Abstract
1. Introduction
One of the main goals of inertial confinement fusion (ICF)[
A crucial issue concerns the uniformity of the capsule irradiation. A successful capsule implosion requires a very uniform irradiation and capsule target; otherwise, the imploding shell suffers the growth of dangerous hydrodynamic instabilities (Richtmyer–Meshkov[
Alternative schemes are currently under study, such as fast ignition induced by laser accelerated electrons[
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Due to its indirect drive design, the LMJ facility does not provide a favourable laser beam configuration for direct drive irradiation. Nevertheless, this large facility is very attractive for direct drive studies because of its large available energy. In this context, this paper aims to chart a path starting from the current characteristics of the LMJ facility and exploring the potential of the shock ignition scheme. After summarizing the main characteristics of the LMJ facility in Section
2. The Laser MegaJoule configuration
The configuration of the Laser MegaJoule facility considered in this paper consists of 176 high-power laser beams. These beams are grouped in 44 identical quads, each one composed by four beams. 40 quads are distributed into the spherical experimental chamber in four axial symmetric rings, and the two hemispheres are rotated by . The two rings closer to the polar axis have an angle of and with each having 10 quads; another 20 quads are located in the rings at and , as shown in Figure
The polar coordinates of the quads have been optimized for the indirect drive scheme. Nevertheless, this laser beam distribution could be helpful also in the direct drive shock ignition scheme. Indeed, as already mentioned, this scheme involves two laser pulses: one for the capsule compression and a second one for the fuel ignition. Thus, there are several options in order to use the LMJ facility as a direct drive facility in the context of the SI scheme. Hereafter, we will consider two options: option A, where 20 quads of the second ring are devoted to the compression of the capsule and with the other 24 quads driving the high-power shock ignition pulse; and option B, with only 10 quads of the second ring devoted to the foot pulse while together with the remaining 34 quads contributing to the drive and the igniting pulse. The main difference between the two options concerns the role of the different quads in the partition of laser power during the low-power foot pulse, the main drive of the compression phase, and the shock ignition phase.
Of course, the choice of the irradiation configuration also has consequences on the irradiation uniformity. Details of these configurations are given in the temporal power pulse sketched in Figure
The division of tasks among the different quads also allows implementing in a natural way both the polar direct drive (PDD)[
It is worth noting that the Orion facility in the UK is composed of 12 beams: two laser beams provide 500 J each at (1054 nm) in a short pulse of 0.5 ps and the other ten provide a total energy of 5 kJ (, nm) in 1–5 ns long pulses. The angular positions of these ten beams are indicated by gray circles in Figure
3. Shock ignition calculations
A relatively large direct drive capsule characterized by an initial aspect ratio has been considered. This capsule is part of a family of capsules that have been recently studied[
Here, the capsule has been used in the context of the shock ignition scheme, and a series of 1D numerical calculations has been performed with the hydro-radiative MULTI code[
In Figure
The radial position where the density is equal to the critical value [] has also been calculated, and it is shown by the dashed red curve in Figure
Two parametric studies have been performed, varying the starting time, , of the shock ignition pulse and the maximum incident power, . In a first case, we used the usual laser wavelength nm during the whole calculations and for each couple of parameters, and , the final energy gain has been calculated. In a second set of calculations the laser wavelength has been doubled during the shock ignition pulse – i.e., when – providing the gain . The colour maps in Figure
4. Illumination non-uniformity
The shock ignition scheme is less demanding than the central ignition one with respect to the uniformity of the irradiation[
In this section, we analyse some of the behaviour of the irradiation by using the illumination model[
In reality, laser beams suffer from unavoidable errors such as beam-to-beam power imbalance , laser pointing error , and error in the target positioning . These errors are statistical quantities that in the case of the LMJ facility are estimated by the standard deviations: % (beam-to-beam), , and . In the LMJ facility, the laser beams are grouped in quads; thus the power imbalance benefits from a statistical factor which reduces it to (quad-to-quad). The illumination non-uniformity, evaluated taking into account these beam uncertainties, is usually measured as an average value () estimated over a large number of calculations[
As has been already said, the LMJ facility is devoted to the indirect drive scheme. This means that the laser beam directions as well as their intensity profile fit with the
In the first set of calculations we considered the non-uniformity provided by the 20 quads located in the second ring of the LMJ facility (option A in Section
It has been already shown[
The specific configurations given by the 10 or 20 quads located in the second ring () of the LMJ facility are not optimized for direct drive irradiation. Nevertheless, it is worth noting that the polar angle of is relatively close to the optimum value, , as found by Schmitt[
As previously mentioned, to improve the laser–capsule coupling, the polar direct drive technique has been proposed. In this case, the laser beams are re-directed towards the equator by a quantity in order to balance the irradiation between polar and equatorial areas. The displacement is also indicated in the sketch of Figure
Another set of calculations has been performed to evaluate the sensitivity of the illumination non-uniformity with respect to a variation of the beam uncertainties , , and . These calculations use the laser intensity profile envisaged for the LMJ facility (, ), a capsule radius , and a PDD parameter %. The average non-uniformities are shown in Figure
A final detailed parametric study has been performed to evaluate the sensitivity of the average illumination non-uniformity to a variation of the PDD parameter and of the super-Gaussian exponent of the laser intensity profile. As in the previous case, the capsule radius has been set to and the elliptical intensity profile is characterized by the widths and . The average non-uniformity, which takes into account the beam uncertainties, is shown as a function of the parameters and in Figure
5. 3D hydrodynamic simulations
Detailed two-dimensional (2D) hydrodynamic numerical simulations are usually employed to analyse the irradiation, compression, ignition, and thermonuclear burn wave propagation in an ICF capsule. Nevertheless, most actual laser–target configurations are intrinsically three-dimensional (3D) systems, and these have motivated the development of 3D hydrodynamic numerical tools[
A first issue is when a 3D configuration can be correctly described as a 2D axis-symmetric problem. For this purpose, a configuration with a number of laser beams in each ring has been considered. In these calculations, the spherical capsule described in Section
The results of these 3D calculations are summarized in Figures
6. Conclusions
The Laser MegaJoule facility has been considered in the context of the shock ignition scheme. Two laser beam configurations have been analysed. A first option (A) uses 20 quads – 80 laser beams (600 kJ, 200 TW) locate at the second ring of the LMJ facility – for the compression of the capsule, making available the remaining 24 quads – 96 laser beams (720 kJ, 240 TW) – for the additional shock ignition pulse. A second option (B) envisages the possibility to use only 10 quads for the compression phase and 34 quads for the compression and SI phases. The total available laser power is 440 TW at ( nm).
