Three-dimensional Particle-in-cell Research Articles

VSHPIC: a particle-in-cell algorithm based on vector spherical harmonics expansion

The particle-in-cell (PIC) method has been widely used for studying plasma physics. However, fully three-dimensional PIC simulations always require huge computational resources. For problems with near azimuthal symmetry, recent work (Lifschitz et al 2009 J. Comput. Phys. 228 1803–14, Davidson et al 2015 J. Comput. Phys. 281 1063–77, Li et al 2021 Comput. Phys. Commun. 261 107784, Li et al 2022 J. Comput. Phys. 470 111599) has shown that expanding all the quantities defined on the grid in azimuthal harmonics and truncating the expansion can improve the code efficiency. In this paper, we describe a novel parallel algorithm for efficiently simulating three-dimensional near-spherical symmetry problems. Our approach expands all physical quantities in the θ and ϕ directions in spherical coordinates using vector spherical harmonics. The code is capable of simulating three-dimensional asymmetric scenarios by accurately tracking the evolution of distinct individual modes while preserving the charge conservation law. The fundamental dispersion relation of EM waves in the plasma has been obtained using VSHPIC simulation results. The code also shows a well strong scalability up to more than 1000 cores.

Study of particle acceleration by Laguerre–Gaussian ultra intense laser plasma interactions

In this work, simulations of multi-petawatt lasers in the range of ∼0.2 PW–∼100 PW with varying Laguerre–Gaussian (LG) azimuthal modes as well as linearly polarized (LP) and circularly polarized (CP) laser beams striking near critical density targets were studied by using three-dimensional particle in cell (PIC) codes. Particle acceleration mechanisms have a dependence on laser polarization and it affects the energy gained by the particles. It is known that laser pulses can be polarized helically by applying the LG distribution function to the fundamental Gaussian laser profile. In this study, differently polarized laser beams with varying powers were employed to study laser driven particle acceleration and compares accelerated charged particles’ energy spectra and angular distribution. It is seen that LG laser beams can accelerate higher energetic particles due to higher conversion efficiency compared to LP and CP laser beams. It is also seen that LG laser beams can collimate ions with a narrower spread compared to LP and CP beams. Furthermore, ions can have a smaller divergence angle with increasing azimuthal mode index when the laser is LG polarized. We also studied the energy deposition of these particles in a water cell obtained by the PIC codes for different laser parameters by using Geant4 Monte Carlo simulations which suggests that LG laser beam can be useful for the future hadron therapy applications.

Ion effects on the scaling of magnetic field amplification in plasmas with the system size

Magnetic field amplification during the nonlinear stage of the current filamentation instability excited by ultra-relativistic electron beams is investigated with a two-dimensional electromagnetic particle-in-cell (PIC) simulation code, with special attention paid to the effects of plasma ions and the system size. The effect of plasma ions is shown to be significant and enhanced magnetic field amplification and beam energy deposition are found due to plasma cavity expansion and merger. When the system size in the transverse direction (perpendicular to the beam propagation direction) is enlarged by a factor of m, the transverse magnetic field energy is found to increase by a factor of m 2 in the case of the plasma with movable ions, in contrast to m with immovable ions. The results are also confirmed by three-dimensional PIC simulations.

On the control of electron heating for optimal laser radiation pressure ion acceleration

We study the onset of electron heating in intense laser–solid interactions and its impact on the spectral quality of radiation pressure accelerated ions in both hole boring and light sail regimes. Two- and three-dimensional particle-in-cell (PIC) simulations are performed over a wide range of laser and target parameters and reveal how the pulse duration, profile, polarization and target surface stability control the electron heating, the dominant ion acceleration mechanisms and the ion spectra. We find that the onset of strong electron heating is associated with the growth of the Rayleigh–Taylor-like instability at the front surface and must be controlled to produce high-quality ion beams, even when circularly polarized lasers are employed. We define a threshold condition for the maximum duration of the laser pulse that allows mitigation of electron heating and radiation pressure acceleration of narrow energy spread ion beams. The model is validated by three-dimensional PIC simulations, and the few experimental studies that reported low energy spread radiation pressure accelerated ion beams appear to meet the derived criteria. The understanding provided by our work will be important in guiding future experimental developments, for example for the ultrashort laser pulses becoming available at state-of-the-art laser facilities, for which we predict that proton beams with $\sim$ 150–250 MeV, $\sim$ 30% energy spread, and a total laser-to-proton conversion efficiency of $\sim$ 20% can be produced.

