Fig. 1. (a) Deflection rate dθ/dx as a function of time as predicted by Eqs. (16) and (17) for a C⁶⁺ plasma for various Mach numbers in the fluid [Eq. (7)] and kinetic formalism [Eq. (5)]. (b) Gaussian deflection rate averaged over τ_SSD = 7 ps, ⟨dθ/dx⟩τSSD, as predicted by Eqs. (19) and (23) (using S = 1 and β = 1) for He²⁺ (blue lines), C⁶⁺ (red lines), and Au⁵⁰⁺ (black lines) plasmas as a function of Mach number. Three-dimensional predictions obtained with Hera are superimposed as red circles (see Appendix A for details). The laser has an averaged intensity of I₀ = 1 × 10¹⁵ W/cm², and λ₀ = 0.35 µm, f_♯ = 8, T_e = 2.5 keV, T_i = 1 keV, and n_e = 0.1n_c.

Fig. 2. Temporal evolution of the beam centroid defined though Eq. (2) in the 3D Parax simulations with ω_m = 2π × 14.25 GHz and for Δ = 5.1 for TSSD ⊥v_d (blue line), LSSD (green line), TSSD ‖v_d (red line), and TSSD ⊥v_d + PS (magenta line).

Fig. 3. Centroid displacement after 2 mm of propagation as predicted by the theory (lines) and by simulations (symbols). The plasma is composed of He²⁺ with T_e = 2 keV, Z_iT_e/T_i = 8, M₀ = 0.9, and I₀ = 2 × 10¹⁴ W/cm². In (a), the 2D centroid displacement Δy is plotted as a function of f_♯ with a fixed T_SSD = 3.7 ps (or a modulation depth of Δ = 9), while in (b), f_♯ = 8.88 is fixed and T_SSD varies. In (a), the predictions from the ray tracing scheme and from Eq. (19) with S = 1.7 and β = 2 are shown by the triangles and the black solid line, respectively. A curve of y ∝ 1/f_♯ is superimposed as the black dashed line, while the results of the 3D paraxial Parax simulation for the case TSSD ⊥v_d are shown by the black circles. In (b), the predictions of Eq. (45) of Ref. 65 are shown by the red dashed line. The cases TSSD ‖v_d, TSSD ⊥v_d, and LSSD correspond to (S, β) = (1.7, 1), (1.7, 2), and (S, β) = (6, 0.8), respectively. The case with PS corresponds to (S, β) = (0.8, 3).

Fig. 4. Hydrodynamic simulation performed with the Troll code³⁸ for the Hybrid B NIF Shot N181209,^55–57 illustrated in the frame of the inner cone (y = 0 is the main inner cone axis), at 6 ns. (a) Total power of main laser drive. (b) Averaged local ionization number ⟨Z_i⟩ (the color map is saturated to 10). (c) Electron temperature T_e (keV). (d) Normalized electron density n_e/n_c (the color map is saturated to n_e/n_c = 0.3). (e) Mach number of the y component of the flow velocity, v_⊥/c_s.

Fig. 5. (a) and (b) Intensity profiles log(I[W/cm²]). (c) and (d) Power deposited in the plasma through inverse bremsstrahlung, S_B (W/cm²). (e) 2.5D deflection rate dθ_SSD/dx (mrad/mm) as predicted by Hera. (a), (c), and (e) show the results predicted by the 2.5D beam bending rate [Eq. (24) with (S, β) = (1.7, 1)], while (b) and (d) show the results from the classical ray tracing scheme. The material boundaries are superimposed as black lines.

Fig. 6. Intensity profiles (W/cm²) of a 3D Gaussian laser pulse propagating through a flowing (M₀ = 0.8) fully ionized carbon plasma at 10% to the critical density as predicted by 3D Hera simulations. (a) Intensity profile in the (x, y) plane, with the flow velocity along the y axis and the main laser direction along the x axis. (b) Intensity profile in the exit plane.

Fig. 7. Comparison between the fit of Eq. (D1) (dashed lines) and the exact calculations (solid lines) for different values of τ_SSD. The parameters in (a)–(c) correspond to a gold plasma (Z_Au = 50, λ_mfp/f_♯λ₀ = 1.47) with γ₀ = 0.007, 0.02, and 0.05, respectively, and those in (d) to a carbon plasma (Z_C = 6, λ_mfp/f_♯λ₀ = 71) with γ₀ = 0.05. T_e = 3 keV and n_e = 9 × 10²⁶ m⁻³.