High-gain direct-drive target design for the Laser Mégajoule

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1 INSTITUTE OF PHYSICS PUBLISHING and INTERNATIONAL ATOMIC ENERGY AGENCY NUCLEAR FUSION Nucl. Fusion 44 (2004) PII: S (04) High-gain direct-drive target design for the Laser Mégajoule B. Canaud 1,a, X. Fortin 1, F. Garaude 1,C.Meyer 1,b, F. Philippe 1, M. Temporal 2, S. Atzeni 3 and A. Schiavi 3 1 Département de Physique Théorique et Appliquée, CEA/DAM Ile-de-France BP12, F Bruyères-le-Châtel, France 2 ETSII, Universidad de Castilla-La Mancha, Ciudad Real, Spain 3 Dipartimento di Energetica, Università di Roma La Sapienza and INFM, Via A. Scarpa 14, I Roma, Italy benoit.canaud@cea.fr Received 29 December 2003, accepted for publication 16 August 2004 Published 1 October 2004 Online at stacks.iop.org/nf/44/1118 doi: / /44/10/005 Abstract In the context of the French Laser-Mégajoule fusion-research programme, a direct-drive target design is developed. It is based on the use of a CH-foam ablator filled with cryogenic deuterium tritium. One-dimensional optimization leads to a potential gain of 60 with a 1.5 MJ laser. The hydrodynamic stability of the implosion is investigated at the ablation front and the hot-spot surface by means of modelling and two-dimensional simulations. The effect of irradiation non-uniformities on low-mode (time-dependent) implosion asymmetries is studied by two-dimensional hydrodynamics simulations with three-dimensional laser-light raytracing. The effect of beam focal shapes on hot spot low-mode asymmetries is also addressed. PACS numbers: Bc 1. Introduction The feasibility of fusion energy production by laser driven inertially confined deuterium tritium (DT) targets has been studied since the 1970s [1,2]. At present, much effort is being devoted to the construction of large laser facilities such as the Laser Mégajoule (LMJ) in France [3] and the National Ignition Facility (NIF) in the USA [4,5], which aim at the achievement of thermonuclear ignition. Two different approaches are investigated to reach high-gain fusion: indirect drive and direct drive. In the first case [5], laser energy is converted into x-rays inside a hohlraum and the implosion of the fuel capsule is driven by the thermal x-ray bath confined in the hohlraum. In direct drive [1, 6, 7], the laser beams heat the capsule directly and drive the spherical implosion. Indirect drive is known to be less efficient than direct drive. On the other hand, direct drive is usually considered to be more sensitive to hydrodynamic instabilities. Even if the LMJ has been designed primarily for indirect drive, direct drive will also be addressed as an alternative way of providing high-gain fusion. In this paper, we present a direct-drive target-design for the LMJ, which has been developed recently. a Author to whom any correspondence should be addressed. b Present address: Département de Lasers de Puissance, CEA/DAM CESTA, BP2, F Le Barp, France. A direct-drive target for LMJ was presented a few years ago [8, 9]. It consisted of a cryogenic DT layer and a CH plastic ablator. Unfortunately, this target was found to be very sensitive to Rayleigh Taylor instability (RTI) at the ablation front. Indeed, the ablation velocity of CH is rather low, due to the relatively high CH density, and such a low velocity results in modest ablative reduction of the classical RTI growth rate. In order to increase the ablative stabilization while maintaining a high laser absorption rate, an interesting solution, first studied by Sacks and Darling [10] and recently re-examined [11 15], is to use a CH-foam ablator filled with DT ice. The structure of such an ablator material consists of small-sized open cells in which the DT is trapped during the target filling and cooling. This structure allows for good mechanical properties, high absorption efficiency (due to the presence of carbon atoms) and low density, close to the low-density natural limit of pure DT ice. The low density results in high ablation velocity and thus significant ablative stabilization of RTI. The proposed target design consists of a cryogenic DT-shell, surrounded by a thick DT-wetted foam, with the central void filled with DT gas. The outer low-density CH-foam (ρ 10 mg cm 3 ) is filled with frozen DT (ρ = 0.25gcm 3 ) and the resulting total average density is ρ 0.258gcm 3. The inner part of the capsule is filled with a low density DT gas at 0.3 mg cm 3. The ablator is surrounded by /04/ $ IAEA, Vienna Printed in the UK 1118

2 High-gain direct-drive target design for the LMJ 1642 μm 1641 μm 1439 μm 1275 μm Figure 1. High-gain direct-drive LMJ-target designs. The grey part indicates the area where pulses produce more than 80% of the design gain. a thin ( 1 µm) plastic layer to sustain the target assembly (see figure 1). Before analysing and optimizing the target performance, in section 2 we analyse the heterogeneous aspect of the foam and its impact on shock propagation. The foam density has been chosen to be sufficiently low so as not to alter significantly the shock velocity and the shock chronometry. One-dimensional target optimization is then presented in section 3. In section 4, the hydrodynamic stability of the capsule is analysed. We consider both the RTI developing at the ablation front during inward acceleration and the RTI at the inner surface of the shell during the final deceleration. The ablation-front RTI is analysed by the Goncharov Betti Takabe [16 18] model applied to one-dimensional flow. The stability of the inner surface (i.e. hot spot front) is examined by means of the modified-takabe-like [18, 19] expression for the growth rate, following [20], and by means of full two-dimensional hydrodynamic simulations as in [21]. A time-resolved analysis of the two-dimensional RTI growth rate shows a disagreement with the Takabe growth rate. In section 5 we discuss how the LMJ focal shape is adapted to the target design to keep the irradiation uniformity below 1% during the whole implosion. The results of two-dimensional hydrodynamic simulations with three-dimensional laser raytracing are presented in section 6. Such a realistic treatment of laser interaction shows a reduction in the efficiency of laser coupling to the target, and then a reduction in the implosion velocity. This requires a rescaling of the target design. We also show how hot spot low-mode asymmetries are related to the ablation front mode spectrum, which is dependent on the focal shapes of the beams. 