2.1 The evolution of the symmetrical CPFs

The left panel in Figure 2 shows the typical interferograms of the evolution of the CPFs, which are produced by two laser-bunches ablating CH foils^{[Reference Yuan, Li, Liu, Zhang, Zhong, Zheng, Dong, Chen, Sakawa, Morita, Kuramitsu, Kato, Takabe, Rhee, Zhu, Zhao and Zhang22–Reference Yuan, Li, Su, Liao, Yin, Zhu and Zhang24]}. The right panel shows the corresponding density profile obtained by the Abel inversion. Before shock formation ( $t<2~\text{ns}$ ), we measure the free expansion of the plasma flow with electron density of $10^{19}~\text{cm}^{-3}$ and flow velocity of $1000~\text{km}\cdot \text{s}^{-1}$ . During shock formation ( $t\sim 2~\text{ns}$ ), both flows interact with each other and result in the sudden fringes shift at the midplane, indicating that a large-density jump ( $n_{\text{shock}}/n_{\text{flow}}\sim 3$ ) is generated. After that ( $t>3~\text{ns}$ ), the shifted fringes become smooth and filament structures parallel to the flow direction appear in the central region, indicating that the excited shock is perhaps dissipated by the growing filamentary structures ( $t>5~\text{ns}$ ). Since the MFP ( $\unicode[STIX]{x1D706}_{ii}$ ) scales as $1/Z^{4}$ , the $\unicode[STIX]{x1D706}_{ii}$ for hydrogen ions ( $Z=1$ ) will be higher than that for carbon ions ( $Z=6$ ). Here we just use $Z=6$ to calculate the lower limit of MFP, whose value is estimated as 180 mm, indicating that ions from inter-flows can freely interpenetrate each other. Consequently, these observed features in the CPFs should be induced by plasma collective behaviors, instead of the hydrodynamic stagnation.

Figure 2. The evolution of the counter-streaming flows obtained by a Nomarski interferometer. The red circle in the raw images stands for the laser focal spot. When both flows coming from the opposing foils interpenetrate each other at the midplane at 2 ns shown in Abel inversion image, the plasma density increases by the unanticipated factor of 3 (from $1\times 10^{19}~\text{cm}^{-3}$ to $2.8\times 10^{19}~\text{cm}^{-3}$ ), indicating that a shock has been generated. The width is measured as about $300~\unicode[STIX]{x03BC}\text{m}$ , much smaller than the MFPs. Obviously, it is collisionless. Subsequently ( ${\sim}3{-}5~\text{ns}$ ), the collisionless shock is dissipated by the growing filamentation structures.

Relevant works^{[Reference Stockem, Fiuza, Bret, Fonseca and Silva25, Reference Kato and Takabe26]} show that two candidate instabilities can be self-generated under such experimental conditions ( $n_{e}\sim 10^{19}~\text{cm}^{-3}$ , $\unicode[STIX]{x1D710}\sim 10^{8}~\text{cm}\cdot \text{s}^{-1}$ , $T_{e}\sim 0.2{-}1~\text{keV}$ ). One is electrostatic instability ^{[Reference Forslund and Shonk27]}, and the other one is Weibel-type/filamentation instability^{[Reference Weibel28, Reference Fox, Fiksel, Bhattacharjee, Chang, Germaschewski, Hu and Nilson29]}. Simulation results show that both instabilities are competitive. The filamentation instability developed later would destroy the evolution of the electrostatic instability^{[Reference Kato and Takabe26]}. This evolution is consistent with the observed experimental results. In order to measure the whole process of the ES, we should suppress the growth of the Weibel-type instability. According to the growth rate of the Weibel-type instability,

(1) $$\begin{eqnarray}\unicode[STIX]{x1D6FE}\propto \frac{\unicode[STIX]{x1D710}}{c}\sqrt{\frac{4\unicode[STIX]{x1D70B}n_{i}Z^{2}e^{2}}{m_{i}}}\propto \frac{Ze}{Acm_{p}}\sqrt{8\unicode[STIX]{x1D70B}fE_{L}},\end{eqnarray}$$

we have two methods to adjust the Weibel-type instability. One is symmetrical case (Case I), replacing the target material, i.e., changing the value, $(Z/A)\sqrt{f}$ . The other one is unsymmetrical case (Case II), changing the driven laser energy $E_{L}$ . Here we define the symmetrical case and unsymmetrical case according to the driven laser conditions instead of the plasma flow conditions. The symmetrical case stands for two bunches heating the target foils, and the unsymmetrical case stands for only one bunch heating the target foil.

