2 Model and simulation

First, we review the traditional RPA process in a one-dimensional simulation to help us to design the target in the single reflection mechanism (SRM). A circularly polarized laser arrives at the target at $t=20T$ (see Figure 1(a)), where $T=\unicode[STIX]{x1D706}/c$ and $\unicode[STIX]{x1D706}=1~\unicode[STIX]{x03BC}\text{m}$ is the laser wavelength. $c$ is the speed of light in vacuum. In the simulation, the laser amplitude has a triangular profile in time (linear up-ramp $t_{\text{up}}=2.2T$ and linear down-ramp $t_{\text{down}}=2.2T$ ). The pressure of the circularly polarized laser stably pushes electrons forward, such that the electrons are piled up at the front of the laser, forming a compressed electron layer. Previous simulations have demonstrated that the velocity of such a compressed electron layer is uniform because the laser has a linear front^{[Reference Wang, Shen, Zhang, Ji, Wen, Xu, Yu, Li and Xu24, Reference Wang, Zhang, Wang, Zhao, Xu, Yu, Yi, Shi, Zhang, Xu, Liu, Pei and Shen41]}. In this case, the velocity of such an electron layer $(v_{\text{CEL}})$ can be calculated from the balance between the laser pressure force $(2a^{2}(1-v_{\text{CEL}})/(1+v_{\text{CEL}}))$ and electrostatic force $(2(\unicode[STIX]{x1D70B}n_{0}v_{\text{CEL}}t)^{2})$ . In the following calculations, the length, time, velocity, density, charge and field are normalized by $\unicode[STIX]{x1D706},\unicode[STIX]{x1D706}/c,c,\unicode[STIX]{x1D714}_{\text{L}}^{2}m_{\text{e}}/4\unicode[STIX]{x1D70B}\text{e}^{2},-e$ and $e/m_{\text{e}}\unicode[STIX]{x1D714}_{\text{L}}c$ , respectively, for the theoretical compact calculation:

(1) $$\begin{eqnarray}2a^{2}(1-v_{\text{CEL}})/(1+v_{\text{CEL}})=2(\unicode[STIX]{x1D70B}n_{0}v_{\text{CEL}}t)^{2},\end{eqnarray}$$

(2) $$\begin{eqnarray}\displaystyle v_{\text{CEL}} & = & \displaystyle \frac{1-3\unicode[STIX]{x1D705}^{2}}{3+3\unicode[STIX]{x1D705}^{2}}+\sqrt[3]{M+\sqrt{M^{2}+N^{3}}}\nonumber\\ \displaystyle & & \displaystyle +\,\sqrt[3]{M-\sqrt{M^{2}+N^{3}}},\end{eqnarray}$$

where $M=-(27\unicode[STIX]{x1D705}^{4}-36\unicode[STIX]{x1D705}^{2}+1)/[27(1+\unicode[STIX]{x1D705}^{2})^{3}],N=(15\unicode[STIX]{x1D705}^{2}-1)/[9(1+\unicode[STIX]{x1D705}^{2})^{2}]$ and $\unicode[STIX]{x1D705}=a_{0}/\unicode[STIX]{x1D70B}t_{\text{up}}n_{0}$ . Here $a_{0}=eE_{\text{L}}/m_{\text{e}}\unicode[STIX]{x1D714}_{\text{L}}c\approx 94$ is the dimensionless peak amplitude of laser, $E_{\text{L}}$ is the amplitude of the laser electric field, $\unicode[STIX]{x1D714}_{\text{L}}$ is the laser frequency, and $m_{\text{e}}$ and $e$ are the electron rest mass and charge, respectively. $a=(a_{0}/t_{\text{up}})(t-v_{\text{CEL}}t)$ is the normalized laser amplitude at the electron layer. The foil density is $n_{0}=50n_{\text{c}}$ and the foil thickness is $d=0.5~\unicode[STIX]{x03BC}\text{m}$ . Then $v_{\text{CEL}}\sim 0.183c$ can be obtained according to Equation (2). Here an idealized triangular laser profile is used in the present scheme, mainly for simplicity of the calculation and simulation. For the case of a Gaussian temporal profile, the velocity of the compressed electron layer is no longer uniform, and the calculation becomes complex, but can be solved by the time-dependent model used in our previous work^{[Reference Wang, Shen, Zhang, Ji, Yu, Yi, Wang and Xu42]}.

Figure 1. Electric field $E_{x}$ (blue solid line) and $E_{y}$ (red dash line), electron density (black dash–dot line), proton density (cyan dot line) at (a) $t=20T$ , (b) $t=22.5T$ and (c) $t=25T$ . (d) Trajectories of electrons (red solid line) and protons (gray solid line) in the simulations. (e) Phase space distributions of protons at $t=22.7T$ (red circles). The black solid line represents the velocity distribution at the end of the HB stage for protons initially at different positions of the foil ( $v_{\text{end}}$ versus $x_{\text{initial}}$ ).