2. Modelling the foam ablator The usual approach to deal with foams in a Lagrangian hydrodynamics code is to assume their homogeneity. However, previous studies [22, 23] have shown that the heterogeneity of a CH-foam filled with DT ice can affect the shock velocity for a wide range of low foam densities due to energy stored in residual vorticity. One consequence is an undercompression of the foam for long time scales. In order to evaluate the impact of heterogeneity on shock propagation in our target design we have performed numerical simulations with the two-dimensional Eulerian code HERA, which uses adaptative mesh refinement (AMR) and an interface tracking method [24]. In our design, the foam is assumed to have an initial density of 10 mg cm 3, before filling by DT. The foam structure appears like a solid fibre tangle with open cells. To model this aspect we have considered a Gaussian distribution in size of transversely-infinite dense-ch cylinders, spatially immersed in a two-dimensional Cartesian DT-ice slab. The positions of the cylinders are randomly chosen keeping an average open cell size of 1 µm and a mean density of 10 mg cm 3. A SESAME equation of state [26] is used for both materials. A strong shock is launched and propagates through the mixture with velocity and pressure relevant to our target design. The evolution of the mixture behind the shock and the shock velocity have been compared to a homogeneous case as described in [24]. This comparison shows no noticeable difference in the shock velocity, which is confirmed by an analytical approach [24]: D/D h < 0.5%. Here, D h is the homogeneous shock velocity and D is the difference in velocities between the heterogeneous and homogeneous cases. The modification of the shock velocity would only affect, in a first approximation, the shock chronometry of the design and would be taken into account in our one-dimensional optimization by adapting the laser pulse shape. But, we emphasize that the increase in the shock velocity is so small that if only slightly disturbs the shock chronometry by less than 70 ps and can be neglected. Moreover, the heterogeneity generates fluctuations of density in the ablator behind the shock, which could seed RTI at the ablation front. However, the acceleration of the ablation front occurs late after the first shock has passed through the foam. The fluctuations in density relax to very low levels (well below 1%) with a mixing time that is small compared to the time scales involved in the implosion. We can, therefore, assume, initially, that the foam can be considered as a homogeneous medium at the ablation front. 1119

3 B. Canaud et al Figure 2. Capsule gain ( ) and implosion velocity (- ---) versus the drive duration. The one-dimensional design position is shown. 3. One-dimensional target optimization The one-dimensional-optimization is based on simulations by the hydrodynamics code FCI2 [25]. Material properties are described by the tabular SESAME [26] equation of state. The laser energy is deposited by inverse bremsstrahlung absorption along radially incident rays propagated with a ray-tracing algorithm. Flux-limited thermal flux (with harmonic limitation to a fraction f = 0.06 of the free streaming value), multigroup radiative transfer, thermonuclear burn and multigroup fusionproduct transport are included in these simulations. Target specifications and pulse shape are shown in figure1. The laser pulse has been shaped in two parts to ensure an isentropic implosion: a first part (foot), with low power, launches a first shock in the pellet, placing the DT ice on the desired isentrope α. (Here, α is defined as the ratio of the material pressure to the Fermi degenerate pressure at the same density and zero temperature.) The second part of the laser pulse is the main drive, which accelerates the shell up to the desired implosion velocity. This last part is defined by four temporal points (t i,p i ), i {1, 4} and a drive delay t drive adjusted to provide the highest gain. The optimization of the laser pulse shape has been done with one-dimensional calculations and a constrained Monte Carlo procedure of temporal selection [27]. In this procedure, each point is randomly chosen uniformly in a bounded interval shown by dotted lines in figure 1 under the constraint t i <t i+1. This procedure is computationally intensive but gives useful information on gain variations with pulse shape. For instance, keeping the previous constraint, any laser pulse defined by four temporal points chosen in the grey part of the plot in the rising part (see figure 1) will produce a one-dimensional gain G>80 for implosion velocities between and cm s 1. Here, following Levedahl and Lindl [28], we define G as the ratio of the fusion yield to the energy absorbed by the capsule. Moreover, the effect of drive duration t drive on the implosion velocity and capsule gain is summarized in figure 2. Here, it can be seen that G 100 is achieved with an implosion velocity of about cm s 1 and a main drive pulse duration between 1.75 and 2.25 ns. Figure 3. Shock propagation