2. Baseline direct-drive target design

The target considered here (Temporal et al. Reference Temporal, Canaud and Ramis2021a, Reference Temporal, Piriz, Canaud and Ramis2024; Temporal, Canaud & Ramis Reference Temporal, Canaud and Ramis2021b; Brandon et al. Reference Brandon, Canaud, Temporal and Ramis2014) is a cryogenic spherical layer of thickness Δ = 198 ${\rm \mu}$m, inner radius R_i = 593 ${\rm \mu}$m and total mass of 300 ${\rm \mu}$g DT at the density of 0.25 g.cm⁻³. The initial aspect ratio of the capsule $A = {R_i}/\varDelta$ is 3. A plastic (CH) shell covers this cryogenic DT layer with a thickness of 30 ${\rm \mu}$m for a density of 1.07 g.cm⁻³ and acts as a laser light absorber. The central part of the target is filled with DT gas at saturation vapour pressure (density 0.3 mg.cm⁻³).

The capsule is directly irradiated by a laser pulse at wavelength λ = 0.35 ${\rm \mu}$m (3ω) and laser light is assumed to propagate radially. Refraction is neglected and the absorption is calculated via inverse bremsstrahlung. A description of the capsule and the laser pulse are given in figure 1. The absorbed laser pulse is characterized by a low-power foot at the constant power of P ₁ = 0.6 TW until the time t ₁ followed by a Kidder-like ramp (Kidder Reference Kidder1976) such as $P(t) = {P_0}{[1 - {(t/\tau )^2}]^a}$ that grows until the time t ₂ up to a maximum power P ₂, of the drive, that stays constant for a relatively short drive of 500 ps. Thus the pulse depends on the four parameters t ₁, t ₂, P ₂ and τ; whilst P_o and a are parameters calculated setting P(t ₁) = P ₁ and P(t ₂) = P ₂.

Figure 1. Sketch of the directly driven capsule and absorbed laser power as a function of time for the reference case (blue) and for the shock-ignition spike (red). Green (χ), grey (f_e) and dashed (T _he) lines characterize hot electrons sources (see § 3).

The 1-D hydrodynamics code Multi-IFE (Ramis & Meyer-ter-Vhen Reference Ramis and Meyer-ter-Vehn2016) is used to optimize the parameters t ₁, t ₂, P ₂ and τ. The MULTI code deals with multi-group radiation transport and tabulated equations of state, and ion and electron thermal conduction that assumes a harmonically limited (8 %) Spitzer heat flux (Spitzer Reference Spitzer1962) at the free-flow limit.

The absorbed laser pulse (see blue curve in figure 1) results of a trade-off between minimizing the imploding kinetic energy (E_k) and minimizing the drive power and whole laser energy, while maximizing the thermonuclear output fusion energy (E_F). It is characterized by the parameters t ₁ = 6.2 ns, t ₂ = 15.4 ns, τ = 22.4 ns and a top power drive of P ₂ = 242 TW. Consequently, P_o = 0.25 TW and α = −10.7. The corresponding absorbed energy is E_a = 355 kJ. For this reference case, the hydrodynamic calculation provides an output fusion energy of E_F = 27.4 MJ, a maximum kinetic energy of E_k = 11.7 kJ, a peak implosion velocity of V = 251 ${\rm \mu}$m ns⁻¹, a stagnation time of t_s = 16.09 ns, a bang time (defined as the time of maximum fusion power) of t_b = 16.19 ns and an in-flight adiabat (defined as the ratio of DT pressure over Fermi pressure, taken during the deceleration phase) of $\alpha = {P_{\textrm{DT}}}/{P_{\textrm{Fermi}}} = P[\textrm{Mbar}]/2.32{\rho ^{5/3}}[\textrm{g.}\textrm{c}{\textrm{m}^{ - \textrm{3}}}]\; \sim \; 1.12$. The calculation related to this case, hereafter the reference case, neglects the production and transport of hot electrons and the loss of the power triggered by parametric instabilities.

We show in figure 1 an example of an ignition peak (in red) that can be considered. In all the calculations, the assumption will be made of an ignition peak in the form of a sixth-order super-Gaussian (quasi-square) temporal laser pulse centred at time t _SI with maximum power P _SI and full width at 1/e, Δ.

