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# Simulation and theory

Plasma-wakefield generation, wave-breaking and electron acceleration in a tenuous plasma can generally not be fully treated by analytic theory owing to a highly nonlinear motion of many billions of electrons at relativistic momenta and the complicated interplay between high-intensity laser pulses and large amplitude plasma wakes. Therefore, only numerical simulations provide means for virtually full-featured and instructive theoretical investigations.

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Particle-in-cell calculations

In the case of kinetic particle calculations for plasma-wakefield acceleration, particle-in-cell (PIC) codes have proven to be powerful tools, which have helped to advance the field tremendously. PIC codes describe the laser-plasma-interaction environment in close approximation to the actual system. In a real plasma, many individual particles interact with each other by self-consistently generated electro-magnetic fields. In PIC simulations, these fields are discretized on a multi-dimensional spatial grid, whereas particles move continuously. Furthermore, the number of particles is significantly reduced compared to an actual plasma by typically merging 10^{3} to 10^{5} physical particles of the same species into one macroparticle, which features the same charge-to-mass ratio. Those macroparticles may be viewed as finite-sized clouds of an ensemble of real particles at similar speed. Such a macroparticle approach is mandatory to reduce the requirements on computational power. This simplified PIC picture is intuitive, leads to an applicable description of plasma-acceleration physics, and by today’s standards represents the most complete manner of accessing wakefield acceleration in theory. The working principle of PIC codes may briefly be summarized in the following way. Starting from initial (macro)particle distributions and currents, and from discrete, initial conditions representing the state of the electro-magnetic fields on the grid, Maxwell’s equations are solved. Thus, the field grid is modified and subsequently acts on the particles. Consequently each particle is moved inside the simulation volume according to the calculated solution of its equation of motion driven by the electro-magnetic grid potentials. The generated particle streams represent currents and set up new charge-density distributions. This again requires the solution of Maxwell’s equations and initiates the next calculation cycle.

In that process, the duration of the discrete time-step, for which the particle motion and field propagation is calculated, can usually not be chosen arbitrarily. It must obey the Courant-Friedrichs-Lewy condition, which couples spatial and temporal resolution and takes care of physically correct dispersion effects in laser propagation direction. This coupling of the discretization of space and time has an important consequence. Information only needs to be exchanged between neighboring grid cells, which makes PIC simulations perfectly suitable for large-scale parallelization without causing too substantial data flow. Nowadays, PIC calculations are extensively used in most branches of laser-plasma physics.

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OSIRIS 2.0 and 3.0

Our group is deploying the particle-in-cell codes OSIRIS 2.0 and 3.0 in close collaboration with the developing OSIRIS Consortium, consisting of the University of California in Los Angeles (UCLA, United States) and the Instituto Superior Técnico de Lisboa (IST, Portugal). The framework is installed and run on high-performance computers at the computing centers of DESY in Hamburg and Zeuthen.

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**HiPACE: a quasi-static particle-in-cell code**

The Highly efficient Plasma Accelerator Emulation (HiPACE) code has been created and developed by T. Mehrling. It is a relativistic, electromagnetic, three-dimensional and fully parallelized particle-in-cell (PIC) code which uses the quasi-static approximation to efficiently simulate a variety of beam-driven plasma-wakefield acceleration scenarios.

HiPACE exploits the disparity of time scales in the interaction of highly relativistic particle beams with plasma to decouple beam and plasma evolution. This enables time steps which are many times greater than those used in full PIC codes and thus, allows for a significant reduction of the computational time (core hours).

Comparisons to the fully explicit PIC code OSIRIS show the capability of the quasi-static PIC code to consistently simulate problems in beam-driven plasma acceleration while reducing the required number of core hours by orders of magnitude.

The code is officially presented in this article.

This work outlines the physical basis, describes the numerical implementation and assesses the parallel performance of the code which in combination lead to high computational efficiency.

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Publications

*"HiPACE: a quasi-static particle-in-cell code."*T. Mehrling et al., Plasma Phys. Control. Fusion 56 084012 (2014). [link].*"High-quality electron beams from field-induced ionization injection in the strong blow-out regime of beam-driven plasma accelerators"*A. M. de la Ossa et al. Nucl. Instrum. Meth. A 740, 231-235 (2013). [link].*"Beam-driven plasma-based acceleration of electrons with density down-ramp injection at FLASHForward"*J. Grebenyuk et al. Nucl. Instrum. Meth. A 740, 246-249 (2013). [link].*"High-Quality Electron Beams from Beam-Driven Plasma Accelerators by Wakefield-Induced Ionization Injection."*A. M. de la Ossa et al. Phys. Rev. Lett. 111, 245003 (2013). [link].*"Transverse emittance growth in staged laser-wakefield acceleration"*T. Mehrling et al. Phys. Rev. ST Accel. Beams 15, 111303 [link].*"Simulations of laser-wakefield acceleration with external electron-bunch injection for REGAE experiments at DESY"*J. Grebenyuk et al. AIP Conf. Proc. 1507, 688 (2012) [link].*"Self-modulation of long electron beams in plasma at PITZ"*A. M. de la Ossa et al. AIP Conf. Proc. 1507, 588 (2012) [link].