2 Laser and target parameters

We carried out 3D simulations with the PIC code EPOCH ^[ Reference Arber, Bennett, Brady, Lawrence-Douglas, Ramsay, Sircombe, Gillies, Evans, Schmitz, Bell and Ridgers ^{28 ]} to investigate the interaction of two counter-propagating relativistic laser pulses incident on a subwavelength thin foil and the resulting foil evolution. The simulation box size is $x\times y\times z=20{\lambda}_0\times 20{\lambda}_0\times 20{\lambda}_0$ , with $800\times 400\times 400$ cells, where ${\lambda}_0=1\ \unicode{x3bc} \mathrm{m}$ is the laser wavelength. Each cell contains 27 electrons and 27 ions. Open boundary conditions are used for both the fields and particles. Thanks to rapid ionization by the laser prepulse, the thin foil is modelled by a fully ionized cold hydrogen plasma layer located at $-0.05{\lambda}_0<x<0.05{\lambda}_0$ and $-10{\lambda}_0<y<10{\lambda}_0$ . The plasma density is ${n}_\mathrm{e}={n}_\mathrm{p}=50{n}_\mathrm{c}$ , where $n_{\mathrm{c}}=m_{\mathrm{e}}\omega_0^2\varepsilon_0/e^2$ is the critical density, ${\omega}_0$ is the laser frequency, ${\varepsilon}_0$ is the permittivity in vacuum and $-e$ and ${m}_\mathrm{e}$ are the electron charge and rest mass, respectively. The two identical laser pulses propagate along the $\pm x$ -directions and are focused at the centre of the $0.1{\lambda}_0$ -thin foil. The space–time profile of the lasers is $a_0\ \mathrm{exp}[-(r/r_0)^2]\, \mathrm{sin}^2(\pi t/2\tau_0)$ , where ${r}_0=5{\lambda}_0$ is the focal spot radius, $\tau_0 = 10{T}_0$ is the pulse duration, ${T}_0=3.3$ fs is the laser period and $a_0 = eE_0/m_{\mathrm{e}} \omega_0 c = 20$ is the normalized laser amplitude. Here, ${E}_0$ is the peak laser electric field and $c$ is the speed of light in vacuum. That is, the intensity, power and energy of each laser are $I_0 = 5.5 \times 10^{20}$ W/cm², 215 TW and 5.4 J, respectively.

We shall consider linearly polarized (LP) and circularly polarized (CP) laser pairs. As some laser pairings are physically identical in the interactions, we shall concentrate only on the pairs yLP + yLP, yLP + zLP, LCP + LCP and LCP + RCP for phase differences $\Delta \varphi =0,\pi /2,\pi$ and $3\pi /2$ between the paired lasers, that is, there are 16 distinct cases. For clarity, here yLP and zLP, and LCP and RCP denote $y$ - and $z$ -direction LP lasers, and the left- and right-handed CP lasers, respectively, and the $+$ sign denotes pairing. Moreover, unless otherwise stated, foil- or ion-density distribution shall refer to that on the foil plane.

3 Dynamics of two counter-propagating lasers interacting with ultrathin foil

The left-hand panel of Figure 1 shows the evolution of the side ( $x,y$ ) and foil-plane ( $z,y$ ) distributions of the foil, or ion, density for yLP + yLP laser pair interaction with the foil for different $\Delta \varphi$ values. One can see that the foil is first locally compressed by the focusing lasers. Then, for $\Delta \varphi =0$ and $\pi$ (Figures 1(a) and 1(c)), the compressed foil centre region begins to expand to the left and right symmetrically. The expansion for $\Delta \varphi =\pi$ is weaker since the lasers are out of phase, so that cancellation of their fields occurs. For $\Delta \varphi =\pi /2$ ( $3\pi /2$ ), the laser pulse from the left (right) has a quarter-period phase lead (lag), so that the affected plasma is pushed to the right (left) by the net light pressure^[ Reference Shen and Meyer-ter-Vehn ^{21 ]}. Of interest is that for $\Delta \varphi =0$ , filament-like streaks along $y$ appear in the foil-density distribution, but for $\Delta \varphi =\pi /2,\ \pi$ and $3\pi /2$ the density is azimuthally symmetric. The right-hand panel for the yLP + zLP pair shows that the foil density is always left–right symmetric. For $\Delta \varphi =0$ and $\pi$ , streaks along $x$ (such as that marked by the red box in the Figure 1(g2)) appear. The transverse features for $\Delta \varphi =0$ and $\pi$ are similar, except that the streaks are oriented along the $y=z$ direction for $\Delta \varphi =0$ and along the $y=-z$ direction for $\Delta \varphi =\pi$ . In both cases, the distribution of the high-density region is identical. This is because the laser intensity is lower outside the laser focal spot and the foil is opaque and will be compressed by the laser ponderomotive force, rather than being modulated by the transmitted overlap laser field in the central region of the foil. For $\Delta \varphi =\pi /2$ and $3\pi /2$ , Figures 1(f4), 1(f5), 1(h4) and 1(h5) show that the density distribution has an annular ring and grainy bubble structure, which has also been observed in light-sail experiments^[ Reference Palmer, Schreiber, Nagel, Dover, Bellei, Beg, Bott, Clarke, Dangor, Hassan, Hilz, Jung, Kneip, Mangles, Lancaster, Rehman, Robinson, Spindloe, Szerypo, Tatarakis, Yeung, Zepf and Najmudin ^{11 ]} and simulations^[ Reference Wan, Andriyash, Lu, Mori and Malka ^{17 ]}.