2.2 Collisionless electrostatic shock formation and evolution in the CPFs

2.2.1 Unsymmetrical case

Figure 3 shows the ES in the CPFs at 5 ns and 9 ns^{[Reference Morita, Sakawa, Kuramitsu, Dono, Aoki, Tanji, Kato, Li, Zhang, Liu, Zhong, Takabe and Zhang30]}. In this experiment configuration, the laser-driven high-velocity, low-density left flow interacts with the low-velocity, high-density flow generated by X-ray ionization at the surface of the right foil. Both flow densities are measured by the interferograms, i.e., $n_{e}^{\text{Left}}\sim 2\times 10^{18}~\text{cm}^{-3}$ and $n_{e}^{\text{Right}}\sim 6\times 10^{18}~\text{cm}^{-3}$ . Some structures appear near to $x=3.6{-}3.7~\text{mm}$ at 5 ns, indicating that both flow velocities can be estimated as $\unicode[STIX]{x1D710}^{\text{Left}}\sim 830~\text{km}\cdot \text{s}^{-1}$ and $\unicode[STIX]{x1D710}^{\text{Right}}\sim 230~\text{km}\cdot \text{s}^{-1}$ . The corresponding MFPs are calculated as 25–35 mm, confirming that both flows interaction is collisionless. These observed structures at 5 ns in Figures 3(a) and 3(c) are identified as a large-density jump. However, its value is difficult to measure because of such a larger density gradient blocking the probe propagation. Anyway, such a large-density jump indicates that a collisionless shock has been generated at 5 ns. With the development of the shock, it continues to move from $x=3.6{-}3.7~\text{mm}$ at 5 ns to $x=3.1~\text{mm}$ at 9 ns, i.e., it propagates with a velocity of $\unicode[STIX]{x1D710}_{s}\sim 125{-}150~\text{km}\cdot \text{s}^{-1}$ . The corresponding Mach number is about 9–11, suggesting a strong collisionless shock formation. The typical features of the shock at 9 ns are shown in Figures 3(e) and 3(f), and the density jump decreases to $n_{\text{shock}}/n_{\text{flow}}\sim 3.89\pm 0.85$ with the whole transition region of about $200~\unicode[STIX]{x03BC}\text{m}$ .

To clarify the generation mechanism of the shock, a quasi-one-dimensional particle-in-cell (PIC) simulation is performed under the same experimental conditions. From Figure 4 in Ref. [Reference Morita, Sakawa, Kuramitsu, Dono, Aoki, Tanji, Kato, Li, Zhang, Liu, Zhong, Takabe and Zhang30], a self-generated bipolar electrostatic field mediated shock formation appears at $x/\unicode[STIX]{x1D706}_{e}\sim 26$ in the beginning, and then propagates toward $\pm x$ directions. The incoming ions will be slowed down or reflected back into upstream. Consequently, ions are accumulated into the interaction region until the density jump conditions are fulfilled^{[Reference Morita, Sakawa, Kuramitsu, Dono, Aoki, Tanji, Kato, Li, Zhang, Liu, Zhong, Takabe and Zhang30, Reference Sakawa, Morita, Kuramitsu and Takabe31]}.

Figure 3. Collisionless shock formation and evolution in unsymmetrical CPFs^{[Reference Morita, Sakawa, Kuramitsu, Dono, Aoki, Tanji, Kato, Li, Zhang, Liu, Zhong, Takabe and Zhang30]}. (a) and (b) show the interferogram obtained at 5 ns and 9 ns, respectively. (c) and (d) are the corresponding shadowgraphs of (a) and (b). (e) and (f) show the density profile and the intensity profile at 9 ns.

2.2.2 Symmetrical case

Figure 4 shows the ES formation and evolution in the CPFs at 6 ns and 10 ns. The CH target material is replaced by the Cu. In this experiment configuration, two similar laser-produced plasma flows interact with each other ( $n_{e}\sim 1\times 10^{19}~\text{cm}^{-3}$ , $2\unicode[STIX]{x1D710}\sim 7.5\times 10^{7}~\text{cm}\cdot \text{s}^{-1}$ ). Two overlapped shocks are formed in the central region and then evolve into two separated shocks from the downstream to upstream. The MFP (16–500 mm) is also much larger than the shock width ( $400{-}700~\unicode[STIX]{x03BC}\text{m}$ ). According to the heuristic model proposed by Park et al. in Ref. [Reference Park, Ryutov, Ross, Kugland, Glenzer, Plechaty, Pollaine, Remington, Spitkovsky, Gargate, Gregori, Bell, Murphy, Sakawa, Kuramitsu, Morita, Takabe, Froula, Fiksel, Miniati, Koenig, Ravasio, Pelka, Liang, Woolsey, Kuranz, Drake and Grosskopf32], the width of the ES transition region can be estimated as $L_{ES}=K\frac{\unicode[STIX]{x1D710}}{\unicode[STIX]{x1D714}_{pi}}\frac{W}{T_{e}}\approx (600{-}800)~\unicode[STIX]{x03BC}\text{m}$ , in agreement with our measurement ( $L\sim 450{-}800~\unicode[STIX]{x03BC}\text{m}$ ). However, the width of the Weibel-type shock should be estimated as $L_{EM}=K\frac{\unicode[STIX]{x1D710}}{\unicode[STIX]{x1D714}_{pi}}\approx 15~\text{mm}$ , much larger than our target separation. The theoretical analysis shows that the observed shocks should be electrostatic. The corresponding PIC simulations also confirm our understanding. As shown in Figure 5 in Ref. [Reference Yuan, Li, Liu, Zhong, Zhu, Li, Wei, Han, Pei, Zhao, Li, Zhang, Liang, Wang, Weng, Li, Jiang, Du, Ding, Zhu, Zhu, Zhao and Zhang20], a strong bipolar electrostatic field resulted from the instability excites the shocks. Large temperature and density gradients between upstream and downstream drive the shock from downstream to upstream.

