Dynamics of the boundary layer created by the explosion of a dense object in an ambient dilute gas triggered by a high power laser

The dynamics of the boundary layer in between two distinct collisionless plasmas created by the interaction between a dense object modeling a cluster and a short laser pulse in the presence of an ambient gas is studied with two dimensional relativistic particle-in-cell simulations, which are found to be described by three successive processes. In the first phase, a collisionless electrostatic shock wave, launched near the cluster expansion front, reflects the ambient gas ions at a contact surface as a moving wall, which allows a particle acceleration with a narrower energy spread. In the second phase, the contact surface disappears and the compressed surface of the ambient gas ions passes over the shock potential, forming an overlapping region between the cluster expansion front and the compressed surface of the ambient gas. Here, another type of nonlinear wave is found to be evolved, leading to a relaxation of the shock structure, while continuing to reflect the ambient gas ions. The nonlinear wave exhibits a bipolar electric field structure that is sustained for a long timescale coupled with slowly evolving ion dynamics, suggesting that a quasistationary kinetic equilibrium dominated by electron vortices in the phase space is established. In the third phase, a rarefaction wave is triggered and evolves at the compressed surface of ambient gas. This is because some of the ambient gas ions tend to pass over the potential of the bipolar electric field. Simultaneously, a staircase structure, i.e., a kind of internal shock, is formed in the cluster due to the deceleration of cluster ions. Such structure formations and successive dynamics accompanied by the transitions from the shock wave phase through the overlapping phase to the rarefaction wave phase are considered to be a unique nature at the boundary layer created by an explosion of a dense plasma object in an ambient dilute plasma.

Figure 1

Snapshots of the 2D images for charge density distributions of (a) the cluster electrons ${\rho }_e^{cl}$, (b) the cluster ions ${\rho }_i^{cl}$, and (c) the ambient gas ions ${\rho }_i^{ag}$ at $t=86.7\mathrm{fs}$ in the case of the ambient gas density $n_{ag}=0.024n_c$ and the laser amplitude $a_0=2.19$. Densities are normalized by the initial charge density of the cluster ions and displayed on a log scale.

Figure 2

Snapshots of the 2D images for charge density distributions of (a)–(d) the cluster electrons ${\rho }_e^{cl}$, (e)–(h) the cluster ions ${\rho }_i^{cl}$, and (i)–(l) the ambient gas ions ${\rho }_i^{ag}$ at (a), (e), (i) $t=133\mathrm{fs}$; (b), (f), (j) $t=200\mathrm{fs}$; (c), (g), (k) $t=300\mathrm{fs}$; and (d), (h), (l) $t=500\mathrm{fs}$ in the case of the density $n_{ag}=0.024n_c$ and the amplitude $a_0=2.19$. Densities are normalized by the initial charge density of the cluster ions and displayed on a log scale. White dashed lines show positions of the compressed surface of ambient gas ions in the $x$ direction. Orange dashed lines in (c) and (d) show positions of the ringlike structure of cluster electrons in the $+x$ direction.

Figure 3

(a)–(d) The 1D cross sectional views for charge density distributions of cluster ions ${\rho }_i^{cl}$ (black solid line), ambient gas ions ${\rho }_i^{ag}$ (blue solid line), cluster electrons ${\rho }_e^{cl}$ (black dotted line), ambient gas electrons ${\rho }_e^{ag}$ (blue dotted line), and the resulting electric field $E_y$ (red solid line) along the $+y$ direction from the center of the cluster at (a) $t=133\mathrm{fs}$, (b) $t=200\mathrm{fs}$, (c) $t=300\mathrm{fs}$, and (d) $t=500\mathrm{fs}$. The maximum value of the ambipolar electric field $E_m^{am}$ and the maximum negative value of the bipolar electric field $E_m^{bi}$ are also shown. Densities are normalized by the initial density of the cluster ions. (e) The time evolution of charge density distributions of cluster ions (black solid line) and ambient gas ions (blue solid line). Dotted lines show the initial charge density distributions of cluster ions (black) and ambient gas ions (blue).

Figure 4

(a1)–(f1) Schematic views of the density distribution of cluster ions (black solid lines) and ambient gas ions (blue solid lines). Schematic views of the phase space distributions of cluster ions (green thick lines) and ambient gas ions (red thick lines) corresponding to (a1)–(f1) are shown in (a2)–(f2). A peaking structure is marked by ($*$). Open and/or filled circles in (b2) and (c2) represent locations of ions.

Figure 5

(a) Temporal evolutions for the location of the expansion front of the cluster ions $r_f^{cl}$ (black solid line, A), the location exhibiting the peak density of ambient gas ions $r_{f\left(B\right)}^{ag}$ (blue solid line, B), and that exhibiting the front of ambient gas ions $r_{f\left(D\right)}^{ag}$ (green solid line, D). Here, A, B, and D represent the same fronts as those in Fig. 12. The charge density of the peaking structure of the ambient gas ions (blue dotted line) along the $+y$ direction from the center of the cluster is also shown. The black dashed line represents the temporal evolution for the location of the expansion front of the cluster ions in a vacuum $r_f^{cl\left(v\right)}$ as a reference. The green dotted line represents the intensity of the laser pulse. The pulse peak passes through the center of the cluster at $t=86.7\mathrm{fs}$. The charge density of the peaking structure of the ambient gas ions is normalized by the initial charge density of the ambient gas ions. (b) Temporal evolutions for the maximum value of the ambipolar electric field $E_m^{am}$ (red dotted line) and the maximum negative value of the bipolar electric field $E_m^{bi}$ (red solid line). The intensity of the laser pulse is also shown (green dotted line).

