Particle-in-cell Code Research Articles

Diagnosis of fast electron transport by coherent transition radiation

Transport of fast electrons in overdense plasmas is of key importance in high energy density physics. However, it is challenging to diagnose the fast electron transport in experiments. In this article, we study coherent transition radiation (CTR) generated by fast electrons on the back surface of the target by using 2D and 3D first-principle particle-in-cell (PIC) simulations. In our simulations, aluminum targets of 2.7 g cc−1 are simulated in two different situations by using a newly developed high order implicit PIC code. Comparing realistic simulations containing collision and ionization effects, artificial simulations without taking collision and ionization effects into account significantly underestimate the energy loss of electron beams when transporting in the target, which fail to describe the complete characteristics of CTR produced by electron beams on the back surface of the target. Realistic simulations indicate the diameter of CTR increases when the thickness of the target is increased. This is attributed to synergetic energy losses of high flux fast electrons due to Ohm heating and colliding drags, which appear quite significant even when the thickness of the solid target only differs by micrometers. Especially, when the diagnosing position is fixed, we find that the intensity distribution of the CTR is also a function of time, with the diameter increased with time. As the diameter of CTR is related to the speed of electrons passing through the back surface of the target, our finding may be used as a new tool to diagnose the electron energy spectra near the surface of solid density plasmas.

Improved modellisation of laser–particle interaction in particle-in-cell simulations

A new method named B-TIS (Bourgeois & Davoine, J. Comput. Phys., vol. 413, 2020, 109426) has recently been proposed for suppressing the influence of numerical Cherenkov radiation that appears in particle-in-cell (PIC) simulation of laser wakefield acceleration (LWFA). However, while this method provides good results when applied to the already accelerated electrons, we show here that it cannot model correctly most of the plasma electron bulk interacting with the laser field. We thus investigate in this paper the origins of this limitation and propose an improved method for which this limitation is removed. This new method, named B-TIS3, can now be applied to a much broader variety of problems and improve the performance in comparison with the standard PIC algorithm. We show that, for an electron interacting directly with a laser pulse, this new technique offers greater accuracy in terms of momentum and motion than the conventional scheme used in many PIC codes. These improvements translate into more faithful energy spectrum and electric charge for the accelerated beam in simulations of vacuum laser acceleration (VLA) or LWFA involving direct laser acceleration (DLA) at low plasma density. This new method, easy to implement and not computationally demanding, should then prove useful to study in depth and help develop novel VLA, DLA and LWFA techniques.

Particle-in-Cell Modeling of Omega Experiments on Ablation of Plasmas

The large electric fields due to charge separation generated by a laser pulse propagating in a plasma can accelerate the electrons to very high energies. The hydrodynamics models that are based on the local thermodynamic equilibrium assuming a Maxwellian distribution of velocities cannot describe the evolution of groups of hot electrons of different energy ranges and kinetic particle-in-cell (PIC) simulations are required. The SMILEI PIC code is used to investigate the generation of hot electrons in plasmas produced by laser pulses in the Omega extended performance (EP) experiments. In 2-D PIC simulation using SMILEI code, a Gaussian-shaped laser pulse of 100-ps duration with the intensity of 10 <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$^{15}$</tex-math> </inline-formula> W/cm <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$^{2}$</tex-math> </inline-formula> and a wavelength of 351 nm impinges onto a plasma with the electron density and temperature profiles. The electromagnetic field, electron, and total energies of a plasma as a function of time are evaluated. The evolution of number density, electromagnetic fields, and energy spectra of electrons is studied. The time evolution of hot electrons in a plasma is investigated. These PIC results provide insights into the contribution and importance of the kinetic effects in the expansion dynamics of ablated plasmas, thereby helping to analyze and explain the Omega experimental data and computational fluid dynamics results.

