Fig. 1. Dispersion relation for the case g = 0.002c²/λ.

Fig. 2. (a) and (b) Time evolution of spatial proton density ln[ñp(k,t)] in the transverse direction. (c) and (d) Transverse momentum distribution of protons ln[n_p(y, P_y)] at t = 25 T. (e) and (f) Corresponding transverse drift distance of protons ln[n_p(y, δ_y)] at t = 25 T. (a), (c), and (e) are for a cold target; (b), (d), and (f) are for a warm target.

Fig. 3. (a) and (b) Comparison of growth rates at low and high electron temperature for the LWM k = 1.9k₀ and the SWM k = 3.8k₀, respectively. (c) Temporal evolution of averaged proton longitudinal momentum P̄x (blue, left axis) within the whole simulation box, electron temperature (black, right black axis), and proton beam energy spread Δɛ/ɛ₀ (red, rightmost axis). Solid and dashed lines represent the LET and HET cases, respectively.

Fig. 4. (a) Proton density profile at t = 22 T. (b) Proton phase space (y, P_y) distribution at t = 22 T. (c) and (d) Electron energy distributions at t = 15 and 22 T. (e) Time evolution of electron energy spectrum. In this simulation, the target thickness l₀ = 0.5λ and the electron density n_e = 40n_c. Other simulation parameters are the same as in Fig. 2. The initial electron temperature is set as 10 eV.

Fig. 5. (a) Proton density profile at t = 25 T (n_e0 = 20n_c). The inset shows the transverse electron (green line) and proton (black line) density profiles. (b) Growth of the dominant mode (k = 1.9k₀) for electron and protons. The inset shows the Fourier analysis of electron mode growth. (c) Time evolution of transverse electron density fluctuation (n_e − n_e0). (d) and (e) The same as (a) and (b) but for the case n_e0 = 80n_c. (f) Ratio of the growth rate from simulations and that predicted by RTI theory (red line) and LS termination time (blue line) vs electron density. The dominant mode wavenumbers are 1.9k₀, 1.4k₀, 1.1k₀, and 1.4k₀, respectively.

Fig. 6. (a) Temporal evolution of proton average longitudinal momentum P̄x within the whole simulation box for the cases n_e0 = 20n_c, 40n_c, 60n_c, and 80n_c. The dashed lines show the accelerations in the LS phase, which are 0.012c²/λ, 0.006c²/λ, 0.004c²/λ, and 0.003c²/λ. (b) Proton energy spectra at t = 50 T from 2D simulations with n_e = 80n_c (blue), 60n_c (green), 40n_c (orange), and 20n_c (red). The inset shows the time evolution of the proton cutoff energy for the case n_e = 20n_c.

Fig. 7. (a) Dominant frequency of electron density modulation ω_m and (b) LS acceleration termination time τ_LS obtained from a series of 2D PIC simulations. The laser intensity a₀ and the areal density σ = n_e0l₀ are varied. For each laser intensity case, the areal density is increased from σ_op to 4σ_op.