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.

Inference of m-NLP data using radial basis function regression with center-evolving algorithm

An efficient technique is presented to infer space plasma density and satellite potentials from Langmuir probe measurements, using multivariate radial basis function (RBF) regression. This inference technique goes beyond analytic approaches which have been developed over nearly a century, and which remain in use in most lab and space plasma experiments. The method is assessed by applying it to synthetic data sets constructed with three-dimensional particle in cell (PIC) simulations of fixed-bias needle Langmuir probes proposed by Jacobsen, to determine a plasma parameter, independently of the temperature. Our approach follows machine learning techniques, whereby models are constructed on training data sets consisting of the simulated collected currents as a function of voltage, corresponding to known physical parameters such as plasma density and temperature, and satellite potential. Compared to standard approaches used in RBF regression, our approach proves to be particularly efficient when working with large training sets, by implementing an evolutive selection of optimal centers. Program summaryProgram title: CERBFCPC Library link to program files:https://doi.org/10.17632/6bskxt2xjj.1Code Ocean capsule:https://codeocean.com/capsule/3524673/Licensing provisions: MITProgramming language: Fortran 90Nature of problem: When constructing a radial basis function (RBF) regression inference model, the determination of the optimal set of centers in a given training data set can be very time consuming, owing to the large number of possible combinations of centers among entries in the set. The center evolving approach implemented in this program, greatly reduces the training time of a model, with minimal impact on model inference accuracy.Solution method: A new search strategy is implemented for finding an optimal set of centers used in radial basis function regression, which is particularly fast and efficient when working with large data sets. Instead of performing a single exhaustive search among all possible combinations of a given number of centers among a training set, successive exhaustive searches are made in randomly selected small subsets of the training sets, while carrying over optimal centers found in one search, to the next one. Using modest subsets, with a number of entries of order 20, RBF models can be constructed with inference accuracies close to ones obtained with more time consuming exhaustive searches.

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.

Enhanced coherent transition radiation from midinfrared-laser-driven microplasmas

We present a particle-in-cell (PIC) analysis of terahertz (THz) radiation by ultrafast plasma currents driven by relativistic-intensity laser pulses. We show that, while the I0{lambda }_{0}^{2} product of the laser intensity I0 and the laser wavelength λ0 plays the key role in the energy scaling of strong-field laser-plasma THz generation, the THz output energy, WTHz, does not follow the I0{lambda }_{0}^{2} scaling. Its behavior as a function of I0 and λ0 is instead much more complex. Our two- and three-dimensional PIC analysis shows that, for moderate, subrelativistic and weakly relativistic fields, WTHz(I0{lambda }_{0}^{2}) can be approximated as (I0λ02)α, with a suitable exponent α, as a clear signature of vacuum electron acceleration as a predominant physical mechanism whereby the energy of the laser driver is transferred to THz radiation. For strongly relativistic laser fields, on the other hand, WTHz(I0{lambda }_{0}^{2}) closely follows the scaling dictated by the relativistic electron laser ponderomotive potential {mathscr{F}}_{{text{e}}}, converging to WTHz ∝ {I}_{0}^{1/2}{lambda }_{0} for very high I0, thus indicating the decisive role of relativistic ponderomotive charge acceleration as a mechanism behind laser-to-THz energy conversion. Analysis of the electron distribution function shows that the temperature Te of hot laser-driven electrons bouncing back and forth between the plasma boundaries displays the same behavior as a function of I0 and λ0, altering its scaling from (I0λ02)α to that of {mathscr{F}}_{{text{e}}}, converging to WTHz ∝ {I}_{0}^{1/2}{lambda }_{0} for very high I0. These findings provide a clear physical picture of THz generation in relativistic and subrelativistic laser plasmas, suggesting the THz yield WTHz resolved as a function of I0 and λ0 as a meaningful measurable that can serve as a probe for the temperature Te of hot electrons in a vast class of laser–plasma interactions. Specifically, the α exponent of the best (I0λ02)α fit of the THz yield suggests a meaningful probe that can help identify the dominant physical mechanisms whereby the energy of the laser field is converted to the energy of plasma electrons.