Concept of the generalized reduced-order particle-in-cell scheme and verification in an axial-azimuthal Hall thruster configuration

The reduced-order particle-in-cell (PIC) scheme is a novel modeling approach that enables computationally efficient electrostatic kinetic simulations of plasma. In our previous publications, we demonstrated that a proof-of-concept implementation of this novel PIC scheme resolves the multi-dimensional plasma processes and their interactions in a Hall thruster in a manner close to traditional electrostatic PIC codes. In this work, we extend our efforts on this topic and present a mathematically mature formulation for the dimensionality reduction of Poisson’s equation in the Vlasov–Poisson system, which enables the generalized reduced-order ‘quasi-multi-dimensional’ PIC scheme. The applicability of the dimensionality-reduction approach to solve general 2D Poisson problems is numerically verified. Next, we present several reduced-order quasi-2D (Q2D) simulations of a well-defined axial-azimuthal simulation case from the literature using approximation orders of the 2D problem whose computational costs are 2%–15% of a full-2D simulation. It is shown that these reduced-order simulations allow us to recover the same characteristics, behaviors and effects reported in the literature regarding the azimuthal instabilities in Hall thrusters. Moreover, in terms of the time-averaged plasma properties, it was found that, when increasing the approximation order, the error associated with the Q2D simulations’ predictions decreases from 15% to 4% for the electric field and from 20% to 2% for the ion number density. We have additionally discussed a series of sensitivity analysis results, including the influence of the initial number of macroparticles per cell on the predictions of the Q2D simulations. According to the detailed results and analyses presented, we conclude that the generalized reduced-order PIC scheme serves as a rigorous foundation for eventual cost-effective and comprehensive three-dimensional kinetic studies of the physics in Hall thrusters and similar electrostatic plasma technologies.

Verification of the generalized reduced-order particle-in-cell scheme in a radial–azimuthal E × B plasma configuration

In this article, we present an in-depth verification of the generalized electrostatic reduced-order particle-in-cell (PIC) scheme in a cross electric and magnetic field configuration representative of a radial–azimuthal section of a Hall thruster. The setup of the simulations follows a well-established benchmark case. The main purpose of this effort is to demonstrate that our novel PIC scheme can reliably resolve the complex two-dimensional dynamics and interactions of the plasma instabilities in the radial–azimuthal coordinates of a Hall thruster at a fraction of the computational cost compared to full-2D PIC codes. To this end, we first present the benchmarking of our newly developed full-2D PIC code. Next, we provide an overview of the reduced-order PIC scheme and the resulting “quasi-2D” code, specifying that the degree of order reduction in the quasi-2D PIC is defined in terms of the number of “regions” along the simulation’s directions used to divide the computational domain. We compare the predictions of the quasi-2D simulation in various approximation degrees of the 2D problem against our full-2D simulation results. We show that, with an increase in the number of regions in the Q2D simulations, the quasi-2D results converge to the 2D ones. Nonetheless, we also highlight that a quasi-2D simulation that provides a factor of 5 reduction in the computational cost resolves the underlying physical processes in an almost indistinguishable manner with respect to the full-2D simulation and incurs an L2-norm error of only about 2% in the ion number density and below 1% in the electron temperature.

High-flux positron generation via the ultra-intense laser irradiating density-modulated plasmas

To investigate plasma density during the Breit–Wheeler positron generation, a comparative study of four plasma targets is performed via the PIC (particle-in-cell) code EPOCH. When an ultra-intense laser (2.8×1023 Wcm−2) is incident, more positrons with high energy are generated in the increasing density plasmas. The positron yield is already 1.5×108 with a cutoff energy of 2 GeV at t=37T0. It is demonstrated that increasing density plasmas will enhance gamma photon radiation and positron generation. In increasing density plasmas, under-dense plasmas favor electron acceleration, and over-dense plasmas will induce laser reflection. Cross sections of the Compton back-scattering and the BW positron generation are both increased via high-energy electrons colliding with the reflected laser. In addition, increasing the laser intensity will directly enhance positron generation. This investigation will further facilitate high-flux positron generation and application.

