{12}$ V m${}^{-1}$, and $\Phi {}^{\text{peak}}=3.8$ MeV (top); $n_f^{\text{peak}}=4.6\times {10}^{27}$ ${\mathrm{m}}^{-3}$ with refluxing and $n_f^{\text{peak}}=2.4\times {10}^{27}$ ${\mathrm{m}}^{-3}$ without refluxing (bottom). $Q/Q_{\text{max}}$ represents the normalized value of a general physical quantity.

Figure 5

An illustration of the mechanism responsible for the lateral spreading of the sheath electric field responsible for TNSA. The sheath field spreads up the target with the increasing intersection of the expanding cloud of fast electrons and the plane of the rear.

Figure 6

The sheath-field profile along the rear surface with the magnetic field turned on ($B$—solid lines) and off (no $B$—dashed lines) after 250 fs (red) and 500 fs (blue).

Figure 7

Particle tracks for 1-MeV electrons injected at various angles. The tracks display each particle’s motion in the time interval 50–100 fs.

Figure 8

Logarithm of the fast-electron number density after 1 ps (top). Relative change in fast-electron number density and background electron temperature ($Q/Q_{\text{max}}$) along the rear of the target after 1 ps (bottom). The maximum number density and temperature on the rear surface were $4.6\times {10}^{27}$ ${\mathrm{m}}^{-3}$ and 690 eV with recirculation and $3.8\times {10}^{27}$ ${\mathrm{m}}^{-3}$ and 530 eV without recirculation.

Figure 9

Comparison between the lateral sheath-field profile for injected distributions with average half angles of ${30}^{°}$ and ${60}^{°}$ after 100 fs (top) and between the sheaths for an injected Gaussian $m=2$ and super-Gaussian $m=50$ (middle). The densities reconstructed for each case (dashed lines) compared to that from the simulation (solid lines) after 100 fs (bottom).