Initially, the protons lag behind the compressed electron layer because the proton mass $m_{\text{i}}=1836m_{\text{e}}$ is much greater than the electron mass. At the end of HB stage, the fastest protons reach the compressed electron layer (see Figures 1(b) and 1(d))^{[Reference Wang, Shen, Zhang, Ji, Wen, Xu, Yu, Li and Xu24]} and the LS stage starts. It should be noted that the protons initially in the target center are accelerated faster than the others at the end of the HB stage in this case^{[Reference Wang, Shen, Zhang, Ji, Wen, Xu, Yu, Li and Xu24]}. The velocities of these protons are mainly distributed around $0.4c$ from $x_{\text{initial}}=20.2~\unicode[STIX]{x03BC}\text{m}$ to $20.25~\unicode[STIX]{x03BC}\text{m}$ (see Figure 1(e)). It is also noted that a narrow energetic spectrum may be obtained if we selectively accelerate the protons in the middle layer. From Figure 1(e), it can also be seen that the spectrum of the middle proton layer becomes broadened with an increased thickness of the middle proton layer, so a thinner middle proton layer will be better.

Then, the multilayer target is designed to ensure that the middle proton layer can be selectively accelerated. According to Figure 1(e), a proton layer from $x=20.2~\unicode[STIX]{x03BC}\text{m}$ to $x=20.25~\unicode[STIX]{x03BC}\text{m}$ is set between the carbon ( $\text{C}^{6+}$ ) layers (corresponding regions are $20~\unicode[STIX]{x03BC}\text{m}<x<20.2~\unicode[STIX]{x03BC}\text{m}$ and $20.25~\unicode[STIX]{x03BC}\text{m}<x<20.5~\unicode[STIX]{x03BC}\text{m}$ ). The electron density is $n_{0}=50n_{\text{c}}$ and the densities of the proton and $\text{C}^{6+}$ layers are $n_{\text{p}}=50n_{\text{c}}$ and $n_{\text{C}6+}=8.3n_{\text{c}}$ , respectively. Here, the charge-to-mass ratio of $\text{C}^{6+}$ is $1/2$ , which is half of the value for the proton (that is, 1), meaning that protons can be more easily accelerated compared with $\text{C}^{6+}$ ions. So, the $\text{C}^{6+}$ ions in the rear part of the target can be assumed to be at rest when the proton layer arrives. Different from the cases depicted in Figure 1, the trajectories of the protons initially at the middle of the target will cross trajectories of the $\text{C}^{6+}$ ions initially at the rear of the target. Hence, the protons can be accelerated by the charge separation field $E_{x}=E_{0}x(t)/d$ , where $E_{0}$ is the maximum charge separation field, given by $E_{0}=4\unicode[STIX]{x1D70B}en_{0}d$ . The proton velocity can be calculated using the following equation:

(3) $$\begin{eqnarray}\displaystyle v_{\text{p}}\left(t+\text{d}t\right)=\frac{4\unicode[STIX]{x1D70B}^{2}n_{0}x\left(t\right)}{m_{\text{p}}\unicode[STIX]{x1D6FE}_{\text{p}}\left(t\right)}\,\text{d}t+v_{\text{p}}\left(t\right), & & \displaystyle\end{eqnarray}$$

where $m_{\text{p}}=1836$ is the proton mass and $\unicode[STIX]{x1D6FE}_{\text{p}}(t)=1\big/\sqrt{1-v_{\text{p}}^{2}(t)}$ is the relativistic factor for the protons. In addition, the position of the protons can be calculated using the following equation:

(4) $$\begin{eqnarray}\displaystyle x_{\text{p}}\left(t+\text{d}t\right)=\frac{2\unicode[STIX]{x1D70B}^{2}n_{0}x\left(t\right)}{m_{\text{p}}\unicode[STIX]{x1D6FE}_{\text{p}}\left(t\right)}\left(\text{d}t\right)^{2}+v_{\text{p}}\left(t\right)\text{d}t+x_{\text{p}}\left(t\right).\quad & & \displaystyle\end{eqnarray}$$

The dynamics of the middle proton layer can be obtained from Equations (3) and (4) for $v_{\text{p}}(t=20T)=0$ and $x_{\text{initial}}=20~\unicode[STIX]{x03BC}\text{m}+0.5d$ . It should be noted that all these equations are related to the electron density $n_{0}$ , and the dynamics of the different ions are determined by their charge-to-mass ratios. The accelerating scheme will be totally different if their densities change.