Figure 1 Evolution of the foil density for different polarizations of the laser pair: (a)–(d) yLP and yLP and (e)–(h) yLP and zLP, with (a) and (e) $\Delta \varphi =0$ , (b) and (f) $\pi /2$ , (c) and (g) $\pi$ , and (d) and (h) $3\pi /2$ . Here, yLP and zLP denote linear polarization in the $y$ - and $z$ -directions, respectively, and $\Delta \varphi$ is the phase difference between the two lasers. The first three columns of both the yLP (left-hand panel) and zLP (right-hand panel) cases show the axial (with respect to the lasers) foil-density distribution in the $z=0$ plane at $t=16{T}_0$ , $22{T}_0$ and $28{T}_0$ , respectively. The fourth and fifth columns in both the left- and right-hand panels show the transverse density distributions at the axial locations defined by the vertical dashed red lines in the second and third columns (for $t=22{T}_0$ and $28{T}_0$ ), respectively. In all panels, the red lines/curves with arrows show the amplitude and displacement direction of the analytically obtained resultant electric field (shown in Table 1) of the two colliding lasers at $x=0$ , where the subscript ‘r’ here denotes ‘resultant’. The large red centre dot in window (c5) corresponds to $E_\mathrm{r}=0$ , that is, the fields of the two lasers cancelled each other out.

Next we consider counter-propagating CP laser pairs LCP + LCP and LCP + RCP. Figure 2 shows that the foil expands more slowly than that of the yLP + yLP and yLP + zLP pairs, which is expected since the pressure of CP light has only a non-oscillating component. For the LCP + LCP pair, we see in the left-hand panel that the side-view of the foil density is left–right symmetric for all values, and Figures 2(b2), 2(c2) and 2(d2) have similar features to those in Figures 1(e2) and 1(g2). Streaks similar to those in Figures 1(e5) and 1(g5) also appear, and their directions are also $\Delta \varphi$ dependent. The second column of the right-hand panel in Figure 2 shows that for the LCP + RCP pair the foil centre is more tightly compressed. For $\Delta \varphi =0$ and $\pi$ (first and third rows), the density remains symmetric, but it is more rapidly destroyed for $\Delta \varphi =0$ than that for $\Delta \varphi =\pi$ . For $\Delta \varphi =\pi /2$ and $3\pi /2$ (second and fourth rows), the foil centre is pushed to the right and left, respectively, as in the corresponding cases for LP laser pairs. Moreover, in all the cases here, the centre of the foil is rather heavily distorted. The fourth and fifth columns show that the foil density in the transverse direction has a grainy bubble centre and periodic less grainy annular rings outside. The above results show that the affected foil region can expand, be driven forward or backward and form streaks and grainy bubbles and rings. In the following, we shall investigate these features in more detail.

Figure 2 Evolution of the foil-density distribution for the polarization combinations (a)–(d) LCP + LCP and (e)–(h) LCP + RCP with (a) and (e) $\Delta \varphi =0$ , (b) and (f) $\pi /2$ , (c) and (g) $\pi$ , and (d) and (h) $3\pi /2$ . The first to third columns show the longitudinal foil-density distribution in the $z=0$ plane at $t=16{T}_0$ , $22{T}_0$ and $28{T}_0$ , respectively. The fourth and fifth columns show the transverse ion distribution in the planes indicated by the red dashed lines in the second and third columns at $t=22{T}_0$ and $28{T}_0$ , respectively. The red lines and arrows in the fifth column represent the magnitude and direction of the resultant radial electric field at $x=0$ . The dot in panel (g5) indicates ${E}_\mathrm{r}=0$ .

