# How Shock ignition can help to overcome the negative effects of hot electrons in direct-drive high-gain inertial confinement fusion

Mauro Temporal, Benoit Canaud, Rafael Ramis

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Mauro Temporal, Benoit Canaud, Rafael Ramis. How Shock ignition can help to overcome the negative effects of hot electrons in direct-drive high-gain inertial confinement fusion. Journal of Plasma Physics, 2024, 90 (5), pp.905900507. 10.1017/s0022377824001211. hal-05166318

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# How Shock ignition can help to overcome the negative effects of hot electrons in direct-drive high-gain inertial confinement fusion

Mauro Temporal<sup>1,‡</sup>, Benoit Canaud<sup>2,3,†</sup> and Rafael Ramis<sup>4</sup>

<sup>1</sup>École Normale Supérieure Paris-Saclay, Centre Borelli UMR 9010, 4 avenue des Sciences, 91190 Gif sur Yvette, France

<sup>2</sup>CEA, DAM, DIF, F-91297 Arpajon, France

<sup>3</sup>Université Paris-Saclay, CEA, LMCE, 91680 Bruyères-le-Châtel, France

<sup>4</sup>ETSI Aeronáutica y del Espacio, Universidad Politécnica de Madrid, 28040 Madrid, Spain

(Received 13 May 2024; revised 27 August 2024; accepted 30 August 2024)

Hot electrons produced by parametric instabilities can negatively affect the thermonuclear gain of a direct-drive inertial confinement fusion capsule. A Monte Carlo transport is coupled to the hydrodynamics MULTI-IFE code in order to study the hot electron transport in a high-gain target implosion. The thermonuclear energy produced by the directly driven implosion of a reference target drops from 27 MJ without hot electrons to 13.5 kJ when the hot electrons are taken into account. A clear relationship is established between the level of hot electrons produced and the degradation of the thermonuclear yield produced by the target. It is shown that adding a relatively short laser spike (shock ignition) to the main pulse restores the thermonuclear energy released by the implosion. Different shock-ignition windows are presented depending on the duration of the spike. It appears that the longer the duration of the spike, the lower the power required for shock ignition. Finally, a 300 ps-duration spike requires only 50 TW to restore a thermonuclear energy output of 20 MJ.

**Keywords:** fusion plasma, plasma confinement

## 1. Introduction

Laser inertial confinement fusion (ICF) (Atzeni & Meyer-ter-Vehn 2004) uses the implosion of a spherical capsule to achieve the thermodynamic regime necessary to initiate and propagate a thermonuclear burning wave, which is able to release more fusion energy than energy invested in the implosion, producing what is commonly referred to as a high energy gain. Two main approaches are envisaged for the implosion of the target: direct drive (Craxton et al. 2015) and indirect drive (Lindl 1998). In the first case, the lasers

† Email address for correspondence: benoit.canaud@cea.fr

‡ Present address: Instituto de Investigaciones Energéticas (INEI), E.T.S.I.I., and CYTEMA, Universidad de Castilla-La Mancha, 13071 Ciudad Real, Spain.
directly illuminate the outer part of the spherical target while, in the indirect drive, the laser energy is converted into X-rays inside a cavity, usually of gold, and it is such X-rays that drive the implosion.

The capsule is a spherical envelope of low Z (plastic or CH) containing a layer of a mixture of deuterium (D) and tritium (T) in cryogenic form (at temperatures of approximately 19 K). The central part is filled with DT gas at saturation vapour pressure. The outer part consists of a layer, the ablator, that is vapourized by the lasers or the X-rays absorbed in it and transformed into plasma heated to very high temperatures (typically a few keV). This plasma, expelled at very high speed (a few hundred km.s⁻¹) from the outside of the target, exerts an inward radial pressure on the solid part of the target, which starts to move, imploding on itself until it collapses. The inner part, protected from lasers or X-rays, will be compressed and heated to very high temperatures, thus initiating fusion reactions and allowing the release of energy.

