Fig. 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) , (b) and (f) , (c) and (g) , and (d) and (h) . Here, yLP and zLP denote linear polarization in the - and -directions, respectively, and 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 plane at , and , 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 and ), 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 , where the subscript ‘r’ here denotes ‘resultant’. The large red centre dot in window (c5) corresponds to , that is, the fields of the two lasers cancelled each other out.

Fig. 2. Evolution of the foil-density distribution for the polarization combinations (a)–(d) LCP + LCP and (e)–(h) LCP + RCP with (a) and (e) , (b) and (f) , (c) and (g) , and (d) and (h) . The first to third columns show the longitudinal foil-density distribution in the plane at , and , 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 and , respectively. The red lines and arrows in the fifth column represent the magnitude and direction of the resultant radial electric field at . The dot in panel (g5) indicates .

Fig. 3. Results of the analytical model and PIC simulations. (a) Normalized intensity of the component of the resultant laser field for the yLP + yLP case at . The shaded region represents the foil and the dashed lines represent the slopes of at . (b) The impulse of the axial pressure force (blue curve) of the resultant laser light in one laser cycle and the displacement (orange squares) of the foil centre versus for the yLP + yLP case.

Fig. 4. (a)–(d) Evolution of the light pressure force in the axial () direction (solid curves) and (e)–(h) versus for different polarization combinations. In all cases, the transverse force components (dashed curves) and (dotted curves) are also given.

Fig. 5. Averaged distribution of the foil density at for the (a) LCP + LCP case with and (c) LCP + RCP case with . (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 and different values. The slope (dotted lines) of the fastest growing mode shows the maximum growth rate .

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

Fig. 7. (a)–(c) Scaling laws of the electron temperature (left-hand -axis, blue dots) and Lorentz factor (right-hand y-axis, orange squares) versus for the LCP + RCP case with (a) , (b) and (c) from PIC simulations. The straight dashed lines are linear fits of the simulation results. (d)–(f) The fastest growing mode (left-hand -axis, grey dots) and maximum growth rate (right-hand -axis, green squares) of transverse instability versus laser amplitude from PIC simulations for the LCP + RCP case with (d) , (e) and (f) . 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 is from Equation (7).

Fig. 8. Relativistic transparency factor (grey curve), displacement of the foil centre (blue triangles), maximum growth rate (orange squares) and fastest growing mode k_m (green dots) versus the foil thickness from PIC simulations for the LCP + RCP case with . For and , the critical foil thickness for relativistic transparency to occur is .

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