Optimizing laser–plasma interactions for ion acceleration using particle-in-cell simulations and evolutionary algorithms

The development of ultra-intense laser-based sources of high energy ions is an important goal, with a variety of potential applications. One of the barriers to achieving this goal is the need to maximize the conversion efficiency from laser energy to ion energy. We apply a new approach to this problem, in which we use an evolutionary algorithm to optimize conversion efficiency by exploring variations of the target density profile with thousands of one-dimensional particle-in-cell (PIC) simulations. We then compare this ‘optimal’ target identified by the one-dimensional PIC simulations to more conventional choices, such as with an exponential scale length pre-plasma, with fully three-dimensional PIC simulations. The optimal target outperforms the conventional targets in terms of maximum ion energy by 20% and show a noticeable enhancement of conversion efficiency to >2 MeV ions. This target geometry enhances laser coupling to the electrons, while still allowing the laser to strongly reflect from an effectively thin target. These results underscore the potential for this statistics-driven approach to guide research into optimizing laser–plasma simulations and experiments.

Efficient data layouts for a three-dimensional electrostatic Particle-in-Cell code

The Particle-in-Cell (PIC) method is a widely used tool in plasma physics. To accurately solve realistic problems, the method requires to use trillions of particles and therefore, there is a strong demand for high performance code on modern architectures. The present work describes performance results of Pic-Vert, a hybrid OpenMP/MPI and vectorized three-dimensional electrostatic PIC code.The code simulates 3d3v Vlasov–Poisson systems on Cartesian grids with periodic boundary conditions. Overall, it processes 590 million particles/second on a 24-core Intel Skylake architecture, without hyper-threading (25 million particles per second per core).The paper presents extensions in 3d of our preliminary 2d results (Barsamian et al., 2017), with highlights on the difficulties and solutions proposed for these extensions. Specifically, our main contributions consist in proposing a new space-filling curve in 3d (called L6D) to improve the cache reuse and an adapted loop transformation (strip-mining) to achieve efficient vectorization. The analysis of these optimization strategies is performed in two-stages, first on a 24-core socket and second on a super-computer, from 1 to 3072 cores, demonstrating significant performance gains and very satisfactory weak scaling results of the code.

Large-scale parallel computing research on full three-dimensional PIC method based on difference equations

Electromagnetic particle-in-cell (PIC) simulation has been widely used in controlled thermonuclear fusion, space physics, free electron laser, high power microwave and the research of plasma. In this paper, basic algorithms and parallel algorithm used in the PIC simulation are studied respectively. And then a complete code implementation of the algorithms mentioned above, CHIPIC3D, is presented, which is written by the Fortran language to simulate the vacuum microwave source device. At last, several vacuum microwave source device simulation results acquired on the Milky Way high-performance platform are presented.

Development of a 14-vane, double-strapped, 5.8-GHz magnetron oscillator

Experiments on a 14-vane, double-strapped magnetron oscillator were performed to demonstrate high-power, high-efficiency coherent radiation at 5.8 GHz. The double-strapped magnetron was designed by using the Buneman-Hatree resonance condition, electromagnetic simulations and non-linear three-dimensional particle-in-cell (PIC) simulations. Experiments showed an oscillation output power of 5.3 kW at 5.79 GHz, corresponding to a DC-RF conversion efficiency of 57%. The cathode voltage was 9.2 kV, the collected anode current was 1 A, and the external magnetic field is 7.5 kG. Experimental results for the RF power, oscillation frequency, and efficiency were in good agreement with the corresponding values from non-linear three-dimensional PIC simulations.