4 Model for the laser interaction in the foil

As the solid-density foil is only $0.1{\lambda}_0$ thin, it is effectively transparent to the incident laser pulses^[ Reference Vshivkov, Naumova, Pegoraro and Bulanov ^{29 ]}. We first assume that it is fully transparent with reflection ratio $R=0$ to obtain a simple analytical solution of the intensity distribution of the resultant laser field. Since we are interested in the interaction time of less than one laser period, the electric fields of the yLP + yLP laser pair can be represented by the plane electromagnetic (EM) waves:

(1) $$\begin{align}\left\{\!\!\begin{array}{l}{E}_{y,\mathrm{L}}={E}_0\cos \left({\omega}_0t-{k}_0x\right),\\ {}{E}_{y,\mathrm{R}}={E}_0\cos \left({\omega}_0t+{k}_0x+\Delta \varphi \right),\end{array}\right.\end{align}$$

where ${k}_0={\omega}_0/c$ is the wave number, $t=0$ is the time when the peaks of the lasers meet at the foil centre and the subscripts L and R denote being from the left and the right. For simplicity, hereafter the time and space quantities are normalized by ${\omega}_0^{-1}$ and ${k}_0^{-1}$ , respectively. The resultant electric field of the two yLP lasers is then as follows:

(2) $$\begin{align}{E}_y&={E}_{y,\mathrm{L}}+{E}_{y,\mathrm{R}}\nonumber\\ &=2{E}_0\cos \left(t+\Delta \varphi /2\right)\cos \left(x+\Delta \varphi /2\right).\end{align}$$

Since any CP EM wave propagating in the $x$ -direction can be represented by two LP waves polarized in the $y$ - and $z$ -directions, the CP laser pair LCP + LCP can be represented by the following:

(3) $$\begin{align}\left\{\!\!\begin{array}{l}{E}_{y,\mathrm{L}}={E}_0/\sqrt{2}\cos \left(t-x\right),\\ {}{E}_{z,\mathrm{L}}=-{E}_0/\sqrt{2}\sin \left(t-x\right),\\ {}{E}_{y,\mathrm{R}}={E}_0/\sqrt{2}\cos \left(t+x+\Delta \varphi \right),\\ {}{E}_{z,\mathrm{R}}={E}_0/\sqrt{2}\sin \left(t+x+\Delta \varphi \right).\end{array}\right.\end{align}$$

The resultant electric field for the LCP + LCP case is thus as follows:

(4) $$\begin{align}\left\{\begin{array}{@{}c}{E}_y=\sqrt{2}{E}_0\cos \left(t+\Delta \varphi /2\right)\cos \left(x+\Delta \varphi /2\right),\\ {}{E}_z=\sqrt{2}{E}_0\cos \left(t+\Delta \varphi /2\right)\sin \left(x+\Delta \varphi /2\right).\end{array}\right.\end{align}$$

Similarly, the resultant fields of the other pairs can be obtained.

The results for the four laser pairs of interest from the analysis above are summarized in Table 1, and the corresponding simulation results are already shown by the (if any) solid red lines, curves, arrows and dots in the fifth columns of both the left- and right-hand panels of Figures 1 and 2. For example, if the resultant field is LP, the foil-density distribution has a streak pattern, whose direction is the same as that of the polarization from the theory. For the yLP + yLP case with $\Delta \varphi =\pi /2$ ( $3\pi /2$ ), the streaks disappear thanks to the right-hand (left-hand) foil displacement, and for $\Delta \varphi =\pi$ , the density distribution is azimuthally symmetric (since ${{E}_\mathrm{r}=0}$ ). When the resultant field is CP, the density is annularly symmetric, also in agreement with the simple theory (as well as from experiments^[ Reference Palmer, Schreiber, Nagel, Dover, Bellei, Beg, Bott, Clarke, Dangor, Hassan, Hilz, Jung, Kneip, Mangles, Lancaster, Rehman, Robinson, Spindloe, Szerypo, Tatarakis, Yeung, Zepf and Najmudin ^{11 ]}, as mentioned above).