Figure 6

The 1D cross sectional views for (a1)–(a3) the field $E_y$ (red solid line) and the electrostatic potential for ions $e\mathrm{\Phi }$ (purple solid line), (b1)–(b3) the charge density ${\rho }_i^{cl}$ (black solid line) and the phase space distributions of the cluster ions, (c1)–(c3) the charge density ${\rho }_i^{ag}$ (blue solid line) along the $+y$ direction from the center of the cluster at (a1)–(c1) $t=133\mathrm{fs}$, (a2)–(c2) $t=300\mathrm{fs}$, and (a3)–(c3) $t=500\mathrm{fs}$. Electrostatic potentials are set to zero at $y=7\mu \mathrm{m}$. Charge densities are normalized by the initial density of ions. Phase space distributions consist of ions in the region of $y\ge 0$ and $-0.15\le x\le +0.15\mu \mathrm{m}$, and are normalized by the initial charge density of the cluster ions and displayed on a log scale.

Figure 7

Energy spectra of the cluster ions (black solid line) and the reflected ambient gas ions (blue solid line) at (a) $t=133\mathrm{fs}$ and (b) $t=300\mathrm{fs}$. The ion number is evaluated in real value by multiplying the particle weight by the PIC particle. Spectra consist of ions in the region of $y\ge 0$ and $-0.15\le x\le +0.15\mu \mathrm{m}$.

Figure 8

The 1D cross sectional views for (a1)–(a3) the phase space distributions of electrons and the electrostatic potential for electrons $-e\mathrm{\Phi }$ (gray solid line), and (b1)–(b3) the contour plots of electron trajectories along the $+y$ direction from the center of the cluster at (a1), (b1) $t=133\mathrm{fs}$, (a2), (b2) $t=200\mathrm{fs}$, and (a3), (b3) $t=300\mathrm{fs}$. The electrostatic potential is set to zero at $y=7\mu \mathrm{m}$. Roman numerals (I)–(IV) in (b3) represent typical electron trajectories corresponding to the potential $-e\mathrm{\Phi }$ in (a3). Phase space distributions consist of electrons in the region of $y\ge 0$ and $-0.15\le x\le +0.15\mu \mathrm{m}$, and are normalized by the initial charge density of the cluster electrons and displayed on a log scale.

Figure 9

The 1D cross sectional views for (a1)–(a3) the phase space distributions of electrons and the electrostatic potential for electrons $-e\mathrm{\Phi }$ (gray solid line), and (b1)–(b3) the contour plots of electron trajectories along the $+y$ direction from the center of the cluster at (a1), (b1) $t=287\mathrm{fs}$, (a2), (b2) $t=300\mathrm{fs}$, and (a3), (b3) $t=320\mathrm{fs}$. Blue dashed lines indicate the position of the cluster expansion front. The electrostatic potential is set to zero at $y=0$. Phase space distributions consist of electrons in the region of $y\ge 0$ and $-0.15\le x\le +0.15\mu \mathrm{m}$, and are normalized by the initial charge density of the cluster electrons and displayed on a log scale.

Figure 10

(a)–(d) The 1D cross sectional views for charge density distributions of cluster ions ${\rho }_i^{cl}$ (black solid line), ambient gas ions ${\rho }_i^{ag}$ (blue solid line), cluster electrons ${\rho }_e^{cl}$ (black dotted line), ambient gas electrons ${\rho }_e^{ag}$ (blue dotted line), and the field $E_y$ (red solid line) along the $+y$ direction from the center of the cluster at $t=380\mathrm{fs}$ with the ambient gas density of (a) $0.012n_c$, (b) $0.024n_c$, (c) $0.096n_c$, and (d) $0.192n_c$. Densities are normalized by the initial charge density of the cluster ions. (e) The phase space distributions of ions (cluster and ambient gas) with the ambient gas density of $0.192n_c$. The phase space distributions consist of ions in the region of $y\ge 0$ and $-0.15\le x\le +0.15\mu \mathrm{m}$, and are normalized by the initial charge density of the cluster ions and displayed on a log scale.

Figure 11

The time evolution of the 1D cross sectional views for charge density distributions of ambient gas ions ${\rho }_i^{ag}$ (blue lines) and cluster ions ${\rho }_i^{cl}$ (black lines) at the density $n_{ag}=0.192n_c$ along the $+y$ direction from the center of the cluster at various times. The initial density distributions of the ambient gas (blue dotted line) and cluster (black dotted line) ions are also shown. Densities are normalized by the initial charge density of the ambient gas or cluster ions.