Figure 4. Collisionless shock formation and evolution in the symmetrical CPFs. (a) and (b) are the interferograms (lower) and electron density distributions (upper) obtained by the Abel inversion, taken at 6 ns and 10 ns, respectively. (c) and (d) are the electron density profiles plotted along the flow direction^{[Reference Yuan, Li, Liu, Zhong, Zhu, Li, Wei, Han, Pei, Zhao, Li, Zhang, Liang, Wang, Weng, Li, Jiang, Du, Ding, Zhu, Zhu, Zhao and Zhang20]}.

2.3 Weibel/filamentation instability in the symmetrical case

Weibel instability is a promising candidate for creating astrophysical shocks. It is a typical electromagnetic phenomenon, driven by the plasma anisotropy. Under the current laser-plasma conditions, i.e., the electron thermal velocity is larger than the flow velocity and the ion thermal velocity is smaller than the flow velocity, the ions freely interpenetrate each other in the presence of a single thermalized electron background. Therefore, it is called ion–ion driven Weibel-type instability. The signature of Weibel instability is that the filamentary structures form and extend in the axis of flow direction. The self-generated magnetic field grows from linear phase to nonlinear phase until saturation. Although many groups^{[Reference Fox, Fiksel, Bhattacharjee, Chang, Germaschewski, Hu and Nilson29, Reference Ross, Higginson, Ryutov, Fiuza, Hatarik, Huntington, Kalantar, Link, Pollock, Remington, Rinderknecht, Swadling, Turnbull, Weber, Wilks, Froula, Rosenberg, Morita, Sakawa, Takabe, Drake, Kuranz, Gregori, Meinecke, Levy, Koenig, Spitkovsky, Petrasso, Li, Sio, Lahmann, Zylstra and Park33–Reference Huntington, Manuel, Ross, Wilks, Fiuza, Rinderknecht, Park, Gregori, Higginson, Park, Pollock, Remington, Ryutov, Ruyer, Sakawa, Sio, Spitkovsky, Swadling, Takabe and Zylstra35]} have been devoted into investigation of the Weibel-mediated shock formation, the required exact physical conditions for such an occurrence are still unclear.

According to Equation (1), we can find that increasing the flow velocity and density can enhance the growth rate of the Weibel instability. Here we demonstrate the Weibel instability in CPFs generated by CH–Al foils. Figure 5 shows the typical results of evolution of the Weibel-type instability. The left flow (in the interferogram) is generated by one bunch ablation of CH foil, and the right flow is generated by the other bunch ablation of Al foil. Such a configuration can produce a relatively high density ( $n_{e}\sim 3\times 10^{19}~\text{cm}^{-3}$ ) and similar velocity ( $\unicode[STIX]{x1D710}\sim 0.8\times 10^{8}~\text{cm}\cdot \text{s}^{-1}$ ) plasma flows, in comparison with CH–CH case in Figure 2 ( $n_{e}\sim 1\times 10^{19}~\text{cm}^{-3}$ , $\unicode[STIX]{x1D710}\sim 1\times 10^{8}~\text{cm}\cdot \text{s}^{-1}$ ). The typical features of the Weibel instability can be observed in the shadowgraph. At 3.5 ns, periodically distributing filaments parallel to flow direction localize at the midplane with a wavelength ( $\unicode[STIX]{x1D706}$ ) of about $50{-}80~\unicode[STIX]{x03BC}\text{m}$ . The wavelength is almost consistent with the ion inertial scale, $d_{i}=c/\unicode[STIX]{x1D714}_{pi}=c/\sqrt{4\unicode[STIX]{x1D70B}n_{i}\bar{Z}^{2}e^{2}/m_{i}}\approx 70{-}80~\unicode[STIX]{x03BC}\text{m}$ (here the used flows density is $n_{e}\sim (2{-}3)\times 10^{19}~\text{cm}^{-3}$ and $\bar{Z}\sim 10$ ), indicating that the instability is driven by the ions. The linear growth rate of the filamentation instability is approximated to be $\unicode[STIX]{x1D6FE}\approx 0.2\unicode[STIX]{x1D710}/\unicode[STIX]{x1D706}=(2{-}3)\times 10^{9}~\text{s}^{-1}$ . The corresponding linear time of the filaments is about 2–3 ns ( $t=2\unicode[STIX]{x1D70B}/\unicode[STIX]{x1D6FE}$ ), after both flows meeting at the midplane. At 4.5 ns, with the growth of instability, the filaments extend longer in the axial direction. The average space between adjacent filaments becomes $100{-}120~\unicode[STIX]{x03BC}\text{m}$ , indicating that the coalescence process between adjacent filaments occurs and the instability evolves from the linear mode into the nonlinear mode. At 6.5 ns, the larger wavelength of order $150{-}180~\unicode[STIX]{x03BC}\text{m}$ and disordered filaments at the midplane reveal that the instability has fully evolved into the nonlinear mode. The experimental evolution of the Weibel instability is consistent with the theoretical prediction.

