Numerical simulations of laser wakefield accelerators in optimal Lorentz frames
Introduction
In a laser-wakefield accelerator (LWFA) [1], the laser propagates in a transparent plasma for distances that range from a few millimeters to several meters long [2], [3]. The strong nonlinear effects constitute a challenge for theoretical approaches, so that numerical simulations, leveraged on phenomenological scaling laws (e.g., [2], [4], [5]), are critical to explore these scenarios. In this context, full particle-in-cell (PIC) simulations play a very important role, since the algorithm includes all the essential physics, by coupling the full set of Maxwell equations with the Lorentz force on relativistic particles. Nevertheless, analytical scalings [2], [3] indicate that the next generation of LWFA experiments will require propagation distances on the meter scale in order to achieve electron bunch energies above 50 GeV. Since the laser wavelength () still needs to be resolved in full PIC simulations, this implies a very large number of algorithm iterations, and therefore demands large computational resources. With these limitations, the numerical modeling of the next generation of laser systems with standard PIC techniques is not possible: a full-scale three-dimensional PIC simulation in the blowout regime of a 250 J system, to be deployed in several laboratories around the World in the next few years, may require several millions of CPU hours, with particles pushed for time iterations. In this context, several reduced-codes have been developed (e.g., [6], [7], [8]) with physical approximations that enable numerical algorithm simplification and a strong reduction of computational requirements. While this approach is applicable to several relevant scenarios, it may lack important physics in particular configurations, namely in self-injection regimes [4]. Recently, a new approach has been suggested where no physical approximations are performed: it was shown that the time and space scales to be resolved in a numerical simulation may be minimized by using an optimal Lorentz frame, which is typically different from the laboratory frame [9]. The choice of frame is indeed relevant in several scenarios, not only to speed up calculations (as already successfully obtained for free electron laser simulations [10]), but also to simplify the interpretation of the physical processes (as in astrophysical shocks [11], [12]). The potential application and estimated computational speedups were discussed in [9], [13], [14], [15], [16] for LWFA simulations, but no complete implementation details and result benchmarks were reported. In this paper, we fully describe the application of the boosted frame to three-dimensional one-to-one LWFA PIC simulations with intense laser beams propagating in underdense plasmas.
Numerical experiments of laser–plasma interactions are usually performed in the laboratory frame where the plasma is at rest [2], [4], [17]. In this configuration, the laser wavelength is the smallest spatial structure to be resolved, and thus defines the longitudinal grid resolution. This, however, implies an additional computational cost due to an over-resolution of the plasma structures. Fig. 1 depicts how this scale gap may be eliminated if the simulation is performed in a relativistic frame moving in the laser direction, since the laser pulse wavelength increases and the plasma contracts, leading to gains both in space, because the plasma length is shorter, and in time, since the crossing time between the laser and plasma is also smaller.
The theoretical computational gains are proportional to [9] when using a moving window (numerical box follows the interest region around the laser pulse), where is the boost relativistic factor, and β is the boost velocity normalized to the speed of light. The speedup is a conjugation of a cell size increase by the laser stretch of , a contraction of the plasma column by γ, and the drift of the particles with velocity β. We emphasize that, in a moving window configuration, the boosted frame scheme only reduces the number of iterations required; the number of particles to process at each time step is equal to that of the laboratory case, and might actually be set larger to increase particle statistics. For scenarios where a moving window configuration cannot be used, numerical particles are reduced by and the overall computation gain is proportional to .
The implementation of the boosted frame scheme to laser–plasma interactions allows a quick numerical modeling of ongoing experiments, and goes towards a more complete full-scale three-dimensional numerical study of the next generation of LWFA systems, otherwise impossible with current configurations and computational resources available.
This paper is organized as follows. Section 2 introduces the concept of LWFA simulations in relativistic moving frames. Benchmarks between results in different frames and experimental data are presented in Section 3. Section 4 details numerical instabilities that may arise in the boosted frame configuration. Finally, conclusions are given in Section 5.
Section snippets
LWFA simulations in non-laboratory frames
In a LWFA [1], an intense laser pulse, with a normalized vector potential and a central wavelength , propagates in a transparent plasma (, where and , with n the electron density). As it propagates, the laser field generates electron plasma waves, with a phase velocity similar to the group velocity of the laser pulse. The space charge field associated with the electron density modulations can then accelerate the particles that surf the wave. For
OSIRIS benchmarks
The boosted frame scheme was implemented in OSIRIS [23], a three-dimensional fully relativistic PIC code, already used to simulate various LWFA scenarios (see, for instance, [4], [17], [24]). Several simulation runs were performed for benchmark, namely direct laboratory/boost result comparisons, benchmarks with QuickPIC [6] for several LWFA regimes, and direct comparison with experimental data. OSIRIS simulations in boosted frames were also used to model LWFA experiments of self-trapping,
Numerical instabilities
The simulation of a relativistically moving plasma can be a challenge for the stability of the numerical algorithms, and the system is more prone to noise build up than the typical LWFA configuration with the plasma at rest.
A particular problem may arise with current deposition precision, even without a laser present. If uniform, completely cold, electron and ion particle distributions drifting with the same velocity are used, the overall current must be exactly zero in all space, by
Conclusions
We have presented the implementation details to perform LWFA simulations in optimal Lorentz frames, different from the standard laboratory configuration, that enable a new generation of faster numerical experiments. The gain is obtained by bringing together the scales of the laser wavelength and of the plasma structures, as a direct consequence of relativistic frame transformations. Using a moving window that travels at the speed of light, the new scheme typically requires at least the same
Acknowledgements
This work was partially supported by Fundação Calouste Gulbenkian and by Fundação para a Ciência e a Tecnologia, under grants PTDC/FIS/66823/2006 and SFRH/BD/35749/2007 (Portugal). S.F.M. and L.O.S. would like to thank KITP, where a part of this research was concluded, partially supported by the National Science Foundation under Grant No. PHY05-51164. The simulations presented here were produced using the IST Cluster (IST/Portugal), the Dawson cluster (UCLA), and NERSC supercomputers. The
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