When lasers interact with the plasma, either in the cavity or directly on the target, so-called parametric instabilities (stimulated Raman scattering (Drake et al. 1974; Afeyan 1997a), and two plasmon decay (Figueroa et al. 1984; Afeyan 1997b)) are likely to produce very high energetic electrons – so-called supra-thermal or hot electrons – which have the particularity of penetrating deep into the capsule until they degrade the desired performance of the target (Temporal, Canaud & Ramis 2021a). This type of deleterious mechanism is much more significant in direct drive than in indirect drive, due to the fact that when the lasers strike the spherical target directly, the hot electron source is located much closer to the cryogenic DT. Parametric instabilities have multiple effects. Firstly, part of the laser power is involved in generating these instabilities near or below the critical quarter density, and is backscattered without being absorbed (Drake et al. 1974). This amount of laser energy is wasted and unused for ablation and shell velocity setting, thus impairing the laser–target coupling efficiency. Secondly, the hot electrons thus created can preheat the DT fuel (Christopherson et al. 2021), inducing a lowering of the shell density in the vicinity of the hot spot (Temporal et al. 2024) and thus increasing its adiabat. Thirdly, lowering the density and raising the temperature in this part of the target modifies the mean free path of the alphas, disrupting the release of thermonuclear energy (Temporal et al. 2024). All these effects, taken together, can prevent thermonuclear combustion, even if the hot-spot ignition conditions are fulfilled (Atzeni & Meyer-ter-Vehn 2004; Temporal et al. 2021a).

Various solutions (Dodd & Umstadter 2001; Wu et al. 2001; Albright, Yin & Afeyan 2014; Montgomery 2016; Zhou et al. 2018; Zhao et al. 2022) have been proposed to mitigate the detrimental effects of parametric instabilities, and thus, hot electrons, on the burning of a high-gain thermonuclear target. They are essentially all based on desynchronizing the laser coherence with respect to the electron plasma waves, by introducing a spectral width into the laser field, with many constraints on the laser installation.

Failing to eliminate parametric instabilities and their consequences in terms of hot electrons, it seems legitimate to consider the possibility of producing thermonuclear energy while generating hot electrons in the target due to parametric instabilities, by slightly modifying the thermodynamic conditions of the fuel at the moment of ignition. This is the aim of the work presented here. Hot electrons deposit their energy throughout the target, but their main effect is in the immediate vicinity of the hot spot, in the dense part of the shell (Temporal et al. 2024). The idea developed in the present work is to use a well-known ignition technique, shock ignition (SI) (Betti et al. 2007), to restore the target’s thermonuclear energy gain. In fact, the aim here is to exploit shock ignition, not to demonstrate shock ignition of a target that does not ignite, but rather to use,
simultaneously, the dual effect of using an ignition peak to both produce a non-isobaric situation conducive to easier ignition, but also to benefit from the increase, however slight, in implosion speed that the addition of the spike brings about, enabling the kinetic threshold of combustion to be shifted below the target’s kinetic energy.

Shock ignition, first proposed by Betti *et al.* (2007), consists in igniting an ICF target in a non-isobaric way, where ignition is traditionally said to be isobaric or quasi-isobaric (Atzeni & Meyer-ter-Vehn 2004). The method used consists of adding a sub-nanosecond laser peak to the main pulse of an isentropic implosion, slightly modifying the latter to keep the peak implosion velocity constant. This method has met with real success, especially in numerical calculations, and many complementary studies have been carried out over the past two decades (Canaud & Temporal 2010; Schmitt *et al.* 2010; Atzeni *et al.* 2014), culminating in the use of this technique in the design of an inertial fusion reactor, the subject of the HiPER project (Dunne 2006; Ribeyre *et al.* 2008; Atzeni *et al.* 2009). Very quickly, the use of high laser powers, and therefore high intensities, led to the question of parametric instabilities and therefore of the hot electrons created by the ignition spike (Piriz *et al.* 2012). Some studies have demonstrated the benefits of energy transport in the ablator by these supra-thermal electrons (Klimo *et al.* 2014; Colaïtis *et al.* 2016; Cristoforetti *et al.* 2017; Llor Aisa *et al.* 2017; Trela *et al.* 2018). However, this is not the subject of the study presented here even if it is still a subject of actual research (Baton *et al.* 2020; Tentori *et al.* 2021; Barlow *et al.* 2022; Tentori, Colaïtis & Batani 2022a,b).

This work proposes to use shock ignition to overcome the deleterious effects of hot electrons on the performance of an ICF target. A baseline direct-drive ICF capsule implosion is considered and modelled by means of one-dimensional (1-D) numerical calculations using the Multi-IFE hydrodynamics code (Ramis & Meyer-ter-Vhen 2016). Hot electrons generated by parametric instabilities are transported through the target in which they deposit their energy with the consequence of significantly affecting the capsule's performance (Temporal *et al.* 2021a), although some electrons even appear to 'wander' (Gus'kov *et al.* 2019) in the target, which is likely to reduce their effect. A laser peak of a few hundred picoseconds is then added to the main laser pulse in order to ignite this non-igniting target.