versus time in the initial target reference frame. The lines represent the pressure isocontours. In our design, a first shock is launched at the beginning of the laser-pulse foot and reaches the inner face of the DT ice at approximately 7.8 ns. During the flight of this first shock, the laser pulse ramps up and launches several weak shocks that do not change the entropy of the shell significantly. These successive shocks encounter the rarefaction waves that originate when previous shocks cross the DT ice/gas interface as seen in figure 3. They are slowed down by the returning waves. In conclusion, the one-dimensional optimized laser pulse has a foot of 4.8 ns duration and power of 6 TW, followed by a pulse ramp of 4.8 ns at the end of which the laser power reaches the peak power of 290 TW. The main pulse at 290 TW has a duration of 2.1 ns. The one-dimensional predicted thermonuclear energy output for this design is 90 MJ for a 1 MJ laser-on-target pulse. This corresponds to a burn fraction of 27%. The laser absorption efficiency is evaluated to be 85%. The target is driven at an average in-flight isentrope α if = 2.6. The peak in-flight aspect ratio (IFAR, defined as the in-flight ratio of the centre of mass position of the non-ablated shell to its thickness) is 26 and the convergence ratio (initial radius divided by the hot-spot radius at stagnation) is 27. The capsule gain is G = 103. The ignition margin, which can be defined [28] as the ratio of the kinetic energy of the shell when the target ignites to the maximum in-flight kinetic energy of the shell (see figure 4), is 38%, close to that of the target designed by the Laboratory for Laser Energetics [29] ( 40%). During burn, when G = 1, the margin is still approximately 20%. This margin indicates earlier ignition, occurring well before the stagnation. The thermonuclear ignition and burn indeed stop the implosion, and stagnation occurs about 150 ps before the case simulated without thermonuclear reactions. This effect is important for the target dynamics during the deceleration. For instance, since burn starts before the end of the compression stage, the peak ρ dr decreases from 2.3 g cm 2 without fusion reactions to 1.6 g cm 2 in the 1120

4 High-gain direct-drive target design for the LMJ Figure 4. Excess kinetic energy for the non-burning ( ) and the burning (- ---)targets. The thermonuclear gain ( ) is also plotted. The origin of time is the beginning of the deceleration (12.1 ns). Figure 5. YON versus the time delay (days) between the manufacturing and the implosion. presence of reactions. On the other hand, the peak density of the compressed shell increases from 550 to 1200 g cm 3 when fusion reactions are taken into account. Indeed, a portion of the ignited target is compressed by the kinetic energy in excess and by the shocks associated to ignition. The large mass of DT ice ( 2.4 mg) contained in this target leads to a dependence of target behaviour on the time interval between fabrication and shot, since tritium decays into He 3 with a half-life of years. Even in a period as short as one month, the helium produced via tritium β-decay is trapped in the target and diffuses towards the capsule centre increasing the density of the gas to a level sufficient to deteriorate the target performance. As an example, the density at the centre of the target increases from 0.3 to 1.06 mg cm 3 over 1 month. Figure 5 shows the modification of the target YON (yield over new) versus time (in days) between the target fabrication and its use. It can be seen that the thermonuclear gain is not really affected during the first ten days, while it decreases strongly between 10 and 25 days, reaching 50% of the initial value after 20 days. Twenty-seven days after the manufacturing of the capsule, the gain becomes as small as Target hydrodynamic stability 4.1. Ablation front RTI The choice of the CH-foam ablator is motivated by the need to improve the hydrodynamic stability of the target during the acceleration phase. Indeed, a design based on a CH-foam is expected to be more stable than one using a dense CH ablator, due to the increased ablation velocity, which approaches that of an All DT target. The one-dimensional simulations (see figure 6) show that during the foot and part of the ramp of the laser pulse the ablation front travels at constant velocity. The acceleration rises to 13 µmns 2 at t = 5 ns, then decreases, and increases Figure 6. One-dimensional flow chart of the acceleration stage of the implosion. again to about 100 µmns 2. This produces conditions for the growth of the RTI seeded by laser non-uniformity (imprint) and target-surface roughness. The ablation velocity grows with laser power and a peak value of 3 µmns 1 is reached at t = 9 ns. We have studied the ablation front RTI of the considered target by means of a post-processor applied to the unperturbed flow computed by one-dimensional FCI2 simulations. The post-processor is based on the Goncharov Betti model [16,17], which gives a fully analytical expression of the ablative RTI growth rate in the linear regime. For simplicity, and without 1121