Next we consider the origins of the different foil-density profiles. From the second and fifth columns in Figures 1 and 2, one can see that the filamentous streaks structure (marked by the red boxes) along the $x$ -direction exists only if the resultant laser field is LP and polarized in the $z$ -direction. The oscillating component of the light pressure force of the resultant LP laser light drives directed periodic perturbations that lead to density striation. Strictly speaking, the foil centre can only be smoothly pushed to the left or right by the net light pressure of two laser pulses. The steady-state model of Shen and Meyer-ter-Vehn^[ Reference Shen and Meyer-ter-Vehn ^{21 ]} shows that the foil motion is determined by the net light pressure $\Delta P={P}_{\mathrm{left}}-{P}_{\mathrm{right}}$ on the foil that is related to $\Delta \varphi$ , where ${P}_{\mathrm{left}}$ and ${P}_{\mathrm{right}}$ are the light pressure on the left- and right-hand surfaces of the foil, respectively. However, this model is limited to continuously incident CP lasers and 1D geometry, and there are only stationary analytical solutions for two lasers with the same rotation direction. In addition, transverse instabilities are excluded. Here, we present a more general analysis of our results. Note that even though the foil is relativistically transparent, it cannot be ignored, that is, the reflection coefficient $R>0$ and the transmission coefficient $T<1$ . Since the resultant electric field is nonuniform longitudinally, there is a pressure difference $\Delta P$ . For the case yLP + yLP, the normalized intensity of the $y$ -component ${E}_y^2/{E}_0^2$ of the resultant laser light along the $x$ -direction at $t=0$ is given in Figure 3(a) for different $\Delta \varphi$ values. The direction of $\Delta P$ is determined by the slope $k{\left|{}_{x=0}={\partial}_x{E}_y^2\right|}_{x=0}$ , where $k$ also represents the difference in the momentum flow of the EM field between the left- and right-hand laser pulses since the EM momentum $\left|\overrightarrow{g}\right|$ scales as ${E}_y^2$ ( $\left|\overrightarrow{g}\right|=\left|\overrightarrow{E}\times \overrightarrow{B}\right|/{\mu}_0={E}^2/2{\mu}_0c$ , where ${\mu}_0$ is the permeability in vacuum). We see that for $\Delta \varphi =0$ and $\pi$ , ${\left.k\right|}_{x=0}=0$ and thus $\Delta P=0$ , resulting in the longitudinally symmetrical compression of the foil, as shown in Figures 1(a) and 1(c). However, when $\Delta \varphi =\pi /2$ and $3\pi /2$ , $k<0$ and $k>0$ , corresponding to the cases of $\Delta P>0$ and $\Delta P<0$ , respectively. The foil will be pushed to the right and left, respectively, which is in good agreement

Figure 3 Results of the analytical model and PIC simulations. (a) Normalized intensity of the $y$ component ${E}_y^2/{E}_0^2$ of the resultant laser field for the yLP + yLP case at $t=0$ . The shaded region represents the foil and the dashed lines represent the slopes of ${E}_y^2$ at $x=0$ . (b) The impulse ${I}_\mathrm{f}$ of the axial pressure force ${f}_\mathrm{p}$ (blue curve) of the resultant laser light in one laser cycle and the displacement ${x}_\mathrm{d}$ (orange squares) of the foil centre versus $\Delta \varphi$ for the yLP + yLP case.

Table 1 The resultant electric field of two counter-propagating laser pulses.

with Figures 1(b) and 1(d). In fact, $k={\partial}_x{E}_y^2$ is consistent with the axial ponderomotive force ${f}_\mathrm{p}=-\left({q}^2/4m{\omega}_0^2\right){\partial}_x{E}^2$ of the laser. Since the axial displacement of the foil is the result of continuous action of ${f}_\mathrm{p}$ , the impulse after integration over a laser period

(5) $$\begin{align}{I}_\mathrm{f}={\int}_0^{T_0}{f}_\mathrm{p}\mathrm{d}t=-\frac{q^2}{4m{\omega}_0^2}{\int}_0^{T_0}\frac{\partial {E}^2}{\partial x}\mathrm{d}t\\[-20pt]\nonumber\end{align}$$

should be of interest. Figure 3(b) shows ${I}_\mathrm{f}$ versus the maximum foil displacement ${x}_\mathrm{d}$ at $t=22{T}_0$ from the PIC simulations of the yLP + yLP case. We see that these relations are consistent with the discussions above; in particular, the axial displacement of the foil centre depends mainly on the impulse exerted by the resultant laser light on the foil.