Based on Equations (3) and (4), the middle proton layer arrives at the back surface of the target $(x_{\text{initial}}=20.5~\unicode[STIX]{x03BC}\text{m})$ at $t\sim 22.6T$ . To start the LS stage as soon as possible, the compressed electron layer should also arrive at the back of the target simultaneously with the middle proton layer, forming a double layer. So, the velocity of the compressed electron layer should be $v_{\text{CEL}}=d/2.6T\sim 0.2c$ . For the linearly rising-up laser front, the velocity $v_{\text{CEL}}$ during the HB stage can be calculated according to Equations (1) and (2). Considering $a=(a_{0}/t_{\text{up}})(1-v_{\text{CEL}})t$ , the steepness of the laser front can be obtained as

(5) $$\begin{eqnarray}\displaystyle a_{0}/t_{\text{up}}=\unicode[STIX]{x1D70B}n_{0}v_{\text{CEL}}(1+v_{\text{CEL}})^{1/2}(1-v_{\text{CEL}})^{-3/2}. & & \displaystyle\end{eqnarray}$$

Then, $a_{0}/t_{\text{up}}\sim 48$ is obtained for $n_{0}=50n_{\text{c}}$ and $v_{\text{CEL}}\sim 0.2c$ , as shown in Figure 2(a). The peak amplitude $a_{0}$ of the laser can be calculated as ${\sim}96$ , according to $a_{0}=(a_{0}/t_{\text{up}})(1-v_{\text{CEL}})d/v_{\text{CEL}}$ and $t_{\text{up}}\sim 2T$ . Here, the velocity of the proton layer can increase up to $v_{\text{p}}\sim 0.4c$ , which is almost twice the velocity $v_{\text{CEL}}\;({\sim}0.2c)$ . This indicates that the proton layer moves faster than the compressed layer. At the same time, the remaining electrons will move together with the other $\text{C}^{6+}$ ions, because the laser intensity begins to decrease after $t\sim 22.6T$ . The velocity of the $\text{C}^{6+}$ –electron double layer is lower than the velocity of the proton–electron double layer. So, the rear part of the laser can be reflected by the $\text{C}^{6+}$ –electron double layer. Only the carbon ions are heated and spread extensively, reducing the multidimensional instabilities of the proton layer to a certain extent. Ultimately, the high-quality proton bunch can be selectively accelerated to $E_{\text{p}}\sim 100~\text{MeV}$ at the end of the HB stage (see Figure 2(b)). It is believed that the single reflection mechanism can be realized only during the HB stage according to Equations (3)–(5). It should be noted that Equations (3)–(5) cannot be applied for the special case $t_{\text{up}}=0T$ , which should be specially solved by the time-dependent models^{[Reference Wang, Shen, Zhang, Ji, Yu, Yi, Wang and Xu42]}.

Figure 2. (a) Relation between the velocity of the compressed electron layer $v_{\text{CEL}}$ and the steepness of the laser front $a_{0}/t_{\text{up}}$ according to Equation (5) for $n_{0}=50n_{\text{c}}$ . (b) Evolutions of the velocity, $v_{\text{p}}$ (black solid line), and energy, $E_{\text{p}}$ , of the proton layer during the HB stage.

Two-dimensional particle-in-cell simulations are carried out to verify the theoretical expectations of the single reflection mechanism. A multilayer target is designed as shown in Figure 3. The hydrogen layer lies in the middle $(20.2~\unicode[STIX]{x03BC}\text{m}\leqslant x\leqslant 20.25~\unicode[STIX]{x03BC}\text{m})$ of the foil. The carbon layer lies in the regions $20~\unicode[STIX]{x03BC}\text{m}<x<20.2~\unicode[STIX]{x03BC}\text{m}$ and $20.25~\unicode[STIX]{x03BC}\text{m}<x<20.5~\unicode[STIX]{x03BC}\text{m}$ . The hydrogen and carbon atoms are assumed to be ionized to $\text{H}^{+}$ and $\text{C}^{6+}$ before the main pulse is incident on the target. The electron density is $n_{0}=50n_{\text{c}}$ . The densities of the $\text{H}^{+}$ and $\text{C}^{6+}$ layers are $n_{\text{p}}=50n_{\text{c}}$ and $n_{\text{C}6+}=8.3n_{\text{c}}$ , respectively. The laser amplitude has a triangular profile in time (linear up-ramp $t_{\text{up}}=2T$ and linear down-ramp $t_{\text{down}}=2T$ ) with a peak value $a_{0}=96$ . The pulse waist is $10~\unicode[STIX]{x03BC}\text{m}$ (FWHM). The simulation box size is 50 $~\unicode[STIX]{x03BC}\text{m}(x)\times 60~\unicode[STIX]{x03BC}\text{m}(y)$ , and the number of spatial grids is $8000\times 9600$ . Each is filled with 20 macroelectrons and 20 macroprotons (or $\text{C}^{6+}$ ions).