5 B. Canaud et al Figure 7. Iso-contours of linear Rayleigh Taylor growth rate (ns 1 ) at the ablation front versus the mode number and time. substantial loss of accuracy, such an expression is fitted by a modified form of the well-known modified Takabe formula [2, 18, 19] γ = α kg 1+kL min βkv a, (1) where v a, g and L min are the ablation velocity, the ablation front acceleration and the minimum density-gradient length, respectively. The quantities α and β are parameters depending on the one-dimensional flow quantities. In the present case, the procedure described in [30] leads to α = 0.9 and β = 2.4. The results of our computations are presented in figure 7, in the form of a contour plot of the RTI linear growth rate as a function of mode number and time. The figure shows that the maximum growth rate, γ = 3.5 ns 1, is achieved between 9 and 10 ns for the mode number 550. The growth rate of the fastest growing modes then stays between 3 and 3.5 ns 1 in a time interval of about 1.5 ns. The total amplification of RTI modes is then calculated using the formula A l /A l0 = exp( t stag 0 γ(t)dt) and plotted in figure 8. A maximum amplification of 10 4 is obtained for the mode 160 and is significantly lower than the one obtained with a CH ablator (>10 5 ) [30]. Its effects on target perturbations have then been computed using Haan s model of saturation [31], and assuming for the target the same surface spectrum known as NOVA roughness mode spectrum [32] given by 0.19 A l0 (µm) = (l/4) 4 + (l/60) (l/60) (l/1200). 8 The amplitude of saturation is defined in the Haan model by A sat (t) = 2r a (t)/l 2 where r a (t) is the ablation front radius. The amplitude of deformation is calculated as follows: A l (t) = A lin (t) = A l0 e t 0 γ(t ) dt for A l (t)<a sat (t), A l (t) = A sat (t) ( 1+ln A lin(t) A sat (t) ) (2) for A l (t)>a sat (t). Figure 8. Initial ( ) and final mode amplitudes versus mode number, with ( ) and without ( ) saturation. The total amplification is also plotted ( ). The results are shown in figure 8, and refer to the maximum amplitude of the perturbation, which is achieved just before ( 11.4 ns) the end of the laser pulse and the beginning of the deceleration. The solid curve corresponds to root-mean-square (rms) rugosity with saturation of σ rms = ( l ((2l +1)/4π)A2 l )1/2 13 µm. We have compared the rugosity without saturation on the same plot (σ rms 27 µm). Safe implosion requires the perturbation to be small compared to the shell thickness. Indeed, estimating the relative perturbation as the ratio of a typical amplitude, 2σ rms,to the shell thickness, we find a peak value of 10%. Note that this value is substantially smaller than what would be achieved by the use of a solid CH plastic ablator [30], and fully justifies the choice of a DT-filled low-density CH-foam ablator. Nevertheless, following [29], we plan to complete this preliminary analysis by including the effects of irradiation nonuniformities such as power imbalance, imprint or mispointing and by including the target gain dependance versus the sum-in quadrature σ Deceleration-phase inner surface RTI The deceleration stage begins at time t = 12.1 ns and lasts for about 450 ps. During this phase, part of the DT ice is heated and ablated by the energy flow from the hot spot. This is due to electron conduction and, as the temperature and density become high enough, to the fusion alpha particles leaving the hot spot. Note that the DT-ice mass ablated during the deceleration stage represents roughly 8% of the initial DT-ice mass, and is much larger than the mass of the gas initially contained inside the shell. Figure 9 shows the time evolution of the hot-spot radius R hs, the acceleration g of the shell s inner surface, of the ablation velocity v a, of the minimum value of the density gradient scalelength and of the Froude number defined by Fr = va 2/gL min. R hs is defined as the location of the minimum of the density scalelength. Such a definition represents a good approximation as long as the density profile is steep at the shell interface. It can be seen 1122

6 High-gain direct-drive target design for the LMJ Figure 9. One-dimensional hydrodynamics data of the final stage of implosion. The plotted quantities are defined in the main text. Figure 11. Calculations initialized with a perturbation mode number l = 16. Density profiles (labelled with ρ) and perturbation amplitude as functions of the radius for a set of selected times between 12.2 and 12.5 ns (alternate plain and dashed lines). The black dots indicate the hot-spot surface positions. occurring for l 20, increase in time, and close to stagnation become larger than 10 ns 1. In addition to the above analysis, we have performed two-dimensional numerical simulations using the hydrodynamic code DUED [33 35], which includes both flux-limited electron conduction and diffusion of fusion alpha-particles. The two-dimensional simulations concern the stages of deceleration and burn (12 t 12.5 ns). Initial conditions were taken from one-dimensional FCI2 simulations at t = 12.0 ns. At the beginning of the two-dimensional simulation the DUED mesh was perturbed along the radial direction by the quantity δr(r, θ, t = 0) = A l (r, t = 0)P l [cos(θ)] (3) Figure 10. Iso-contours of linear Rayleigh Taylor growth rate (ns 1 ) at the hot-spot surface versus the mode number and time. that the acceleration increases rapidly, reaching a value of about 5000 µmns 2 at the time of fuel burn. In the same time interval, the ablation velocity of the shell s inner layer increases too. The density gradient scalelength is about L min 1 µm. The Froude number increases to 0.4 at t = 12.3 ns and decreases afterwards. We assume that deceleration phase RTI growth can be described by a Takabe-like expression, i.e. by equation (1) with α = 0.9 and β = 1.4 as used in [20]. The results of the computation are shown in figure 10, presenting growth rate contour levels in the l t plane. We see that at about ns modes up to about l = 120 are unstable. At this stage, the ablation velocity is low (v a 8 µmns 1 ) and, therefore, stabilization only occurs at very large l. Peak growth rates, with l being the mode number, P l the Legendre polynomial and θ the angle between the radial direction and the symmetry axis. The radial part of the perturbation decreases exponentially along the radial direction according to A l (r, t = 0) = A l0 exp[ l r R hs (t = 0) /R hs (t = 0)], where A l0 is