Figure 4 shows the evolution of ${f}_\mathrm{p}$ and the relation between ${f}_\mathrm{p}$ and $\Delta \varphi$ for different polarization combinations. Here, ${f}_{\mathrm{p}y}=-\left({q}^2/4m{\omega}_0^2\right){\partial}_x{E}_y^2$ and ${f}_{\mathrm{p}z}=-\left({q}^2/4m{\omega}_0^2\right){\partial}_x{E}_z^2$ are also given. For the yLP + yLP and yLP + zLP cases, Figures 4(a) and 4(b) show that ${f}_\mathrm{p}$ oscillates periodically with time. The difference between them is that for the yLP + yLP pair, ${f}_\mathrm{p}$ is either greater than or less than 0 (also seen in Figure 4(e)), so the foil may be pushed right or left. For the yLP + zLP case, ${f}_\mathrm{p}$ can be positive or negative, but its integral over one laser period is equal to 0 (i.e., ${I}_\mathrm{f}=0$ ), as shown in Figure 4(b). This results in a longitudinally symmetric distribution of foil density at different $\Delta \varphi$ values. When the two lasers are CP, the superposition of ${f}_{\mathrm{p}y}$ and ${f}_{\mathrm{p}z}$ (i.e., ${f}_\mathrm{p}$ ) is constant for each $\Delta \varphi$ , as seen in Figures 4(c) and 4(d). In the LCP + LCP case, ${f}_\mathrm{p}\equiv 0$ (seen in Figures 4(c) and 4(g)) and the foil density is symmetrically distributed. Since there is no oscillating term in ${f}_\mathrm{p}$ , the expansion is slower than that due to the LP laser pair. For the LCP + RCP case, the relationship between ${f}_\mathrm{p}$ and $\Delta \varphi$ is fixed, as shown in Figure 4(h). When $\Delta \varphi$ is not an integral multiple of $\pi$ , such as $\pi /2$ and $3\pi /2$ , ${f}_\mathrm{p}\ne 0$ and the foil is pushed longitudinally to one side, as shown in Figures 2(f) and 2(h). It should be mentioned that despite the simplicity of the above model, for the LCP + RCP polarization combination, both the relations between ${f}_\mathrm{p}$ (or ${I}_\mathrm{f}$ ) and $\Delta \varphi$ , $t$ and the longitudinal motion of the thin foil are consistent with the 1D steady-state model of Shen and Meyer-ter-Vehn^[ Reference Shen and Meyer-ter-Vehn ^{21 ]}. Moreover, the model is also suitable for other polarization combinations.

Figure 4 (a)–(d) Evolution of the light pressure force ${f}_\mathrm{p}$ in the axial ( $x$ ) direction (solid curves) and (e)–(h) ${f}_\mathrm{p}$ versus $\Delta \varphi$ for different polarization combinations. In all cases, the transverse force components ${f}_{\mathrm{p}y}$ (dashed curves) and ${f}_{\mathrm{p}z}$ (dotted curves) are also given.

5 Transverse instability in foil driven by two counter-propagating laser pulses

We now consider the development of the transverse instability in the foil. The intensity of the 2D Fourier spectrum of the averaged transverse foil density may be used to track the evolution of the instability^[ Reference Wan, Andriyash, Lu, Mori and Malka ^{17 ]}. As examples, we first consider the LCP + LCP case for $\Delta \varphi =\pi /2$ and the LCP + RCP case for $\Delta \varphi =0$ , since their resultant fields are LP and CP, respectively. Figures 5(a) and 5(c) show the transverse distributions of the averaged density in these cases, and their Fourier spectra are given in Figures 5(b) and 5(d), respectively. We see that for the LCP + LCP case with $\Delta \varphi =\pi /2$ , the dominant mode in the $k$ space is ${k}_y={k}_z$ , consistent with the result that the direction of the streaks is the same as that of the polarization of the resultant laser field. For the yLP + yLP and yLP + zLP cases, except for those in Figures 1(c), 1(f) and 1(h), we can draw the same conclusion since the resultant laser fields are LP. It should be noted that this analysis of the very initial evolution of the foil with respect to the incident lasers does not identify the instability leading to foil destruction. In fact, the foil response to the resultant laser field is similar to that driven by a single LP laser. On the other hand, if the resultant field is CP, foil destruction can result from the transverse instability^[ Reference Palmer, Schreiber, Nagel, Dover, Bellei, Beg, Bott, Clarke, Dangor, Hassan, Hilz, Jung, Kneip, Mangles, Lancaster, Rehman, Robinson, Spindloe, Szerypo, Tatarakis, Yeung, Zepf and Najmudin ^{11 ,} Reference Wan, Andriyash, Lu, Mori and Malka ^{17 ]}. As shown in the right-hand panel of Figure 2 for the LCP + RCP case, a grainy bubble structure similar to that observed in the RPA light-sail experiments appears. Figures 5(c) and 5(d) show that the foil density is strongly modulated and the unstable mode has annular spatial distribution. The amplitude of the ring at ${k}_\mathrm{r}={\left({k}_y^2+{k}_z^2\right)}^{1/2}$ (grey circles in Figure 5(d)) then gives the relative magnitude of mode $k$ , as shown in Figure 5(e). For the LCP + RCP case with $\Delta \varphi =0$ , the fastest growing mode and the maximum growth rate of the instability are ${k}_\mathrm{m}=2{k}_0$ and ${\gamma}_\mathrm{m}=0.98{T}_0^{-1}$ , respectively. Note that although bubble structures also appear in Figures 1(f) and 1(h), they are located on the periphery of the laser affected area, and the foil expansion is similar to that of the LP laser pairs. Thus, in the following we shall mainly consider the instability of the LCP + RCP case.