3. Hot electrons effects on burning target

A Monte Carlo package previously coupled to the Multi-IFE code for α-particles (Temporal et al. Reference Temporal, Canaud, Cayzac, Ramis and Singleton2017) and fusion product transport (Temporal, Canaud & Ramis Reference Temporal, Canaud and Ramis2021c) has been adapted for the production, propagation and energy deposition of the hot electrons generated by plasma instabilities (Temporal et al. Reference Temporal, Canaud and Ramis2021a). In this package, the hot electrons are assumed to be generated at the radial position r_c of the quarter-critical plasma density, n_c/4, where n_c is the critical density. Following the work of Froula et al. (Reference Froula, Yaakobi, Hu, Chang, Craxton, Edgell, Follett, Michel, Myatt, Seka, Short, Solodov and Stoeckl2012), the dimensionless parameter $\chi = I{L_n}/{T_e}$, normalized to 10¹⁴ (W.${\rm \mu}$m).(cm² keV)⁻¹, is used locally to estimate the temperature of the hot electrons T _he(χ) as well as the fraction of the laser energy converted to hot electrons f_e(χ). Here, I, L_n and, T_e are the laser intensity, density gradient length and electron temperature, respectively, taken at the quarter-critical density.

In our study, the incoming laser intensity can be greater than 10¹⁵ W cm⁻² while in Froula et al. (Reference Froula, Yaakobi, Hu, Chang, Craxton, Edgell, Follett, Michel, Myatt, Seka, Short, Solodov and Stoeckl2012), laser intensity never exceeds this value. Anyway, in Solodov & Betti (Reference Solodov and Betti2008), measurements have been done above 10¹⁵ W.cm⁻², showing the same trend as in Froula et al. (Reference Froula, Yaakobi, Hu, Chang, Craxton, Edgell, Follett, Michel, Myatt, Seka, Short, Solodov and Stoeckl2012), namely saturation of the hot-electron fraction. The estimate of T _he is made using a numerical fit previously performed from the data of Froula et al. and more precisely from figure 4 in Froula et al. (Reference Froula, Yaakobi, Hu, Chang, Craxton, Edgell, Follett, Michel, Myatt, Seka, Short, Solodov and Stoeckl2012), leading to ${T_{\textrm{he}}}(\chi ) = 100[\textrm{keV}]\chi /750$. Concerning f_e, a linear fit per part is made (Temporal et al. Reference Temporal, Canaud and Ramis2021a) from the experimental data presented in Froula et al. (Reference Froula, Yaakobi, Hu, Chang, Craxton, Edgell, Follett, Michel, Myatt, Seka, Short, Solodov and Stoeckl2012). These data show a saturation of the hot-electron fraction f_e for χ larger than 750. The saturation level f _sat is used here as a free parameter. An example of χ and f_e is shown in figure 1 for the directly driven capsule without SI spike assuming a given saturation level (f _sat = 5 %).

Figure 2 shows the variation of f_e with χ for ten f _sat saturation levels between 1 % and 15 %. The parameters χ, T _he and f_e are calculated at each hydrodynamic time step dt and used to create a population of electrons characterized by a Maxwellian distribution of temperature T _he and total energy ${E_{\textrm{he}}} = {f_e}(\chi ){P_L}\,\textrm{d}t$, where P_L is the absorbed laser power. The electron emission is assumed to be isotropic. During the hydrodynamic time step, the absorbed laser power is reduced by a factor (1 − f_e) in order to ensure the conservation of the total energy. The hot electrons are assumed to move in a straight line over a distance Δs where 1 % of their energy is deposited, so that $\Delta s = 0.01\ast \varepsilon /(\textrm{d}\varepsilon /\textrm{d}s)$. The process proceeds step by step until the kinetic energy of the electrons is less than or equal to the temperature of the plasma.

Figure 2. Hot-electron energy fraction (f _he) evaluated for fifteen saturation values (f _sat = 1 % to 15 %) as a function of the parameter χ.

In order to take into account the angular diffusion, at the end of each step Δs, the direction of the electrons is deflected by an angle θ and rotated by a random angle $\phi \in [0,2{\rm \pi}]$ around the incident direction Δs, where θ is the angle between the incident direction and the new direction and it follows a Gaussian distribution characterized by a standard deviation σ_θ (see (2) of Temporal et al. Reference Temporal, Canaud and Ramis2021a). The stopping power dε/ds (Atzeni, Schiavi & Davies Reference Atzeni, Schiavi and Davies2008; Solodov & Betti Reference Solodov and Betti2008; Robinson et al. Reference Robinson, Strozzi, Davies, Gremillet, Honrubia, Johzaki, Kingham, Sherlock and Solodov2014) used to calculate the energy lost by the electrons as well as the calculation of σ_θ are detailed in Robinson et al. (Reference Robinson, Strozzi, Davies, Gremillet, Honrubia, Johzaki, Kingham, Sherlock and Solodov2014) and Temporal et al. (Reference Temporal, Canaud and Ramis2021a).