the maximum initial amplitude located at the hot-spot radius R hs. As regards the numerical accuracy, a detailed parametric study has confirmed the adequacy of the spatial mesh and of the integration time step used in the present RTI analysis. A series of simulations have been performed varying the mode number l between 2 and 64 and considering a very small amplitude single-mode perturbation, A l0 = 10 3 µm. In figure 11 the density profile and the perturbation amplitude A l (r, t) are shown as a function of the capsule radius r for the case l = 16, and for a set of selected times between t = ns and the stagnation time t = ns. Black dots mark the hot-spot radius. It appears that the maximum perturbation (i.e. the maximum of A l (r, t)) is always located at a radius r m inside the hot spot (r m <R hs ). It is also found that the distance R hs r m increases from 0 to a maximum of about 12 µm at about t = ns and then decreases to 8 µmatthe 1123

7 B. Canaud et al M Figure 12. Normalized amplitude of the perturbation versus time for mode 12 ( ), mode 24 ( ), mode 40 ( ), and mode 64 ( ). Mode 64 is shown to be never amplified during the last stage ( 200 ps) of the deceleration and can be a signature of the cut-off. stagnation time. These results are analogous to those detailed in [21], where an indirect-drive target was considered. Figure 9 shows a strong variation of hydrodynamic data (acceleration, ablation velocity, minimum gradient length and Froude number) and suggests a time-resolved analysis of the deformation growth at the hot-spot surface. In figure 12 we show the evolution of the perturbation at the hot-spot surface for different values of l. We see that mode l = 64 is stable. Modes l = 12, 24 and 40 show nearly exponential growth for t 12.3 ns. Anyhow, the amplification of the perturbation at r(t) = R hs (t) remains at a low level ( 4). Assuming the amplitude of the mode number l grows roughly exponentially, A l (r = R hs,t) = A l (t) = A l0 exp( t 0 γ(t ) dt ), we have computed the time-resolved growth rate defined by γ(t)= dln(a l (t)/a l0 )/dt. The result is shown in figure 13. We observe four stages in the evolution. At first, 12 t 12.1 ns, the mode amplitudes decrease in time. There is no RTI. Then, for 12.2 t 12.3 ns, a mode spectrum appears with quasi-zero growth rates for low modes. At t = 12.2ns modes up to l = 20 are stable and the maximum of γ is located around the mode number 35, at a low level ( 2 ns 1 ). In the subsequent 100 ps the maximum of γ(l)moves to lower modes (l 20), keeping approximately the same value. After t = 12.3 ns and in the following 150 ps, γ increases quickly and the spectrum maintains the same shape. The maximum stays located around the mode l = 20 and reaches 9ns 1 at ns, while the growth of higher-l modes is substantially reduced. Modes with l 64 are stable. We point out that this is the first simulation clearly showing the cut-off of the RTI spectrum first predicted by [18]. The last stage concerns the last 50 ps. At this time the target has ignited and burns. The RTI becomes strongly Figure 13. Growth rate of the perturbed amplitude evaluated at the hot-spot surface at different times. disturbed by the thermonuclear reaction and the target disassembly. In order to compare the dispersion relation with theoretical models, we have focused our attention on the third stage, i.e. the time interval 12.3 ns t ns, and considered three different times: t = 12.35, 12.4 and ns. To compare these results with equation (1) we have used time-resolved values of the acceleration, ablation velocity and density gradient scalelength. Moreover, following [20], we have set α = 0.9 and β = 1.4. The results are shown in figure 14. Dots represent DUED results and the plain line the Takabelike formula. We see that this curve has the same qualitative behaviour (i.e. with a maximum and a cut-off ) as the simulation results, but with locations of the maximum and the cut-off different from the two-dimensional results. We tried to adjust both the constants (α and β) without any success, the maximum and the cut-off being impossible to get at the same time. From the previous results, we can conclude that the Takabe formula overestimates the growth of the instability and does not match the two-dimensional simulation results for the growth rate of the perturbation at the hot spot boundary r = R hs (t). To estimate the overall growth of the perturbation during deceleration, we have carried out a calculation analogous to that performed for the acceleration stage. The NIF ice roughness spectrum [36] used to initialize the perturbation of the hot-spot surface is given by 1 A l,m (µm) = A l0 = 3l l. 4 To estimate the total growth of the deformation, the final mode amplifications obtained from the previous DUED calculations are applied to the ice roughness spectrum. We have verified that the instability never reaches the saturation regime defined by the dashed curve in figure 15, where A sat = min(2r hs /l 2,R hs /10l). In the same plot we have plotted the initial mode spectrum (dotted curve), the total growth of the deformation deduced from DUED monomode calculations (dots) and the general formula A l0 exp( γ t), 1124