Figure 5 Averaged distribution of the foil density at $t=21{T}_0$ for the (a) LCP + LCP case with $\Delta \varphi =\pi /2$ and (c) LCP + RCP case with $\Delta \varphi =0$ . (b), (d) 2D Fourier transform of the density distribution in (a) and (c), respectively. (e) Evolution of transverse instability of the LCP + RCP case for $\Delta \varphi =0$ and different ${k}_\mathrm{r}$ values. The slope (dotted lines) of the fastest growing mode ${k}_\mathrm{m}$ shows the maximum growth rate ${\gamma}_\mathrm{m}$ .

Figure 6 shows the evolution of the fastest growing mode of the transverse instabilities in all the 16 cases. As shown in Figures 6(a)–6(d), the instability grows rapidly, then more slowly and the foil is eventually broken. The maximum growth rates ${\gamma}_\mathrm{m}$ are shown in Figure 6(e). For the yLP + yLP case, ${\gamma}_\mathrm{m}$ is positively correlated with ${E}_\mathrm{r}$ , as depicted by the lengths of the red arrows in the fifth column of the left-hand panel of Figure 1. For $\Delta \varphi =0$ and $\pi$ , ${\gamma}_\mathrm{m}$ is maximum and minimum, respectively. For $\Delta \varphi =0$ and $\pi$ of the yLP + zLP case, and for all $\Delta \varphi$ values of the LCP + LCP case (as shown in the fifth column of the right-hand panel of Figure 1 and the fifth column in the left-hand panel of Figure 2), ${\gamma}_\mathrm{m}$ remains roughly the same since ${E}_\mathrm{r}$ is similar in these cases. However, for the yLP + zLP case with $\Delta \varphi =\pi /2$ and $3\pi /2$ , ${\gamma}_\mathrm{m}$ is somewhat smaller and ${E}_\mathrm{r}$ is also smaller, as depicted by the arrow lengths and circle diameters in the fifth column of the right-hand panel in Figure 1. Similar behaviour can be found in the LCP + RCP case, as shown in the fifth column of the right-hand panel of Figure 2. In fact, the fifth columns of the right-hand panel of Figure 2 and the left-hand panel of Figure 1 for the LCP + RCP and yLP + yLP cases, respectively, indicate that the dependence of ${\gamma}_\mathrm{m}$ and ${E}_\mathrm{r}$ on $\Delta \varphi$ is similar. For all 16 cases, ${\gamma}_\mathrm{m}$ is smallest if the laser pair is out of phase, or $\Delta \varphi =\pi$ . As a result, the instability also develops slowest and the foil can thus be most tightly compressed.

Figure 6 (a)–(d) Evolution of the fastest growing mode of transverse instability for all 16 cases. Here, the maximum growth rate ${\gamma}_\mathrm{m}$ (i.e., the slope) is labelled in (a)–(d). To compare intuitively, ${\gamma}_\mathrm{m}$ is also shown by a histogram, as seen in (e) and (f).