Figure 5. The evolution of the filamentation instability in CPFs. Interferogram with magnification of 2.5 times shows initial conditions of both flows at 3.5 ns. A series of shadowgraph with larger magnification of 4 times is applied to measure the evolution of the Weibel instability. The intensity profile shows the typical features of the evolution of the filaments.

2.4 Other applications of the CPFs

The neutron yield in CPFs is an important tool to distinguish between collisionless and collisional effects. Neutrons generation in CPFs can originate from three sources: (i) the laser-induced fireballs from the target foils, (ii) the counter-propagating ions interaction with each other, and (iii) the scattering ions interaction with the ions from intra-flows. Here we carry out two neutron generation experiments for comparison^{[Reference Zhao, Chen, Li, Fu, Rhee, Li, Zhu, Li, Liao, Zhang, Han, Liu, Huang, Ma, Li, Xiong, Huang, Fu, Zhu, Zhao and Zhang36]}. One is for collisionless case (foil targets)—high-velocity, low-density counter-streaming flows—and the other one is collisional case (K-shaped targets)—low-velocity, high-density counter-streaming jets. The neutrons of 2.45 MeV are produced by deuterium–deuterium (DD) nuclear reactions ( $\text{D}+\text{D}\rightarrow 3\text{He}+n(2.45~\text{MeV})$ ). The typical signal of the 2.45 MeV neutrons is shown in Figure 6. Since the nuclear cross section changes sharply at the limits of 2 MeV of incident ions, it can be regarded as similar for both cases at the typically laser-produced flow velocity region ( $10^{7}{-}10^{8}~\text{cm}\cdot \text{s}^{-1}$ ). However, the density difference is very large. From the optical measurement, we can find that the flows are with density of $10^{19}~\text{cm}^{-3}$ , while the jets are with density larger than $10^{21}~\text{cm}^{-3}$ (roughly $n_{jet}\sim 100n_{\text{flow}}$ ). When we compare both cases, we can find that the neutron yields in collisional case ( $10^{6}$ )^{[Reference Zhao, Chen, Li, Fu, Rhee, Li, Zhu, Li, Liao, Zhang, Han, Liu, Huang, Ma, Li, Xiong, Huang, Fu, Zhu, Zhao and Zhang36]} is only 5–10 times as much as that in collisionless case ( $(1{-}5)\times 10^{5}$ )^{[Reference Zhang, Zhao, Yuan, Fu, Bao, Chen, He, Hou, Li, Li, Li, Liao, Rhee, Sun, Xu, Zhao, Zhu, Zhu, Zhang and Zhang37]}. One possible explanation is that the ions scattered by the self-generated electromagnetic field interaction with the ions from intra-flows will effectively enhance the neutron yields. This laser-produced CPF provides us not only a new tool to distinguish Coulomb collisions from the collisionless effects, i.e., random electromagnetic fields originated from instabilities, but also a platform for studying the nuclear astrophysics, such as the abundance of the $^{26}$ Al^{[Reference Prantzos and Diehl38]} and $^{6,7}$ Li^{[Reference Cyburt, Fields, Olive and Yeh39]}.

Figure 6. The observed neutron signals from D–D nuclear reaction in collisional case^{[Reference Zhao, Chen, Li, Fu, Rhee, Li, Zhu, Li, Liao, Zhang, Han, Liu, Huang, Ma, Li, Xiong, Huang, Fu, Zhu, Zhao and Zhang36, Reference Zhao, Zhang, Yuan, Li, Li, Rhee, Zhang, Li, Zhu, Li, Han, Liu, Ma, Li, Tao, Li, Guo, Huang, Fu, Zhu, Zhao, Chen, Fu and Zhang40]}.