New applications for the Boris Spectral Deferred Correction algorithm for plasma simulations

The paper investigates two new use cases for the Boris Spectral Deferred Corrections (Boris-SDC) time integrator for plasma simulations. First, we show that using Boris-SDC as a particle pusher in an electrostatic particle-in-cell (PIC) code can, at least in the linear regime, improve simulation accuracy compared with the standard second order Boris method. In some instances, the higher order of Boris-SDC even allows a much larger time step, leading to modest computational gains. Second, we propose a modification of Boris-SDC for the relativistic regime. Based on an implementation of Boris-SDC in the RUNKO PIC code, we demonstrate for a relativistic Penning trap that Boris-SDC retains its high order of convergence for velocities ranging from 0.5c to >0.99c.

Machine-learning-based models in particle-in-cell codes for advanced physics extensions

In this paper we propose a methodology for the efficient implementation of machine learning (ML)-based methods in particle-in-cell (PIC) codes, with a focus on Monte Carlo or statistical extensions to the PIC algorithm. The presented approach allows for neural networks to be developed in a Python environment, where advanced ML tools are readily available to proficiently train and test them. Those models are then efficiently deployed within highly scalable and fully parallelized PIC simulations during runtime. We demonstrate this methodology with a proof-of-concept implementation within the PIC code OSIRIS, where a fully connected neural network is used to replace a section of a Compton scattering module. We demonstrate that the ML-based method reproduces the results obtained with the conventional method and achieves better computational performance. These results offer a promising avenue for future applications of ML-based methods in PIC, particularly for physics extensions where a ML-based approach can provide a higher performance increase.

Time evolution and finite element representation of Phase Space Zonal Structures in ORB5

Phase Space Zonal Structures (PSZSs), obtained by averaging out dependencies on angle-like variables in the energetic particle (EP) distribution function, play a fundamental role in regulating EP transport induced by Alfvén instabilities in burning plasmas, acting as a slowly varying nonlinear equilibrium state. Therefore, they are of great interest for the development of reduced models for the description of EP heat and particle transport on long time scales, comparable with the energy confinement time, for future burning plasma experiments. In this work, we propose an efficient finite element based projection of the time evolution of the PSZS, suited for global particle-in-cell (PIC) gyrokinetic (GK) codes. The resulting algorithm has been implemented in the global GK PIC code ORB5. PSZSs can not only be used for validating reduced models, but also as a diagnostic tool for characterizing the nonlinear interaction in phase space between Alfvén instabilities and EPs in global GK simulations.

Laser Wakefield Photoneutron Generation with Few-Cycle High-Repetition-Rate Laser Systems

Simulations of photoneutron generation are presented for the anticipated experimental campaign at ELI-ALPS using the under-commissioning e-SYLOS beamline. Photoneutron generation is a three-step process starting with the creation of a relativistic electron beam which is converted to gamma radiation, which in turn generates neutrons via the γ,n interaction in high-Z material. Electrons are accelerated to relativistic energies using the laser wakefield acceleration (LWFA) mechanism. The LWFA process is simulated with a three-dimensional particle in cell code to generate an electron bunch of 100s pC charge from a 100 mJ, 9 fs laser interaction with a helium gas jet target. The resultant electron spectrum is transported through a lead sphere with the Monte Carlo N-Particle (MCNP) code to convert electrons to gammas and gammas to neutrons in a single simulation. A neutron yield of 3×107 per shot over 4π is achieved, with a corresponding neutron yield per kW of 6×1011 n/s/kW. The paper concludes with a discussion on the attractiveness of LWFA-driven photoneutron generation on high impact, and societal applications.

Kinetic theory of particle-in-cell simulation plasma and the ensemble averaging technique