The characteristics of the foil instability can be estimated from the 3D relativistic two-fluid theory of Wan $et\ al.$ ^[ Reference Wan, Andriyash, Lu, Mori and Malka ^{17 ]}. In the linear stage, the maximum growth rate is as follows:

(6) $$\begin{align}\left\{\begin{array}{@{}l}{\gamma}_{\mathrm{m},\mathrm{RT}}\simeq {\left(\sqrt{\kappa /2}{\alpha}_{\mathrm{in}}{\omega}_\mathrm{pe}/{v}_{\mathrm{osc}}\right)}^{1/2},\\\\[-7pt] {}{\gamma}_{\mathrm{m}, \mathrm{ei}}\simeq 2{\left({\omega}_\mathrm{pi}^2{\omega}_\mathrm{pe}\right)}^{1/3}{\left(\kappa {m}_\mathrm{e}/{m}_\mathrm{i}\right)}^{1/6},\end{array}\right.\\[-20pt]\nonumber\end{align}$$

where $\kappa=(2c^2-v_{\mathrm{osc}}^{2})/2\gamma_{\mathrm{e}}c^2$ , ${v}_{\mathrm{osc}}={eE}_0/{\gamma}_0{m}_\mathrm{e}{\omega}_0$ is the electron quiver velocity in the resultant field, ${\gamma}_\mathrm{e}={T}_\mathrm{e}/{m}_\mathrm{e}{c}^2+1$ is the Lorentz factor of the electrons, ${T}_\mathrm{e}$ is the electron temperature, ${\gamma}_0={\left(1+{a}_0^2/2\right)}^{1/2}$ for LP and $\gamma_0={\left(1+{a}_0^2\right)}^{1/2}$ for CP light, ${\alpha}_{\mathrm{in}}={P}_0/{m}_\mathrm{i}{n}_\mathrm{e}{d}_0$ is the axial ion speed, ${P}_0=\left(1+R-T\right){I}_0/c$ is the light pressure, ${m}_\mathrm{i}$ is the ion mass and ${\omega}_\mathrm{pe}={\left({e}^2{n}_\mathrm{e}/{\varepsilon}_0{m}_\mathrm{e}\right)}^{1/2}$ and ${\omega}_\mathrm{pi}={\left({e}^2{n}_\mathrm{i}/{\varepsilon}_0{m}_\mathrm{i}\right)}^{1/2}$ are the electron and ion plasma frequencies, respectively. According to the model of relativistic-induced transparency (RIT)^[ Reference Vshivkov, Naumova, Pegoraro and Bulanov ^{29 ]}, $R={\epsilon}_0^2/{a}_0^2$ and $T=1-{\epsilon}_0^2/{a}_0^2$ when ${a}_0>{\epsilon}_0>1$ , where ${\epsilon}_0={\omega}_\mathrm{pe}^2{d}_0/2{\omega}_0c$ characterizes the optical properties of the subwavelength foil. The wave number of the fastest growing mode of transverse instability is given by the following:

(7) $$\begin{align}{k}_\mathrm{m}=\sqrt{2\kappa }{\omega}_\mathrm{pe}/{v}_{\mathrm{osc}}.\\[-18pt]\nonumber\end{align}$$

The above analysis is for a single laser beam. To extend it to the interaction of a counter-propagating laser pair with a thin foil, it is necessary to replace ${P}_0$ with the combined pressure $\Delta P$ . Since the foil is nearly transparent to the incident light, the two lasers effectively merge into a resultant light bunch and the interaction of the two lasers with the foil can be approximated by that between the resultant light field and the thin foil. Since ${x}_\mathrm{d}<{\lambda}_0$ (see Figures 2(f3) and 2(h3)), ${E}_0$ , ${a}_0$ and ${I}_0$ can be replaced by those of the resultant laser field. Thus, ${k}_\mathrm{m}$ and ${\gamma}_\mathrm{m}$ for given laser and plasma parameters can be obtained.

