Optimal laser intensity profiles for a uniform target illumination in direct-drive inertial confinement fusion Download: 698次
1. Introduction
In the direct-drive (DD) inertial confinement fusion (ICF)[1, 2] context a spherical capsule containing the deuterium–tritium (DT) nuclear fuel is irradiated by laser beams. The final goal is to generate energy gain via a nuclear fusion reaction: . The external shell of the capsule absorbs a fraction of the incoming laser energy producing a plasma; the plasma temperature () increase provides the outward expansion of the low-density corona and launches a series of inward shock waves. These shock waves compress the DT payload in a high-density shell that implodes and reaches stagnation. In the classical central ignition scheme, the high-density shell confines a small amount () of DT fuel – called a hot-spot – which is heated to high temperature () and compressed to areal densities comparable with the -particle range (), thus providing the ignition of the thermonuclear fusion reactions. Recently, the new shock ignition (SI) scheme[3] has been proposed. In the SI scheme the fuel is first compressed by the usual DD technique, then a high-power laser pulse (hundreds of TW) is used to launch a strong shock wave which provides the fuel ignition.
In all ICF schemes the capsule irradiation must be very uniform in order to inhibit growth of dangerous hydrodynamic instabilities that can prevent a successful fuel compression. In the promising SI scheme the requirements in terms of irradiation uniformity are less stringent in comparison with the classical central ignition scheme. Nevertheless, the irradiation uniformity represents one of the major constraints in ICF and a great deal of effort has been dedicated to its optimization.
In this paper we propose a numerical method to calculate the optimal laser intensity profiles of a generic number of laser beams irradiating a spherical target. Analytical optimization of the laser intensity profiles has been already performed for configurations based on the geometry of the Platonic solids[4–7]. These analyses always provide axially symmetric laser intensity profiles where all the laser intensities are equal. These solutions can be applied to laser configurations such as Gekko XII[8] () or Omega[9] (60 beams) but are not suitable for laser configurations like the National Ignition Facility (NIF)[10], the Laser MegaJoule (LMJ)[11] or the smaller Orion[12] facility where the locations of the beams are optimized for the indirect-drive[13] ICF scheme. Indeed, in these latter cases the optimal laser intensity profiles must be adapted to the laser configuration. As a consequence, the laser intensity profiles are not necessarily equal to each other or axially symmetric.
2. Numerical method to optimize the intensity profiles
The proposed numerical method allows us to find the laser intensity profiles that optimize the illumination uniformity for a given laser configuration. These calculations are performed within an illumination model in which laser refraction is neglected, photons propagate linearly and the results only apply to the low-power foot-pulse that characterizes the first few ns of an ICF irradiation, the so-called imprint phase. Thus, the solution guarantees the uniformity of the first shock wavefront[14].
In the past, optimizing methods have usually been based on analytical or numerical parametric studies looking for the laser parameters that minimize the illumination nonuniformity. In contrast, in the present case a sort of predictor–corrector method is used: in a first step, trial laser intensity profiles are used to evaluate the – imperfect – irradiation of the spherical target; in a second step, the laser intensities are recalculated using the results of the first step in order to provide perfect illumination uniformity.
The model problem is characterized by a spherical target of radius irradiated by laser beams. The target centre is located at the origin of a Cartesian coordinate system and the laser beam directions are characterized by the unitary vector defined by their polar angles and (see the details of the geometry in Figure
Fig. 1. Spherical target and main coordinate system []; vector direction of a generic surface element and versor of the th laser beam, ; coordinate system [, ] for the th laser intensity profile.
The elementary surface , located at the polar coordinates , is irradiated by a given number () of laser beams and receives a total laser intensity . This intensity is given by the contributions of all the incoming intensities associated with the laser profiles multiplied by the scalar product to account for projection of the surface area. Thus, for the laser beams the irradiation of the spherical target surface is given by
Of course, the condition to obtain is to generate a perfectly uniform irradiation over the whole target surface, i.e., to realize , where is the desired intensity over the target surface. This could be done by a simple re-normalization of the laser intensity profiles, by
The optimal laser intensity profiles, solution of the coupled Equations (
3. Profiles for a two-ring 2D irradiation configuration
As a first example we considered a two-dimensional (2D) axially symmetric laser configuration where the beams can be approximated by two annular rings at the co-latitudes and ; this is achieved by imposing that . The two optimal intensity profiles and have been calculated for different values of . In this perfectly symmetric case, the two solutions are equal and are just rotated by , . The laser intensity profiles normalized to one and corresponding to the annular ring of the north hemisphere are shown in Figure
Fig. 2. Optimal laser intensity profiles (north hemisphere) for an axially symmetric beam configuration. The intensity profiles have been normalized to one () and the scale colour ranges from 0 to 1. Full dots correspond to the north pole and the grey curve is the equator projection on the focal planes.