We derive the kinetic theory of fluctuations in physically and numerically stable particle-in-cell (PIC) simulations of electrostatic plasmas. The starting point is the single-time correlation at the start of the simulation between the statistical fluctuations of the weighted densities of macroparticle centers in the plasma particle phase-space. The fluctuations are associated with different initial conditions, typically due to the random initial conditions (in velocity space) of the macroparticles/simulation plasma, assigned according to their initial distribution of probability. The single-time correlations at all time steps and in each spatial grid cell are then determined from the Laplace–Fourier transforms of the discretized Klimontovich-like equation for the macroparticles and Maxwell’s equations for the fields, as computed by modern PIC codes. We recover the expressions for the electrostatic field and the plasma particle density fluctuation autocorrelation spectra as well as the kinetic equations describing the average evolution of PIC-simulated plasma particles, first derived by Langdon (1970b Proc. 4th Conf. Numerical Simulation of Plasmas) using a test macroparticle approach perturbing a discretized Vlasovian plasma and then averaging the obtained physical quantity over the initial macroparticle velocity distribution. We generalize and extend these results to the modern algorithms in PIC codes using arbitrary macroparticle weights. Analytical estimates of statistical fluctuation amplitudes are derived as a function of the plasma simulation parameters, using the central limit theorem in the limit of a large number of macroparticles per cell. The theory is then used to analyze the ensemble averaging technique of PIC simulations where statistical averages are performed over ensembles of PIC simulations, modeling the same plasma physics problem but using different statistical realizations of the initial distribution functions of the macroparticles. This method is illustrated by linear Landau damping uncovering (from noise, which is usually considered numerical) the physical fluctuations driven by a single small amplitude electrostatic wave perturbing a PIC simulation plasma in equilibrium.

Integrating a ponderomotive guiding center algorithm into a quasi-static particle-in-cell code based on azimuthal mode decomposition

High fidelity modeling of plasma based acceleration (PBA) requires the use of three dimensional, fully nonlinear, and kinetic descriptions based on the particle-in-cell (PIC) method. In PBA an intense particle beam or laser (driver) propagates through a tenuous plasma whereby it excites a plasma wave wake. Three-dimensional PIC algorithms based on the quasi-static approximation (QSA) have been successfully applied to efficiently model the interaction between relativistic charged particle beams and plasma. In a QSA PIC algorithm, the plasma response to a charged particle beam or laser driver is calculated based on forces from the driver and self-consistent forces from the QSA form of Maxwell's equations. These fields are then used to advance the charged particle beam or laser forward by a large time step. Since the time step is not limited by the regular Courant-Friedrichs-Lewy (CFL) condition that constrains a standard 3D fully electromagnetic PIC code, a 3D QSA PIC code can achieve orders of magnitude speedup in performance. Recently, a new hybrid QSA PIC algorithm that combines another speedup technique known as an azimuthal Fourier decomposition has been proposed and implemented. This hybrid algorithm decomposes the electromagnetic fields, charge and current density into azimuthal harmonics and only the Fourier coefficients need to be updated, which can reduce the algorithmic complexity of a 3D code to that of a 2D code. Modeling the laser-plasma interaction in a full 3D electromagnetic PIC algorithm is very computationally expensive due the enormous disparity of physical scales to be resolved. In the QSA the laser is modeled using the ponderomotive guiding center (PGC) approach. We describe how to implement a PGC algorithm compatible for the QSA PIC algorithms based on the azimuthal mode expansion. This algorithm permits time steps orders of magnitude larger than the cell size and it can be asynchronously parallelized. Details on how this is implemented into the QSA PIC code that utilizes an azimuthal mode expansion, QPAD, are also described. Benchmarks and comparisons between a fully 3D explicit PIC code (OSIRIS), as well as a few examples related to laser wakefield acceleration, are presented.

Characterization of sub-picosecond laser-produced fast electrons by modeling angularly resolved bremsstrahlung measurements with 3D hybrid particle-in-cell code