Considering that the bubble structure in the LCP + RCP case is similar to that observed in the RPA experiments and simulations^[ Reference Palmer, Schreiber, Nagel, Dover, Bellei, Beg, Bott, Clarke, Dangor, Hassan, Hilz, Jung, Kneip, Mangles, Lancaster, Rehman, Robinson, Spindloe, Szerypo, Tatarakis, Yeung, Zepf and Najmudin ^{11 ,} Reference Wan, Andriyash, Lu, Mori and Malka ^{17 ]}, we use this case as an example for analysing the transverse instability in more detail. Figures 7(a)–7(c) show the electron temperature ${T}_\mathrm{e}$ and ${\gamma}_\mathrm{e}$ versus ${a}_0$ for $\Delta \varphi =0$ , $\pi /2$ and $\pi$ . The results for $\Delta \varphi =3\pi /2$ are exactly the same as those of $\Delta \varphi =\pi /2$ . We can see that ${T}_\mathrm{e}$ and ${\gamma}_\mathrm{e}$ grow linearly with ${a}_0$ for both $\Delta \varphi =0$ and $\pi /2$ . For the latter, they are higher since the foil experiences higher net light pressure. For $\Delta \varphi =\pi$ , both ${T}_\mathrm{e}$ and ${\gamma}_\mathrm{e}$ remain almost unchanged in the range $5<{a}_0<20$ , since ${f}_\mathrm{p}=0$ and the foil remains symmetrically compressed. With larger ${a}_0$ the foil becomes more relativistically transparent and its axial distortion becomes more pronounced, resulting in significantly larger ${T}_\mathrm{e}$ and ${\gamma}_\mathrm{e}$ . Figures 7(d)–7(f) show the dependence of ${k}_\mathrm{m}$ and ${\gamma}_\mathrm{m}$ on ${a}_0$ for different $\Delta \varphi$ values. Since for $\Delta \varphi =\pi$ , ${E}_\mathrm{r}$ should be less than ${E}_0$ but not null (as the foil is not fully transparent), we set ${E}_\mathrm{r}={E}_0/2$ in the corresponding theory. We can see that for larger ${a}_0$ , ${k}_\mathrm{m}$ and ${\gamma}_\mathrm{m}$ first decrease and increase (rising stage, blue shaded) with ${a}_0$ , respectively, and then become nearly level (saturation stage, orange shaded). For ${a}_0\le 20$ , the behaviour agrees well with that given by Equations (6) and (7) for the RTI. For the $\Delta \varphi =\pi$ case, ${\gamma}_\mathrm{m}$ is smallest and ${E}_\mathrm{r}$ is near null. Figures 7(d)–7(f) show that for less phase-matched laser pairs, the threshold light intensity is larger and the growth rate is smaller, suggesting that the instability can be manipulated by tailoring the phases of the paired lasers.

Figure 7 (a)–(c) Scaling laws of the electron temperature ${T}_\mathrm{e}$ (left-hand $y$ -axis, blue dots) and Lorentz factor ${\gamma}_{\mathrm{e}}$ (right-hand y-axis, orange squares) versus ${a}_0$ for the LCP + RCP case with (a) $\Delta \varphi =0$ , (b) $\pi /2$ and (c) $\pi$ from PIC simulations. The straight dashed lines are linear fits of the simulation results. (d)–(f) The fastest growing mode ${k}_\mathrm{m}$ (left-hand $y$ -axis, grey dots) and maximum growth rate ${\gamma}_\mathrm{m}$ (right-hand $y$ -axis, green squares) of transverse instability versus laser amplitude ${a}_0$ from PIC simulations for the LCP + RCP case with (d) $\Delta \varphi =0$ , (e) $\pi /2$ and (f) $\pi$ . For comparison, the theoretical results from Equation (6) are also given: a solid green curve for RTI and a dashed green curve for the electron-ion (ei) coupling effect, and the grey curve for ${k}_\mathrm{m}$ is from Equation (7).

Several theoretical models have been proposed to explain the RTI. We have used Wan et al.’s model^[ Reference Wan, Andriyash, Lu, Mori and Malka ^{17 ]} since there the effect of electron temperature is included, which can be important in the problem here. It is also helpful for identifying which RTI and/or electron coupling effect governs the development of transverse foil instability.

7 Summary

To summarize, we have investigated the dynamics of two counter-propagating relativistic fs laser pulses interacting with ultrathin foil. It is found that the transverse feature of the foil depends on the polarization direction and intensity distribution of the resultant laser field, and its longitudinal motion is determined by the impulse of the longitudinal light pressure force of the resultant laser light. For the LCP + RCP case, a grainy bubble and ring structure, characteristic of thin-foil RTI, appears in the foil density. The maximum growth rate of the instability increases with the resultant laser intensity, first rapidly and then slowly. When the two lasers are out of phase, the instability is weakest. Our results should be helpful for understanding counter-propagating laser pair interaction with ultrathin foils, which has been proposed for the production of ultra-bright $\gamma$ -rays^[ Reference Shen and Meyer-ter-Vehn ^{22 ,} Reference Luo, Zhu, Zhuo, Ma, Song, Zhu, Wang, Li, Turcu and Chen ^{23 ]}, electron–positron pairs^[ Reference Bell and Kirk ^{34 –} Reference Lobet, Ruyer, Debayle, d’Humières, Grech, Lemoine and Gremillet ^{37 ]}, pulsed neutrons^[ Reference Macchi ^{24 –} Reference Feng, Shao, Jiang, Zou, Hu, Zhang, Yang, Yin, Ma and Yu ^{27 ]}, ultra-intense light^[ Reference Shen and Yu ^{18 ]}, etc.