In these frames the grey curves show the projection of the equator in the focal plane, while the full grey dots localize the projection of the north pole. In these images the laser intensity has been normalized to 1 and the colour scale ranges from 0 to 1. For small polar angles, e.g., , the laser beams are closer to the -axis, thus the surfaces of the polar areas are highly irradiated with a nearly orthogonal angle of incidence; on the contrary, for larger angles, it is the equatorial belt that will be over-irradiated in comparison with the polar regions. To compensate for this unbalanced irradiation the optimal laser intensity profile provides different intensities in correspondence to the equatorial and polar target areas. Specifically, at smaller angles (see, e.g., ) a maximum intensity is directed towards the equator, while at larger angles (e.g., ) the laser intensity is higher in proximity to the polar areas. It is worth noticing that the laser intensity profile becomes circular (axially symmetric) when equals the Schmitt angle (Ref. [6]). This is not surprising; indeed, as shown by Schmitt, the angle is the best co-latitude if the axisymmetric beam intensity profile is given by with .
4. Optimal profile for some LMJ configurations
An LMJ configuration consisting of 40 quads has been considered in this paper. The quad of the LMJ is composed of a bundle of four laser beams and provides a maximum laser energy (power) of 30 kJ (10 TW) at (). The polar coordinates of the 40 quads are shown in Figure
Fig. 3. Polar coordinates of 40 quads of the LMJ facility. Quads for the configurations A and B; red quads of the north hemisphere (C) and blue quads of the south hemisphere (D).
Fig. 4. Optimal laser intensity profiles normalized to one (north hemisphere) for the LMJ configurations A–D. The power imbalance is given by the parameter and the laser intensity scale colour varies linearly from 0 to 1.
The optimization method has been applied to the four laser configurations A–D. As above, all the trial intensities are given by the scalar product: . The optimal intensity profiles provided for these configurations are shown in Figure
The configuration A has only two beams per hemisphere, and their optimized intensity profile shows three zones at higher intensity: one located below the equator and the other two closer to the pole. The intensity profile for the ten beams of configuration B is shown in Figure
The NIF configuration has been also analysed, providing four optimal laser intensity profiles. In this case the 48 quads of the NIF facility are located at four rings in each hemisphere: four quads at , four at , eight at and eight at . The method of optimization, initialized with the trial intensity , produces intensity profiles similar to those found for configuration D. These calculations also supply the optimal power imbalances , , , while the maximum power () is assigned to the laser beams located at the larger angle .
5. Conclusions
In conclusion, we developed a general method to calculate the optimal laser intensity profiles that optimize the illumination nonuniformity of a spherical target. The method can be used for any DD laser configuration accounting for a general number of laser beams, provided that the beams irradiate the whole target surface. In some sense this is a kind of predictor–corrector method that consists of two steps: firstly, initialized by a set of trial laser intensity profiles, the imperfect surface irradiation is calculated; then, the beam profiles are recalculated in order to correct the previously estimated nonuniform illumination.
A set of four laser configurations based on the LMJ facility has been considered. In these cases, the optimal intensity profiles have been individuated using axially symmetric trial profiles. The resulting optimal intensity profiles are not axially symmetric and their shapes look like to those envisaged by the PDD technique; in addition, these calculations also predict the optimal beam-to-beam power imbalance. These results assume perfect beam-to-beam power imbalance, neglecting laser pointing errors and target positioning uncertainties; deviation from these idealized assumptions would damage the uniformity of the target illumination.
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Article Outline
Mauro Temporal1, Benoit Canaud2, Warren J. Garbett3, Rafael Ramis4. Optimal laser intensity profiles for a uniform target illumination in direct-drive inertial confinement fusion[J]. High Power Laser Science and Engineering, 2014, 2(4): 04000e37.