Hard x-rays produced by intense laser-produced fast electrons interacting with solids are a vital source for producing radiographs of high-density objects and implosion cores for inertial confinement fusion. Accurate calculation of hard x-ray sources requires a three-dimensional (3D) simulation geometry that fully models the electron transport dynamics, including electron recirculation and the generation of absolute photon yields. To date, 3D simulations of laser-produced bremsstrahlung photons over tens of picoseconds and code benchmarking have not been performed definitively. In this study, we characterize sub-picosecond laser-produced fast electrons by modeling angularly resolved bremsstrahlung measurements for refluxing and non-refluxing targets using the 3D hybrid particle-in-cell (PIC), Large Scale Plasma code. Bremsstrahlung radiation and escaped electron data were obtained by focusing a 50-TW Leopard laser (15 J, 0.35 ps, 2 × 1019 W/cm2) on a 100-μm-thick Cu foil and a Cu with a large plastic backing (Cu–CH target). Data for both the Cu and Cu–CH targets were reproduced for simulations with a given set of electron parameters. Comparison of the simulations revealed that the hard x-ray emission from the Cu target was significantly longer in duration than that from the Cu–CH target. The benchmarked hybrid PIC code could prove to be a powerful tool in the design and optimization of time- and angular-dependent bremsstrahlung sources for flash x-ray and gamma-ray radiography.

Time-Domain Simulation of Helical Gyro-TWTs With Coupled Modes Method and 3-D Particle Beam

A new self-consistent time-domain model for the simulation of gyrotron traveling-wave tubes with a helically corrugated interaction space (helical gyro-TWTs) is presented. The new model links classical methods using the approach of slowly varying variables together with an expansion of the electromagnetic field in eigenmodes and advanced full-wave particle-in-cell (PIC) solvers. The aim is to significantly reduce the required calculation time compared to full-wave PIC solvers, while less strict assumptions are introduced as in the classical approaches of slowly varying variables. For the first time, the classical theory of coupled circular waveguide modes for the description of the operating electromagnetic eigenmode in the helical interaction space is combined with a 3-D PIC representation of the electron beam. This allows the simulation of the beam–wave interaction over a broad bandwidth and at arbitrary harmonics of the cyclotron frequency. In addition, arbitrary electron beams (with spreads, offsets of the guiding center from the symmetry axis, and so on) can be investigated. The new approach is compared with the full-wave 3-D PIC code CST Microwave Studio. A good agreement of the simulation results is achieved, while the computing time is significantly reduced.

CBETor: a hybrid-kinetic particle-in-cell code for cross-beam energy transfer simulation

The parametric instability related to ion motion and the resulting cross-beam energy transfer are important aspects in the physics of inertial confinement fusion. The numerical simulation of the above physical problems still faces great technical challenges. This paper introduces a 2D hybrid-kinetic particle-in-cell (PIC) code, CBETor. In this code, the motion of ions is described by the kinetic method, the motion of electrons is described by the simplified fluid method and the propagation of laser in plasma is described by solving the wave equation. We use CBETor and the popular fully kinetic PIC code EPOCH to simulate the stimulated Brillouin scattering and cross-beam energy transfer process, respectively. The physical images are in good agreement, but CBETor can significantly reduce the amount of calculation. With the premise of correctly simulating the ion dynamics, our hybrid-kinetic code can effectively suppress the noise of numerical simulation and significantly expand the simulation scale of physical problems. CBETor is very suitable for simulating the physical process dominated by ion motion in the interaction of medium intensity laser and underdense plasma.

A hybrid nodal-staggered pseudo-spectral electromagnetic particle-in-cell method with finite-order centering

Electromagnetic particle-in-cell (PIC) codes are widely used to perform computer simulations of a variety of physical systems, including fusion plasmas, astrophysical plasmas, plasma wakefield particle accelerators, and secondary photon sources driven by ultra-intense lasers. In a PIC code, Maxwell's equations are solved on a grid with a numerical method of choice. This article focuses on pseudo-spectral analytical time-domain (PSATD) algorithms and presents a novel hybrid PSATD PIC scheme that combines the respective advantages of standard nodal and staggered methods. The novelty of the hybrid scheme consists in using finite-order centering of grid quantities between nodal and staggered grids, in order to combine the solution of Maxwell's equations on a staggered grid with the deposition of charges and currents and the gathering of electromagnetic forces on a nodal grid. The correctness and performance of the novel hybrid scheme are assessed by means of numerical tests that employ different classes of PSATD equations in a variety of physical scenarios, ranging from the modeling of electron-positron pair creation in vacuum to the simulation of laser-driven and particle beam-driven plasma wakefield acceleration. It is shown that the novel hybrid scheme offers significant numerical and computational advantages, compared to purely nodal or staggered methods, for all the test cases presented.