8 High-gain direct-drive target design for the LMJ A 0 *exp ( γ t) M Figure 15. Total amplification of the perturbation during the deceleration with a ice roughness spectrum. The long-dashed curve represents the amplitude of saturation. The squares are obtained from two-dimensional DUED calculations. Dotted curve represents the initial roughness spectrum. transit or cavity oscillations. We plan to analyse such processes in the near future. 5. Target designs and focal shape Figure 14. Time-resolved growth rate at the hot-spot surface at three different times: (a) 12.35, (b) 12.4 and (c) ns deduced from DUED calculations (dots with error bars), and obtained from equation (1) (α = 0.9, β = 1.4, ). where γ = 1/t t 0 γ dt is the time-averaged growth rate. These calculations lead to a final rugosity of σ rms 1.7µm for an initial rugosity of 1 µm. The above estimate is only indicative, and probably optimistic, since the hot-spot surface perturbation at the beginning of the deceleration is the same as the initial ice surface perturbation and does not take into account other sources of perturbation such as feed-in from the ablation front, Richtmyer Meshkov instabilities due to shock The LMJ facility will be a 240-beamlet frequency-tripled (3ω = 0.35 µm) Nd : glass laser system with a nominal energy of 1.8 MJ and a maximum power of 600 TW. Four beamlets are assembled in one beam. Beam positioning for direct drive has been defined as a compromise between optimal configuration and technological constraints [37] and is described in detail in [38]. The overall non-uniformity has been investigated for an ideal case of perfect beams, having a super-gaussian spatial shape. An optimum beam pattern has been found, which minimizes the intrinsic rms non-uniformity at a level smaller than 0.25%. The corresponding beam placement (a three-cone-like bunch distribution with equatorial planar quasi-symmetry) and the optimized beam pattern eliminate odd modes and create dominant modes 6 and 8. Using the same assumptions as in [38], we consider axially symmetric beams with m-super- Gaussian transverse profiles [f(r)= exp( (r 2 /d 2 ) m )] where d is the half-width at 1/e. Each beam is assumed to deliver the same energy and laser power. Their axes are centred on the centre of the target. However, the implosion dynamics modifies the rms non-uniformity since the absorption region is displaced during the implosion. The time dependent rms non-uniformity follows a vertical straight line in the (m, d/r) plane and stays below 1% during the implosion history. Finally, beam focal shape and target design are strongly dependent on each other. They have to be optimized in order to achieve high gain using the LMJ incident energy of 1.5 MJ, and with the further constraints of keeping both the irradiation non-uniformities and the laser energy losses at a small level. If P 0 (t) is the on-target power, the beam power is related to P 0 by P(t) = [ 1 Ɣ(1/m,(r2 a (t)/d2 ) m ] 1 ) P 0 (t), (4) Ɣ(1/m) 1125

9 B. Canaud et al Figure 16. Iso-contours (plain curves) of LMJ incident energy (in MJ) and σ = 1% curve versus super-gaussian parameters: m and d/r 0. The grey part indicates where σ>1%. where Ɣ[a,x] and Ɣ[x] are the incomplete and complete Gamma functions [39], respectively. In this formula, the absorption zone is assumed to be localized at radius r a, near the critical density radius r c. The distance between r a and r c increases in time during the implosion, to reach 40 µm, at the end of the laser pulse. This represents at most 3% of r c and can be considered negligible. During the whole implosion, a part of the laser energy is then lost at the side of the target as r a (t) evolves from r 0 to 0.4r 0. The total incident energy is the time integral of P(t)estimated by timeintegrating equation (4). Using P 0 (t) given by the previous one-dimensional optimization, the total laser energy becomes a function of the two parameters m and d/r 0 where r 0 (see figure 16) is the initial radius of the target. Requiring the laser non-uniformity to stay below 1% during the whole implosion, defines a forbidden zone in parameter space (see the grey area in figure 16). It appears that the choice m = 3.1 and d/r 0 = 0.83, indicated by the black dot in figure 16, allows us to deliver an adequate incident energy of 1.5 MJ with non-uniformity σ < 0.9% during the whole implosion. 6. Two-dimensional hydrodynamics calculations with three-dimensional laser ray-tracing Unfortunately, the previous assumption of laser absorption near critical density is altered by the three-dimensional aspect of the problem. In order to understand the influence of the propagation of laser rays in the corona, we have performed two-dimensional hydrodynamics calculations of the previous design with FCI2 and a fully three-dimensional ray-tracing algorithm [40]. In these calculations we have taken into account the beam positioning by dealing with the six conelike aspect of the LMJ. Focal shapes are simulated by a super- Gaussian function with parameters m and d. At each time step, Figure 17. Visualization of the three-dimensional ray-tracing projected onto a two-dimensional plane. rays are randomly shot and transported from a lens to the beam focus through the plasma as seen in figure17. The lenses are placed at the six different cone angles. Rays are refracted by the plasma and energy is deposited by inverse bremsstrahlung along the ray path. We have investigated the implosion of the target in three cases, with different focal shapes f : (a) m = 3.1 and d = 0.83r 0, (b) an optimal case with m = 2 and d = 0.97r 0 and (c) a flat-top case (m = ) with d = r 0. All these shapes give irradiation non-uniformities smaller than 1%. We have then modified the three-dimensional laser pulse in order to take into account the differences in laser target coupling, leading to a required total energy of (a) 1.57, (b) 2.1 and (c) 2.2 MJ. Oblique incidence results in increased refraction on the ray path. The absorption rate is higher than when rays are focused at the target centre (one-dimensional geometry). However, the zone of maximum absorption is displaced outwards from the critical density radius, due to the oblique propagation of rays relative to the density profile. The distance between the radius of maximum absorption in the three-dimensional simulation and the radius of maximum absorption in one-dimensional simulations evolves in time from 160 up to 330 µm during the main drive part of the laser pulse, corresponding to an increase of the absorption surface ( r 2 ) varying between 20% and 80%. The absorbed intensity is reduced by a factor up to 1.7 in three dimensions compared to that in one dimension. As a result, the ablated mass is lower in the two-dimensional simulation than in one dimension even if the total absorbed energy is greater in two dimensions. As a consequence, the implosion velocity decreases between the one- and twodimensional calculations, from to cm s 1 leading to a dramatic reduction of the ignition margin, which now drops below 10%. This important effect on the implosion velocity of threedimensional ray tracing has to be addressed in terms of target 1126