A classical ICF capsule – devoted to the central ignition scheme – has been used in the context of the shock ignition scheme. A set of mono-dimensional numerical simulations has been performed to enlighten some aspect of the shock ignition scheme. For this specific capsule it is found that the threshold power in the shock ignition pulse is about 250 TW. Nevertheless, assuming that all this power is incident to the surface of the critical density provides incident intensity larger than . At these large intensities () we expect saturation of dangerous laser–plasma instabilities (SRS, SBS, and TPD) that modify the laser energy deposition mechanism. In this new regime, a large fraction of the laser energy is transferred to high-energetic electrons, and the photon penetration depth is limited to a quarter of the critical density (), instead of the classical limit, . These physical mechanisms are not included in our numerical tools; however, we performed some calculations to estimate the effect caused by limiting the deposition of the laser energy in the region at lower density (). To mimic this effect, the light wavelength during the shock ignition pulse has been artificially doubled (); thus, because , the critical density becomes a quarter. As expected, this affects negatively the power threshold in the shock ignition pulse that now increases to about 400 TW. This should be considered as a pessimistic estimation. In fact, none of the positive effects associated with the high-energetic electrons are included in our calculations.
The second issue addressed in the paper concerns the irradiation uniformity provided during the first few ns of the foot pulse. First, it has been shown that the elliptical laser intensity profile of the LMJ facility provides better results in comparison to the usually circular profile. The two LMJ options A and B have been considered, taking into account beam uncertainties such as quad-to-quad power imbalance (%), pointing error (), and target positioning (). Both of these configurations cause an over-irradiation of the capsule polar regions in detriment to the equatorial area. In order to improve these schemes, the polar direct drive technique has been applied to optimize the irradiation uniformity. It has been found that for the elliptical laser intensity profile (, , ) expected at the LMJ facility the optimal capsule radius is , and this provides an average illumination non-uniformity of % and 4.9% in case A and case B, respectively. These minimum non-uniformities correspond to the use of a PDD parameter %. This capsule radius is relatively small in comparison to the available LMJ energy and the requirements for typical ignition capsule designs; however, bigger capsules could be envisaged assuming larger focal spots provided by either defocusing of the laser beams or using an alternative set of phase plates.
A 3D version of the code MULTI has been used to perform a set of preliminary hydrodynamic calculations. The LMJ options A and B have been considered in these calculations, and the laser irradiation uniformity has been split into the azimuthal and polar components by means of decomposition in spherical harmonics. For the analysed laser–capsule configuration – with the laser intensity profile that reduces to at the initial capsule radius – it is found that the azimuthal component is negligible in the case of option A (ten beams per hemisphere). This encouraging result seems indicates that a 2D analysis is appropriate in option A, while in the second case, option B, it may not be. Of course these conclusions depend on the beam and capsule sizes, and further investigations are needed for specific configurations.
Finally, the two LMJ options A and B involve the use of 10 or 20 quads located in the second rings characterized by the polar angles and . These options are in many aspects similar to the configuration already available at the Orion facility, where ten laser beams are located at and . In addition, these ten ns-long laser beams operate at the wavelength () as in the LMJ facility. The similarity between the two installations motivates us to stress the opportunity to perform Orion’s experiments addressed to PDD issues of interest also for future direct drive LMJ campaigns. Indeed, although of relatively small energy – 5 kJ in few ns for the ten long-pulse Orion beams – this installation is fully adequate for direct drive experiments that may explore the laser–capsule coupling as well as the uniformity and timing of the first shock wave generated during the low-power (TW) ns-long foot pulse needed to control the initial imprint phase of an ICF implosion, thus helping to underwrite modelling of polar direct drive implosions.
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