The paper is organized as follows. Section 2 presents the baseline design consisting of the target and the laser pulse. Then, § 3 describes how the hot electrons and their transport in the target modify the implosion and the ignition of the capsule. Finally, the § 4 presents how the shock ignition allows restoration of the released thermonuclear energy and § 5 concludes the paper.

## 2. Baseline direct-drive target design

The target considered here (Temporal *et al.* 2021a, 2024; Temporal, Canaud & Ramis 2021b; Brandon *et al.* 2014) is a cryogenic spherical layer of thickness $\Delta = 198$ µm, inner radius $R_i = 593$ µm and total mass of 300 µg DT at the density of 0.25 g.cm$^{-3}$. The initial aspect ratio of the capsule $A = R_i/\Delta$ is 3. A plastic (CH) shell covers this cryogenic DT layer with a thickness of 30 µm 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 $\lambda = 0.35$ µm (3$\omega$) 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 1976) such as $P(t) = P_0[1 - (t/\tau)^2]^a$ that grows until the time $t_2$ up to a maximum power
<figure><img src="image_2.png" /><figcaption>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 ($\chi$), grey ($f_e$) and dashed ($T_{he}$) lines characterize hot electrons sources (see § 3).</figcaption></figure>

$P_2$, 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 $\tau$; whilst $P_o$ and $a$ are parameters calculated setting $P(t_1) = P_1$ and $P(t_2) = P_2$.

The 1-D hydrodynamics code Multi-IFE (Ramis & Meyer-ter-Vhen 2016) is used to optimize the parameters $t_1$, $t_2$, $P_2$ and $\tau$. 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 1962) 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, $\tau = 22.4$ ns and a top power drive of $P_2 = 242$ TW. Consequently, $P_o = 0.25$ TW and $\alpha = -10.7$. 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 $V = 251$ µm 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 $\alpha = P_{DT}/P_{Fermi} = P[\text{Mbar}]/2.32\rho^{5/3}[\text{g.cm}^{-3}] \sim 1.12$. 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
<figure><img src="image_3.png" /><figcaption>FIGURE 2. Hot-electron energy fraction ($f_{he}$) evaluated for fifteen saturation values ($f_{sat} = 1\%$ to $15\%$) as a function of the parameter $\chi$.</figcaption></figure>

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, $\Delta$.

### 3. Hot electrons effects on burning target

A Monte Carlo package previously coupled to the Multi-IFE code for $\alpha$-particles (Temporal et al. 2017) and fusion product transport (Temporal, Canaud & Ramis 2021c) has been adapted for the production, propagation and energy deposition of the hot electrons generated by plasma instabilities (Temporal et al. 2021a). 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. (2012), the dimensionless parameter $\chi = IL_n/T_e$, normalized to $10^{14}$ (W.$\mu$m).(cm$^2$ keV)$^{-1}$, is used locally to estimate the temperature of the hot electrons $T_{he}(\chi)$ as well as the fraction of the laser energy converted to hot electrons $f_e(\chi)$. 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 $10^{15}$ W cm$^{-2}$ while in Froula et al. (2012), laser intensity never exceeds this value. Anyway, in Solodov & Betti (2008), measurements have been done above $10^{15}$ W.cm$^{-2}$, showing the same trend as in Froula et al. (2012), 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. (2012), leading to $T_{he}(\chi) = 100[\text{keV}]\chi/750$. Concerning $f_e$, a linear fit per part is made (Temporal et al. 2021a) from the experimental data presented in Froula et al. (2012). These data show a saturation of the hot-electron fraction $f_e$ for $\chi$ larger than 750. The saturation level $f_{sat}$ is used here as a free parameter. An example of $\chi$ 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 $\chi$ for ten $f_{sat}$ saturation levels between 1% and 15%. The parameters $\chi$, $T_{he}$ 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 $T_{\text{he}}$ and total energy $E_{\text{he}} = f_e(\chi)P_L dt$, 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 $\Delta s$ where 1% of their energy is deposited, so that $\Delta s = 0.01 * \varepsilon/(d\varepsilon/ds)$. The process proceeds step by step until the kinetic energy of the electrons is less than or equal to the temperature of the plasma.