Figure 3. Distributions of (a)–(c) electric field $E_{y}$ , (d)–(f) electron density $n_{\text{e}}$ , (g)–(i) $\text{C}^{6+}$ density and (j)–(l) proton density at $t=21T$ (first row), $t=23T$ (second row) and $t=25T$ (third row).

Figure 3 depicts the detailed progress from the HB to the LS stage as the CP laser irradiates the multilayer target. Initially, the electrons are stably pushed forward because there are no oscillating terms in the expression for the ponderomotive force for CP lasers^{[Reference Robinson, Zepf, Kar, Evans and Bellei19]}, as shown in Figure 3(d). The $\text{C}^{6+}$ ions lag behind the compressed electron layer initially. The charge separation field, $E_{x}$ , then becomes stronger with the increased distance between the electrons and the ions. Both $\text{C}^{6+}$ ions and protons begin to be accelerated by $E_{x}$ . At $t=23T$ , the compressed electron layer has reached the back of the target together with the proton layer, which is almost consistent with the theoretical assumption $(t\sim 22.6T)$ according to Equation (4) (refer to Figures 3(e) and 3(k)). Based on Equation (1), the velocity of the proton layer ( $v_{\text{p}}\sim 0.4c$ ) is much higher than the velocity of the compressed electron layer $(v_{\text{CEL}}=\sim 0.2c)$ . This means the proton layer is accelerated and separated from the compressed electron layer in the LS stage. In addition, the peak decreases after $t=22.6T$ . To maintain the balance between the laser pressure and charge separation forces again, the charge separation field, $E_{x}$ , weakens. This is realized by the $\text{C}^{6+}$ ions reaching the compressed electron layer after $t=22.6T$ . Finally, two double layers are formed (see Figures 3(f)–3(l)). The $\text{C}^{6+}$ –electron layer is heated and spreads extensively in space until the laser is totally reflected at $t=25T$ (see Figure 3(c)). In contrast, the proton–electron layer always moves ahead of the $\text{C}^{6+}$ –electron layer, maintaining a compact high-quality bunch, which verifies the theoretical model (Equations (3)–(5)).

Figure 4. (a) Trajectories of the $\text{C}^{6+}$ layer (black square), the proton layer (blue triangle) and the interface between the laser and the compressed electron layer (red circle). Enlarged plots of the trajectories are shown in (b). (c) Phase space distributions of $\text{C}^{6+}$ ions and protons at $t=25T$ . (d) Energetic spectra for the proton layer in different initial regions at $t=25T$ . Here, protons in the region $-3~\unicode[STIX]{x03BC}\text{m}<y<3~\unicode[STIX]{x03BC}\text{m}$ are considered.

It should be noted that the proton layer has a greatly reduced probability of moving together with the laser–plasma interface in this case, as can clearly be seen from Figures 4(a) and 4(b). Initially, the laser interface moves forward and overtakes the proton layer at $t\sim 21.5T$ . At the end of the HB stage $(t\sim 22.5T)$ , the proton layer overtakes the interface. After $t\sim 22.5T$ , the proton layer continues to move ahead of both the interface and $\text{C}^{6+}$ ions with a constant velocity, because the laser pulse is reflected by the $\text{C}^{6+}$ –electron double layer. As depicted in Figure 4(a), the laser pulse is completely reflected away from the $\text{C}^{6+}$ layer, and does not affect the proton layer. Here, the proton layer is just ‘slingshot-likely’ reflected once by the charge separation field only during the HB stage (refer to Figure 4(c)). Thus, instabilities of the laser–plasma interface are intrinsically reduced. A pure proton beam with a quasimonoenergetic spectrum $({\sim}5\,\%)$ centered at ${\sim}100~\text{MeV}$ is finally generated at $t=25T$ (see Figure 4(d)). From Figures 4(a) and 4(b), it can be found that the proton layer is separated from the carbon layers, and the carbon layer blocks the rear parts of the laser after $t\sim 23T$ , so the spectrum of the proton beam does not change much after $t\sim 23T$ .

The accelerations of the proton layers in the different regions are compared to verify the optimum conditions for the single reflection mechanism, shown in Figure 4(c). The center energy of the proton layer $(20.4~\unicode[STIX]{x03BC}\text{m}<x_{\text{initial}}<20.45~\unicode[STIX]{x03BC}\text{m})$ is only $E_{\text{p}}=39~\text{MeV}$ . The main reason is that the proton layer, initially at rear part of the foil, is accelerated for a shorter time during the HB stage. In contrast, the proton layer initially at the front can be accelerated for a longer time to a higher maximum energy (137 MeV). However, the spectrum spread $({\sim}35\,\%)$ is much worse than the optimum case $({\sim}5\,\%)$ . On the one hand, this is caused by the disturbed charge separation field when the falling part of the laser pulse is reflected. On the other hand, the spectrum spread is intrinsically best for the proton layer in the middle layer, as shown in Figure 1(e).