10 High-gain direct-drive target design for the LMJ Table 1. Comparison of the three designs adapted to the three different focal shapes: (a), (b) and (c). Target #1 #2 #3 Focal shape (a) (b) (c) External radius (µm) Foam thickness (µm) DT ice layer (µm) In-flight adiabat Capsule gain Total laser energy (MJ) Thermonuclear energy (MJ) Figure 18. Thermonuclear energy versus the kinetic energy for the homothetic target family. The first one-dimensional design, the one-dimensional redesign obtained after the scaling, and the results in two-dimensional geometry are indicated. design. In order to keep the implosion velocity constant, the laser energy has to be increased. To ensure an implosion velocity of cm s 1, the total energy in the three different focal shapes described earlier has to be increased to (a) 1.9 MJ, (b) 2.4 MJ and (c) 2.5 MJ, respectively. Since LMJ cannot deliver energy in excess of 2 MJ, we have to adapt the target design to the available energy and to the focal shape. This is done by scaling the size of the target and the shape of the laser pulse as shown in table 1. These designs have been chosen in the homothetic family parameterized by the in-flight isentrope and the implosion velocity. It is built by a homothetic transformation where time and size scale as f 1/3, and f is the homothetic factor defined by f = 1 for the marginally igniting target (G = 1). Energy and mass scale as f, and laser power as f 2/3. The on-target intensity and the IFAR are invariants of the transformation. Figure 18 shows the variation of the capsule gain G versus the shell kinetic energy. This is found to follow a well-known trend [41]. The marginally igniting target requires a kinetic energy of 12 kj. Our two-dimensional designs are placed at a factor f 5in case (a), and f 4 in cases (b) and (c), to be compared with f 6 for the one-dimensional design. The other interest of two-dimensional calculations concerns the relation between low-mode implosion asymmetries and irradiation asymmetries. As shown in figure 19(a) for case (a), the relative importance of different modes evolves in Figure 19. Temporal evolution of the mode spectrum for the on-target irradiation (a) and for the ablation front radius (b). time, resulting in a mixing of low-mode asymmetries. At the beginning of the laser pulse, modes 6 and 4 are dominant. Mode 4 decreases rapidly and inverts its phase earlier than mode 6. During the main drive laser pulse, mode 6 inverts its phase and mode 4 becomes the dominant mode. We have compared this temporal evolution with the time behaviour of the mode spectrum at the ablation front for the same case (a) in figure 19(b). The ablation asymmetry is principally due to mode 4 but the remaining mode 2 still shows a moderate amplitude. Mode 6 appears to have a smaller contribution. But when we look at the time evolution of the hot spot mode spectrum in figure 20, we see that mode 4 and mode 6 are the dominant modes and mode 2 stays at a very low level during the deceleration phase. This behaviour is due to the mixed contributions of the ablation front asymmetries and of the deformations of the DT ice gas interface by the successive shocks. Another interesting comparison is done by analysing the three different cases (a), (b) and (c) with three different focal shapes. Figure 20 shows the computed hot spot mode spectrum for these three cases. Note that even though the irradiation uniformity seems to be better in the case (a), the mode spectrum of the hot spot is better in the case (c). 1127

11 B. Canaud et al Figure 20. Temporal evolution of the hot-spot mode spectrum for focal shapes (a) (open symbols), (b) (grey symbols), and (c) (black symbols). Mode 2 is represented by squares, mode 4 by circles and mode 6 by triangles. 7. Conclusions Various aspects of the design of a direct-drive target for the LMJ have been investigated. The target employs a DT-wetted ablator and a cryogenic DT layer. The foam heterogeneity is found to have no effects on the shock speed. Onedimensional optimization based on a Monte Carlo procedure has led to a baseline target design, which seems almost as hydrodynamically stable as the All DT target [29]. The deceleration phase appears to be more stable than predicted by modified Takabe-like formulae, and compatible with the achievement of ignition with presently achievable DT-ice roughness. Two-dimensional fluid calculations with fully three-dimensional ray-tracing have shown that the absorption zone is displaced in the corona farther out from the critical density than in one-dimensional calculations, leading to a reduction of the absorbed intensity. To keep the implosion velocity at the design value of cm s 1, the target has to be scaled down to a lower mass of DT ice. The relation between focal shape of beams and target performance has also been addressed and the low-mode deformations of the hot spot have been related to the time-integrated ablation front asymmetries. Acknowledgments We wish to thank