Numerical suite for cathodeless plasma thrusters

This work focuses on a numerical suite targeted at cathodeless plasma thrusters. A Global Model is used for preliminary estimation of the propulsive performance. In the plasma source, the code 3D-VIRTUS solves self-consistently plasma transport with a fluid approach, including the electromagnetic wave propagation. Starfish and SPIS are used to handle the magnetic nozzle region via a fully kinetic Particle-in-Cell (PIC) model. Starfish solves the plasma dynamics in a two-dimensional axisymmetric domain and has been coupled to 3D-VIRTUS in order to estimate the propulsive performance. SPIS is a three-dimensional PIC code that has been used to simulate the mutual interactions between the satellite surfaces, the propulsion system plumes and the environmental plasma. Results have been benchmarked against thrust measurements performed on a low-power (50 W range) Helicon Plasma Thruster.

Particle-in-cell model of a relativistic magnetron system driven by a pulse forming network Marx generator

A three dimensional particle-in-cell (PIC) model of a relativistic magnetron (RM) is established using CST Studio Suite. As we adopted a novel technique to imitate pulse forming action of pulser, simulation domain is now fully extended to include a pulser, a transmission line, and a magnetron tube. This feature enables the model to describe a dynamic power coupling process between the pulser and RM in a self-consistent manner. It is demonstrated that model accuracy is strongly dependent on user-defined parameters for an explosive electron emission model. Particularly, emission rise time for space charge formation influences significantly on an initial mode competition phase. Compared to a previous excitation method using a step voltage pulse, the new model gives more reliable results as the intrinsic impedance of the pulser is now considered. Our model is able to estimate the major RM characteristics over a wide range of an external magnetic field. As a result of dynamic power coupling, the RM impedance, the gap voltage, the excitation frequency, and the output power tend to increase at a higher external magnetic field condition. It is suitable to adopt a proposed modeling technique for the virtual assemble and simulation of the system. Hence, it will suggest a practical way to implement and verify complicated structures or innovative designs that could not be solved by existing PIC codes.

PICSAR-QED: a Monte Carlo module to simulate strong-field quantum electrodynamics in particle-in-cell codes for exascale architectures

Physical scenarios where the electromagnetic fields are so strong that quantum electrodynamics (QED) plays a substantial role are one of the frontiers of contemporary plasma physics research. Investigating those scenarios requires state-of-the-art particle-in-cell (PIC) codes able to run on top high-performance computing (HPC) machines and, at the same time, able to simulate strong-field QED processes. This work presents the PICSAR-QED library, an open-source, portable implementation of a Monte Carlo module designed to provide modern PIC codes with the capability to simulate such processes, and optimized for HPC. Detailed tests and benchmarks are carried out to validate the physical models in PICSAR-QED, to study how numerical parameters affect such models, and to demonstrate its capability to run on different architectures (CPUs and GPUs). Its integration with WarpX, a state-of-the-art PIC code designed to deliver scalable performance on upcoming exascale supercomputers, is also discussed and validated against results from the existing literature.

Modeling of Bound Electron Effects in Particle-in-Cell Simulation

To include the bound electron effects in particle-in-cell (PIC) simulation, we propose a model in which the response of the dipole components of partially ionized ions to external electromagnetic fields can be included. Instead of treating the macro-ion particle as a single particle without an internal structure, the ions are considered to have a structure composed of a central nucleus and a bounded electron cloud in our model. The two parts experience the interactions of both the external electromagnetic fields and the internal Coulomb fields. In this way, the laser scattering effects by a partially ionized medium can be modeled properly in the PIC simulation. The laser propagation in a neutral medium and the Bragg scattering of the laser in crystal structure have been simulated with a PIC code modified based on our model as the benchmark. Our model may find applications to study some interesting problems, such as the x-ray laser-driven wakefield acceleration in crystals, the x-ray laser-driven high energy density physics, and intense laser propagation in partially ionized nonlinear optical materials, etc.