A series of calculations of a target implosion were performed using the 1-D Multi-IFE hydro-radiative code and varying the level of saturation (f _sat) of hot-electron production between zero, corresponding to the reference case without hot electrons, and f _sat = 15 %. It is worth noticing that the hot electrons and the induced laser power reduction have significant effects for low saturation levels (above 2 %) and lead to a strong reduction of the thermonuclear energy produced by the implosion.

The generation of parametric instabilities in the corona, accompanied by the creation of hot electrons, has two main consequences for implosion. The first concerns the energy deposition of these hot electrons as they propagate in the target, which will modify the hydrodynamics of the implosion and, in particular, the fuel adiabat. The second concerns the amount of laser energy absorbed by the plasma via inverse bremsstrahlung, which is reduced since some of it is either backscattered, in the case of stimulated Raman scattering, or converted directly into hot electrons, resulting in a reduction in laser–target coupling efficiency.

In order to identify the respective role of each consequence, two additional curves are presented in figure 3.

Figure 3. Thermonuclear output fusion energy E_F (MJ) as a function of the hot-electron saturation.

The first one (blue curve) concerns the reduction of the laser power due to the production of hot electrons without taking into account the energy deposition of the hot electrons. A second curve (green curve) concerns the energy deposition of hot electrons without taking into account the reduction of the laser power. This comparison shows that the transport and energy deposition of hot electrons inside the target have more important deleterious effects than the reduction of the absorbed laser power alone since the threshold is more or less the same when both effects are taken into account, whereas the threshold is 10 % when only the laser reduction is taken into account.

4. Shock ignition of baseline design with hot electrons

In a previous work (Temporal et al. Reference Temporal, Canaud and Ramis2021a), we have shown that the hot-electron energy deposition is more destructive the deeper it affects the target zones. It thus disturbs the hydrodynamic conditions of ignition and combustion of the gain target. The question is therefore whether shock ignition can be sufficiently effective in restoring the burning conditions of the DT. To verify this possibility, a representative case of marginal ignition (see figure 3, case a), corresponding to a saturation of f _sat = 5 % and producing only 13.5 kJ of thermonuclear energy, is considered. A series of calculations was performed by adding to the main laser pulse (blue curve in figure 1) a shock-ignition spike (red curve in figure 1), leading to an increase of the whole energy, characterized by a spike pulse centred at the time t _SI, a maximum spike power P _SI and six different spike time widths at 1/e, Δ = 50, 100, 200, 300, 400 and 500 ps. In these calculations, hot electrons are taken into account even for the spike with the same saturation level (f _sat = 5 %). The spike time t _SI ranges from 15.4 to 15.7 ns and the spike power ranges from 0 to 300 TW. The ignition window, defined as the thermonuclear fusion energy as a function of t _SI and P _SI, is shown in figure 4 for different spike widths. For all widths, there is a region in the parameter space that recovers combustion with thermonuclear energies above 20 MJ. Increasing the width Δ facilitates the ignition of the shock, with lower spike power P _SI.

Figure 4. Thermonuclear fusion energy E_F (MJ) as a function of the spike parameters t_SI (ns) and P _SI (TW). The cases refer to a saturation level of f _sat = 5 % and spike full width at 1/e Δ = 50, 100, 200, 300, 400 and 500 ps.

The grey dots on the graphs of figure 4 show the positions of maximum thermonuclear energy in terms of spike power P _SI for each time t _SI. This is a ridgeline from which to follow the evolution of the thermonuclear energy for each peak power (deduced from the grey points).

It is worth noticing that the case Δ = 500 ps is equivalent to increasing the power of the drive and thus the whole energy involved in the implosion. A series of calculations, carried out by varying the drive power above 242 TW, show that the thermonuclear energy of 27 MJ can be recovered again for an increase of the laser power of the drive of approximately 90 TW more, similarly to the case Δ = 500 ps. Let us notice that, in this case (Δ = 500 ps), the spike time is approximately 15.65 ns, which is equivalent to synchronizing the spike with the main drive.

Figure 5 shows the evolution of the thermonuclear energy as a function of the ridge power of the spike in the case of a full width at 1/e of Δ = 50 ps (refers to the top left of figure 4).

Figure 5. Maximum fusion energy as a function of the spike power P _SI for Δ = 50 ps.

At P _SI = 0 TW (without shock ignition), the implosion produces only 13.5 kJ of thermonuclear energy. By increasing the ridge power P _SI, the thermonuclear fusion energy increases and crosses a threshold that separates the marginally igniting implosions (for a low spike power) from burning implosions (for a sufficiently high spike power) tending asymptotically towards the 27.4 MJ of the reference case for the greatest spike power.