Figure 12

Temporal evolutions for the location of the expansion front $r_f^{cl}$ (black solid line, A) and that of the staircase structure $r_s^{cl}$ (red solid line, C) of the cluster ions, and that exhibiting the peak density $r_{f\left(B\right)}^{ag}$ (blue solid line, B) and that exhibiting the front $r_{f\left(D\right)}^{ag}$ (green solid line, D) of ambient gas ions along the $+y$ direction from the center of the cluster at the density $n_{ag}=0.192n_c$. The inset shows a 1D cross sectional view for the density distributions of ambient gas ions (blue solid line) and cluster ions (black solid line) at $t=400\mathrm{fs}$. The locations A, B, C, and D in the inset correspond to those in the figure. The black dotted line represents the temporal evolution for the location of the expansion front of the cluster ions in a vacuum $r_f^{cl\left(v\right)}$ as a reference.

Figure 13

Snapshots of the 2D images for charge density distributions of (a), (b) the cluster ions ${\rho }_i^{cl}$ and (c), (d) the ambient gas ions ${\rho }_i^{ag}$ at (a), (c) $t=400\mathrm{fs}$ and (b), (d) 600 fs at the density $n_{ag}=0.192n_c$. Densities are normalized by the initial charge density of the cluster ions and displayed on a log scale.

Figure 14

(a) The time $t=t_c$ at which the crossing between the cluster expansion front and the compressed surface of ambient gas ions takes place (the red solid line) and the time $t=t_r$ at which the rarefaction wave is triggered (the red dashed line) at the density $n_{ag}=0.012n_c$, $0.024n_c$, $0.048n_c$, $0.096n_c$, and $0.192n_c$, respectively. The time $t_c^{\left(m\right)}$ represents the crossing time derived by using the function of the electric field modeled as Eq. (4) from the viewpoint of particle motions in the laboratory frame (red dotted line). The maximum energy of ambient gas ions $W_m^{ag}$ at $t=400\mathrm{fs}$ is also shown (blue solid line). (b) The 1D cross sectional views for the ambipolar electric field $E_y$ for the different ambient gas density $n_{ag}$ along the $+y$ direction from the center of the cluster at $t=120\mathrm{fs}$. The field $E_y$ outside the cluster is approximately fitted by the exponential functions with different amplitude $E_0$ and the scale length $L_e$, which is represented by the black dashed line.

Figure 15

Energy distributions of the cluster electrons at $t=93.3\mathrm{fs}$, 100 fs, 107 fs, 113 fs, and 120 fs, respectively. Black dashed lines show the exponential function.

Figure 16

(a) The spatial distributions of ion density (black solid line), electron density (black dotted line), and electric field $\widetilde{E_y}$ (purple solid line) at the initial state, i.e., at $t=0$, in the region $0\le \tilde{y}\le 2$ for ${\tilde{\varepsilon }}_{\max }=7$. The spatial distributions of (b) electric field $\widetilde{E_y}$ and (c) potential energy $-e\tilde{\mathrm{\Phi }}$ for four cases, i.e., ${\tilde{\varepsilon }}_{\max }=4$, 5, 6, and 7. Here, ${\tilde{\varepsilon }}_{\max }$ represents the maximum electron energy that is normalized by the electron temperature $T_e$. The black dashed lines in (b) represent the exponential function for each energy ${\tilde{\varepsilon }}_{\max }$, and the black arrows in (c) represent the location of the electron front ${\tilde{r}}_e$, which is normalized by $r_f$, for each energy ${\tilde{\varepsilon }}_{\max }$.

Figure 17

The spatial distributions of the electric field $\widetilde{E_y}$ (purple solid line) and the potential energy $e\tilde{\mathrm{\Phi }}$ (purple dashed line) for ${\tilde{\varepsilon }}_{\max }=7$. The field ${\widetilde{E_y}}_1$ (brown solid line), which is calculated by using the first term on the RHS of Eq. (A5), and the field ${\widetilde{E_y}}_2$ (orange solid line), which is calculated by using the second, are also shown. The black dashed line shows a nearly exponential function in the intermediate radial region except those near the ion front and electron front.

Figure 18

The spatial distribution of the field $E_y$ at different times for the density (a) $n_{ag}=0.024n_c$ and (b) $n_{ag}=0.048n_c$.

Figure 19

(a) The spatial distribution of the field $E_y$ (red solid line) at $t=93.3\mathrm{fs}$ for the density $n_{ag}$ = (a1) $0.024n_c$ and (a2) $0.048n_c$. The exponential function $\tilde{E}\left(y\right)$ (red dashed line) is also shown. (b) Temporal evolutions for the location of the cluster ion (black solid line), which is initially located at $y=R_0$, and those of the ambient gas ions (blue lines), which are initially located at $y=R_0$, $2R_0$, $4R_0$, $6R_0$, $8R_0$, $10R_0$, or $12R_0$. Note that (b1) and (b2) correspond to (a1) $0.024n_c$ and (a2) $0.048n_c$, respectively. The location $y$ is calculated by using the function $\tilde{E}\left(y\right)$. Red dashed line represents the time $\widetilde{t_c}$ at which the crossing takes place.