In order to take into account the angular diffusion, at the end of each step $\Delta s$, the direction of the electrons is deflected by an angle $\theta$ and rotated by a random angle $\phi \in [0, 2\pi]$ around the incident direction $\Delta s$, where $\theta$ is the angle between the incident direction and the new direction and it follows a Gaussian distribution characterized by a standard deviation $\sigma_{\theta}$ (see (2) of Temporal et al. 2021a). The stopping power $d\varepsilon/ds$ (Atzeni, Schiavi & Davies 2008; Solodov & Betti 2008; Robinson et al. 2014) used to calculate the energy lost by the electrons as well as the calculation of $\sigma_{\theta}$ are detailed in Robinson et al. (2014) and Temporal et al. (2021a).

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 ($f_{\text{sat}}$) of hot-electron production between zero, corresponding to the reference case without hot electrons, and $f_{\text{sat}} = 15\%$. 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.

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. 2021a), 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 $f_{\text{sat}} = 5\%$ 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
<figure><img src="image_4.png" /><figcaption>FIGURE 3. Thermonuclear output fusion energy $E_F$ (MJ) as a function of the hot-electron saturation.</figcaption></figure>

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 $1/e$, $\Delta = 50, 100, 200, 300, 400$ 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 $\Delta$ facilitates the ignition of the shock, with lower spike power $P_{SI}$.

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 $\Delta = 500$ 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 $\Delta = 500$ ps. Let us notice that, in this case ($\Delta = 500$ 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 $\Delta = 50$ ps (refers to the top left of figure 4).

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.
<figure><img src="image_5.png" /><figcaption>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 $\Delta = 50, 100, 200, 300, 400$ and $500$ ps.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>FIGURE 5. Maximum fusion energy as a function of the spike power $P_{SI}$ for $\Delta = 50$ ps.</figcaption></figure>

The threshold described here depends strongly on the duration of the spike, as shown in figure 6, where $\Delta$ 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 $\Delta$ (red dots).
<figure><img src="image_7.png" /><figcaption>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 $\Delta$.</figcaption></figure>

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 $\Delta$. 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 (2010), Canaud, Laffite & Temporal (2011), Temporal et al. (2011), Canaud et al. (2012) and Brandon et al. (2016). 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. (2016), 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), $\Delta = 50$ 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 ($\rho R_{HS}$, $T_{HS}$) plane where $\rho R_{HS}(t)$ 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><img src="image_8.png" /><figcaption>FIGURE 7. Temporal evolution of the hot-spot temperature $T_{HS}$ and areal density $\rho R_{HS}$ for the reference case (green curve), the non-igniting case (a) and the shock-ignited case (b).</figcaption></figure>

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. 2021; Temporal et al. 2021a), 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.

## 5. Conclusions

This work shows the deleterious effects of hot electrons produced by parametric instabilities on the thermonuclear energy production of a high-gain target ($G \sim 77$) in direct-drive ICF and the possibility offered by shock ignition – obtained by adding a spike to the main pulse – to restore the high thermonuclear gain, initially lost by the effect of hot electrons. It is shown that a reference target producing approximately 27 MJ, only produces 13.5 kJ when the hot electrons and their transport in the target are taken into account.
Different levels of saturation of the fraction of laser energy converted into hot electrons are considered using hydrodynamics simulations coupled with a Monte Carlo electron transport package. A direct relationship is established between the level of hot electrons produced and the degradation of the thermonuclear yield. Finally, adding a laser spike to the main pulse restores the thermonuclear energy released by the implosion. Different shock-ignition windows are presented depending on the duration of the spike. It clearly appears that, the longer the duration of the spike, the lower the power required for shock ignition.

A spike power threshold appears, defined as the power required to produce 1 MJ of thermonuclear energy. The threshold on spike power decreases with the duration of the spike until a spike duration of 300 ps when the threshold power saturates around 50 TW. For a spike duration of 300 ps, the energy contained in the spike is very low, around 15 kJ, for a power of around 50 TW.

It is also necessary to put these results into perspective with regard to the method used. Indeed, in the calculations, the laser propagates radially, and its absorption takes place at greater depth than in a situation where a focal spot would be taken into account. As such, the results presented here should be considered more qualitatively than quantitatively.

## Acknowledgements

Editor Louise Willingale thanks the referees for their advice in evaluating this paper.

## Funding

M.T. has been supported by the CEA-ENS LRC-MESO grant no2018-011. R.R. has been supported by the Spanish *Ministerio de Ciencia Innovación y Universidades* project RTI2018-098801-B-100.

## Declaration of interests

The authors report no conflict of interest.

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