The threshold described here depends strongly on the duration of the spike, as shown in figure 6, where Δ varies from 50 to 500 ps, the last corresponding to the main pulse drive duration but not launched at the same time. The threshold is arbitrary defined as the smallest spike power for which the implosion of the target produces 1 MJ. The variation of this threshold is shown in figure 6 as a function of the spike duration Δ (red dots).

Figure 6. Minimum spike power (red dots) and absorbed laser energy (green dots) required for generating fusion energy of 1 MJ as a function of the spike duration Δ.

As shown earlier and inferred from figure 4, the minimum peak power required for producing at least 1 MJ decreases as the spike duration increases and almost saturates at approximately 55 TW for spike durations above 300 ps. For this last duration, the energy involved in ignition is roughly 15 kJ. This suggests that a minimal absorbed energy is required to initiate target combustion. The green curve in figure 6 represents the variation in the absorbed energy contained in the spike as a function of the duration of the peak Δ. This minimum energy to produce 1 MJ thermonuclear is here approximately 10 kJ for a spike duration of 50 ps.

Such a low level of spike power needed to ignite and burn an ICF target is not new and was already shown in Canaud & Temporal (Reference Canaud and Temporal2010), Canaud, Laffite & Temporal (Reference Canaud, Laffite and Temporal2011), Temporal et al. (Reference Temporal, Ramis, Canaud, Brandon, Laffite and Le Garrec2011), Canaud et al. (Reference Canaud, Laffite, Brandon and Temporal2012) and Brandon et al. (Reference Brandon, Canaud, Temporal and Ramis2016). The novelty here is that, even when hot-electron energy deposition is taken into account, it is still possible to ignite the ICF target with relatively low spike power (below 200 TW). In addition, it is worth noticing that shorter spike duration requires lower spike energy (a spike of 50 ps requires only 9 kJ of additional energy against roughly 28 kJ for the spike of 500 ps).

In order to further investigate the hot-spot dynamics and following Brandon et al. (Reference Brandon, Canaud, Temporal and Ramis2016), we focus on the hot-spot thermodynamic paths of three distinct cases : the reference case, a case (a) without shock ignition and with hot electrons (f _sat = 5 %) and a case (b) with both hot electrons (f _sat = 5 %) and shock ignition with an ignition power P _SI = 250 TW (referred to as b in figures 4 and 5), Δ = 50 ps and t _SI = 15.52 ns. In the first case (a), the fusion energy is E_F = 13.5 kJ while case (b) provides 21.3 MJ and gives back almost the 27.4 MJ of the baseline design. For all cases, the thermodynamic paths are evaluated in the (ρR _HS, T _HS) plane where ρR _HS(t) and T _HS(t) are the time-resolved areal density and mass-averaged ionic temperature of the hot-spot volume, respectively. In this analysis, the Lagrangian cells where DT fusion occurs define the hot-spot volume. Figure 7 shows the hot-spot trajectories for the three cases during the deceleration phase (starting at the peak implosion velocity time) to stagnation (black dots, defined as the time of minimum volume for the hot spot) and then ignition or rebound without combustion.

Figure 7. Temporal evolution of the hot-spot temperature T _HS and areal density ρR _HS for the reference case (green curve), the non-igniting case (a) and the shock-ignited case (b).

It can already be seen that the thermodynamic path of the reference case (‘baseline’ design) is clearly different from that of the target perturbed by the hot electrons (a), confirming that the hot electrons, in addition to disturbing the cryogenic DT (Christopherson et al. Reference Christopherson2021; Temporal et al. Reference Temporal, Canaud and Ramis2021a), also manage to modify the thermodynamic path of the hot spot during the deceleration phase, while the laser is switched off. It can be seen that the temperature reached at stagnation by the perturbed target (a) is significantly lower (3 keV vs 4 keV) than that reached by the basic design while the hot-spot areal density is not modified. A simple shock is thus sufficient to close the 1 keV temperature gap and also recompress the dense shell to ignite and burn the target. Indeed, looking at the red curve corresponding to the thermodynamic path of the hot spot of the shock-ignited perturbed target, it is worth noting that the path follows that of the perturbed target up to a certain point and then deviates to follow that of the reference target. Following this end of the path to stagnation, the target ignites and burns almost like to the reference case, confirming that shock ignition overcomes the deleterious effects of hot electrons on a high-gain ICF target. Nevertheless, the maximum areal density of the shock-ignition case stands below the one from the reference case, leading to a lower thermonuclear energy.