C. Bayer, M. Primout and E. Lefebvre for valuable discussions and comments, and H. Jourdren for making the code HERA available to us. One of us (M.T.) was supported by the Ministerio de Ciencia y Tecnologia of Spain (Grant no FTN ). Three of us (B.C., M.T. and S.A.) would like to thank P.A. Holstein especially for fruitful discussions and comments on the RTI section. References [1] Nuckolls J., Wood L., Thiessen A. and Zimmerman G Laser compression of matter to super-high densities: thermonuclear applications Nature [2] Lindl J.D Development of the indirect-drive approach to inertial confinement fusion and the target physics basis for ignition and gain Phys. Plasmas [3] Pellat R Ignition: a dual challenge Inertial Fusion Sciences and Applications 2001 ed K.A. Tanaka et al (Elsevier: Paris) pp [4] Tarter C.B Inertial fusion and high-energy-density science in the United States Inertial Fusion Sciences and Applications 2001 ed K.A. Tanaka et al (Elsevier: Paris) pp 9 16 [5] Lindl J.D The physics basis for ignition using indirect-drive targets on the National Ignition Facility Phys. Plasmas [6] Bodner S.E. et al 1998 Direct drive laser fusion: status and prospects Phys. Plasmas [7] McCrory R.L. et al 2001 Omega ICF experiments and preparation for direct drive ignition on NIF Nucl. Fusion [8] Bayer C. and Juraszek D private communication [9] Fortin X. and Canaud B Direct Drive Laser Fusion Calculations at CEA: Proc. 1st Inertial Fusion Sciences and Applications 1999 ed C. Labaune et al (Elsevier: Paris) pp [10] Sacks R.A. and Darling D.H Direct drive cryogenic ICF capsules employing D-T wetted foam Nucl. Fusion [11] Bodner S.E., Colombant D.G., Schmitt A.J. and Klapisch M High-gain direct-drive target designs for laser fusion Phys. Plasmas [12] Skupsky S. et al 2002 High-gain direct-drive target designs for the National Ignition Facility Inertial Fusion Sciences and Applications 2001 ed K.A. Tanaka et al (Elsevier: Paris) pp [13] Takeda T., Mima K., Norimatsu T., Nagatomo H. and Nishiguchi A Implosion simulations of foam-buffered target for ignition experiments Inertial Fusion Sciences and Applications 2001 ed K.A. Tanaka et al (Elsevier: Paris) pp

12 High-gain direct-drive target design for the LMJ [14] Norimatsu T., Nagai K., Takeda T. and Yamanaka T Foam insulated direct-drive cryogenic target Inertial Fusion Sciences and Applications 2001 ed K.A. Tanaka et al (Elsevier: Paris) pp [15] Bodner S.E., Colombant D.G., Schmitt A.J., Gardner J.H., Lehmberg R.H. and Obenschain S.P Overview of new high-gain target design for a laser fusion power plant Fusion Eng. Des [16] Betti R. et al 1998 Growth rates of the ablative Rayleigh Taylor instability in inertial confinement fusion Phys. Plasmas [17] Goncharov V. N Self-consistent stability analysis of ablation fronts in inertial confinement fusion PhD Thesis University of Rochester, Rochester [18] Takabe H., Mima K., Montierth L. and Morse R.L Self-consistent growth rate of Rayleigh Taylor instability in an ablatively accelerating plasma Phys. Fluids [19] Munro D.H Analytic solutions for Rayleigh Taylor growth-rates in smooth density gradients Phys. Rev. A [20] Lobatchev V. and Betti R Ablative stabilization of the deceleration phase Rayleigh Taylor instability Phys. Rev. Lett [21] Atzeni S. and Temporal M Mechanism of growth reduction of the deceleration-phase ablative Rayleigh Taylor instability Phys. Rev. E [22] Kotelnikov A.D. and Montgomery D.C Shock induced turbulence in composite materials at moderate Reynolds numbers Phys. Fluids [23] Hazak G., Velikovitch A.L. and Gardner J.H Shock propagation in a low-density foam filled with fluid Phys. Plasmas [24] Philippe F., Canaud B., Fortin X., Garaude F. and Jourdren H Effects of microstructure on shock propagation in foams Laser Part. Beams [25] Buresi E. et al 1986 Laser Part. Beams [26] SESAME: The LANL equation of state database 1992 Los Alamos National Laboratory, LA-UR [27] Primout M Optimization of x-ray conversion efficiency of laser-preformed metallic plasma Phys. Plasmas submitted [28] Levedahl W.K. and Lindl J.D Energy scaling of inertial confinement fusion targets for ignition and high gain Nucl. Fusion [29] McKenty P.W. et al 2001 Analysis of a direct-drive ignition capsule designed for the National Ignition Facility Phys. Plasmas [30] Meyer C., Canaud B., Fortin X. and Garaude F Hydrodynamic stability analysis of direct drive high-gain fusion capsules for the laser Mégajoule Poster KP st Annual Conf. on Plasma Physics Division of the American Physical Society (APS) (Monterey, USA) [31] Haan S.W Onset of nonlinear saturation for Rayleigh Taylor growth in the presence of a full spectrum of modes Phys. Rev. A [32] Cook R.C., McEachern R.L. and Stephens R.B Representative surface profile power spectra from capsules used in nova and omega implosion experiments Fusion Technol [33] Atzeni S D Lagrangian studies of symmetry and stability of laser fusion-targets Comput. Phys. Commun [34] Atzeni S The physical basis for numerical fluid simulations in laser fusion Plasma Phys. control. Fusion [35] Atzeni S. and Ciampi M.L Burn performance of fast ignited, tritium-poor ICF fuels Nucl. Fusion [36] Marinak, M.M., Kerbel G.D., Gentile N.A., Jones O., Pollaine S., Dittrich T.R. and Haan S.W Three-dimensional HYDRA simulations of National Ignition Facility targets Phys. Plasmas [37] H. Dumont et al 1995, 1996 CEA private communication [38] Canaud B., Dague N., Bocher J.L. and Fortin X Laser Mégajoule irradiation uniformity for direct drive Phys. Plasmas [39] Abramowitz M. and Stegun I.A Handbook of Mathematical Functions (New York: Dover) p 260 [40] Hedde T CEA private communication [41] Nakai S. and Takabe H Principles of inertial confinement fusion physics of implosion and the concept of inertial fusion energy Rep. Prog. Phys