Physical mechanism of the electron-ion coupled transverse instability in laser pressure ion acceleration for different regimes

In radiation pressure ion acceleration (RPA) research, the transverse stability within laser plasma interaction has been a long-standing, crucial problem over the past decades. In this paper, we present a one-dimensional two-fluid theory extended from a recent work Wan et al. Phys. Rev. Lett. 117, 234801 (2016) to clearly clarify the origin of the intrinsic transverse instability in the RPA process. It is demonstrated that the purely growing density fluctuations are more likely induced due to the strong coupling between the fast oscillating electrons and quasistatic ions via the ponderomotive force with spatial variations. The theory contains a full analysis of both electrostatic (ES) and electromagnetic modes and confirms that the ES mode actually dominates the whole RPA process at the early linear stage. By using this theory one can predict the mode structure and growth rate of the transverse instability in terms of a wide range of laser plasma parameters. Two-dimensional particle-in-cell simulations are systematically carried out to verify the theory and formulas in different regimes, and good agreements have been obtained, indicating that the electron-ion coupled instability is the major factor that contributes the transverse breakup of the target in RPA process.

Figure 1

(a) The schematic model of the hole boring process by radiation pressure. (b) The physical picture of transverse instability within the high-density layer. $z$ represents the longitudinal direction and $x$ and $y$ represent the transverse directions. $n_{i1}$ and $E_{x,y1}$ represent the ion density and the transverse electric field fluctuations, respectively. $f_p$ represents the ponderomotive force.

Figure 2

The relationship between $k$ and $\mathrm{Im}(\omega$) obtained from Eq. (7) in the case of ${\gamma }_0=1.06$, ${\omega }_{pe}=6.32{\omega }_0$, ${\omega }_{pi}=0.147{\omega }_0$, and $T_e=0.005m_e{c}^2$.

Figure 3

The relationship between $k$ and $\mathrm{Im}(\omega$) obtained from Eq. (12) in the case of ${\gamma }_0=1.06$, ${\omega }_{pe}=6.32{\omega }_0$, ${\omega }_{pi}=0.147{\omega }_0$, and $T_e=0.005m_e{c}^2$.

Figure 4

The relationship between $k$ and $\mathrm{Im}(\omega$) obtained from Eq. (7) (solid blue) and Eq. (13) (dashed red) in the case of ${\gamma }_0=1.5$, ${\omega }_{pe}=6{\omega }_0$, ${\omega }_{pi}=0.13{\omega }_0$, and $T_e=0.005m_e{c}^2$.

Figure 5

The relations between $\xi$ and $\omega$ from Eq. (14) (solid blue) and Eq. (15) (dashed yellow) in the case of ${\gamma }_0=1.5$, ${\omega }_{pe}=6.32{\omega }_0$, and ${\omega }_{pi}=0.147{\omega }_0$.

Figure 6

Schematic drawing of the hole boring process for $a_0<1$. Here $n_{p0}$ is the initial plasma density.

Figure 7

The ion density fluctuations at $t=180{\omega }_0^{-1}$ after laser impinges on the target in the case of $a_0=0.2$, $n_{p0}=10n_c$. (a) The ion density with transverse ripples within the front high-density layer and its line-out distribution at $z=10.38c/{\omega }_0$ (the red dot line). (b) The FFT of the proton density and its line-out distribution at $k_z=0$ (the green dot line). $k_{\mathrm{sim}}$, $k_{\mathrm{num}}$, and $k_{\mathrm{est}}$ are obtained from PIC simulations, from direct numerical solutions of Eq. (7), and from Eq. (21), respectively.

Figure 8

(a) The relationship between $k_m$ and $a_0$ for $n_{p0}=10n_c$. (b) The relationship between $k_m$ and $n_{p0}$ for $a_0=0.2$. $k_{\mathrm{sim}}$, $k_{\mathrm{num}}$, and $k_{\mathrm{est}}$ are obtained from PIC simulations, from direct numerical solutions of Eq. (7), and from Eq. (21), respectively.

Figure 9

(a) The relationship between ${\gamma }_m$ and $a_0$ at $n_{p0}=10n_c$. (b) The relationship between ${\gamma }_m$ and $n_{p0}$ at $a_0=0.2$. ${\gamma }_{\mathrm{est}}$ and ${\gamma }_{\mathrm{sim}}$ are the estimated and simulation growth rates of ion density ripples, respectively.

Figure 10

The ion density fluctuations at $t=120{\omega }_0^{-1}$ after the laser impinges on the target in the case of $a_0=5$, $n_{p0}=10n_c$. (a) The ion density transverse ripples within the high-density layer and its line-out distribution at $z=15.37c/{\omega }_0$ (the black dot line). (b) The FFT of the ion density and its line-out distribution at $k_z=0$ (the green dot line). $k_{\mathrm{sim}}$, $k_{\mathrm{num}}$, and $k_{\mathrm{est}}$ are obtained from PIC simulations, from direct numerical solutions of Eq. (7), and from Eq. (25), respectively.

Figure 11

(a) The relationship between $k_m$ and $a_0$ when $n_{p0}=20n_c$. (b) The relationship between $k_m$ and $n_{p0}$ when $a_0=5$. $k_{\mathrm{sim}}$, $k_{\mathrm{num}}$, and $k_{\mathrm{est}}$ are obtained from PIC simulations, from direct numerical solutions of Eq. (7), and from Eq. (25), respectively.

Figure 12

The ion density fluctuations at $t=200{\omega }_0^{-1}$ after laser impinges on the target in the case of $a_0=5$ and $n_{p0}=3n_c$. (a) The ion density transverse ripples within the front high-density layer and its line-out distribution at $26.24c/{\omega }_0$. (b) The FFT of the ion density and its line-out distribution at $k_z=0$ (the green dot line). $k_{\mathrm{sim}}$, $k_{\mathrm{num}}$, and $k_{\mathrm{est}}$ are obtained from PIC simulations, from direct numerical solutions of Eq. (7), and from Eq. (25), respectively.

Figure 13

The ion density evolution in the case of $a_0=2.5$, $n_{p0}=10n_c$, $d=0.4c/{\omega }_0$ at $t=100{\omega }_0^{-1}$ after the laser impinges on the target (a, b) and in the case of $a_0=1$, $n_{p0}=4n_c$, $d=0.4c/{\omega }_0$ at $t=150{\omega }_0^{-1}$ after the laser impinges on the target (c, d). (a, c) The ion density distributions and their line-out distribution at $z=31.02c/{\omega }_0$ for (a) and $21.57c/{\omega }_0$ for (c) (the black dot line). (b, d) The FFT of the ion density and its line-out distribution at $k_z=0$ (the green dot line). $k_{\mathrm{sim}}$, $k_{\mathrm{num}}$, and $k_{\mathrm{est}}$ are obtained from PIC simulations, from direct numerical solutions of Eq. (7), and from Eq. (25), respectively.