PIC Algorithm with Multiple Poisson Equation Solves During One Time Step

In order to reduce the overall computational time of a PIC (particle-in-cell) computer simulation, an attempt was made to utilize larger time step sizes by implementing multiple solutions of Poisson's equation within one time step. The hope was this would make the PIC simulation stable at larger time steps than an explicit technique can use, and using larger time steps would reduce the overall computational time, even though the computational time per time step would increase. A three-dimensional PIC code that tracks electrons and ions throughout a three-dimensional Cartesian computational domain is used to perform this study. The results of altering the number of times Poisson's equation is solved during a single time step are presented. Also, the size of the time that can be used and still maintain a stable solution is surveyed. The results indicate that using multiple Poisson solves during one time step provides some ability to use larger time steps in PIC simulations, but the increase in time step size is not significant and the overall simulation run time is not reduced

Two-stage acceleration of protons from relativistic laser-solid interaction

A two-stage proton acceleration scheme using present-day intense lasers and a unique target design is proposed. The target system consists of a hollow cylinder with conical inner wall, which is followed by the main target with a flat front and a dishlike flared rear surface. At the center of the latter is a tapered proton layer, which is surrounded by side proton layers at an angle to it. In the first acceleration stage, protons in both layers are accelerated by target normal sheath acceleration. The center-layer protons are accelerated forward along the axis while the side protons are accelerated and focused towards them. As a result, the side-layer protons radially compress as well as axially further accelerate the front part of the center-layer protons in the second stage. Two-dimensional (2D) particle-in-cell (PIC) simulations show that a quasimonoenergetic proton bunch with the maximum energy over 250 MeV and energy spread similar to 17% can be generated when such a target is irradiated with an 80 fs laser pulse with focused intensity 3.1 x 10(20) W/cm(2). Three-dimensional (3D) PIC simulation gives the reduced maximum energy similar to 112 MeV but even smaller energy spread similar to 3% under the same laser conditions due to anisotropic electron acceleration with linearly polarized lasers.

Energy loss and longitudinal wakefield of relativistic short proton bunches in electron clouds

The aim of our study is the numerical computation of the wakefield and energy loss per unit length for relativistic, short ($l10\text{ }\text{ }\mathrm{ns}$) proton bunches interacting with an electron cloud inside the beam pipe. We present analytical expressions for the energy loss in the impulse kick approximation. For the simulation of the wakefields a 2D self-consistent, electrostatic particle-in-cell (PIC) code is employed. Results for the energy loss and for the wakefields are presented for the parameter scope of the CERN LHC and SPS. For selected parameters the results are compared to a three-dimensional (3D) electromagnetic PIC code.

Particle-in-cell mode beam dynamics simulation of the low energy beam transport for the SSC-linac injector

A new SSC-linac system (injector into separated sector cyclotron) is being designed in the HIRFL (heavy ion research facility of Lanzhou). As part of SSC-Linac, the LEBT (low energy beam transport) consists of seven solenoids, four quadrupoles, a bending magnet and an extra multi-harmonic buncher. The total length of this segment is about 7 meters. The beam dynamics in this LEBT has been studied using three-dimensional PIC (particle-in-cell) code BEAMPATH. The simulation results show that the continuous beam from the ion source is first well analyzed by a charge-to-mass selection system, and the beam of the selected charge-to-mass ratio is then efficiently pre-bunched by a multi-harmonic buncher and optimally matched into the RFQ (radio frequency quadrupole) for further acceleration. The principles and effects of the solenoid collimation channel are discussed, and it could limit the beam emittance by changing the aperture size.

Wake effect and stopping power for a charged ion moving in magnetized two-component plasmas: Two-dimensional particle-in-cell simulation

A two-dimensional particle-in-cell (PIC) model is proposed to study the wake field and stopping power induced by a nonrelativistic charged particle moving perpendicular to the external magnetic field in two-component plasmas. The effects of the magnetic field on the wake potential and the stopping due to the polarization of both the plasma ions and electrons are discussed. The velocity fields of plasma ions and electrons are investigated, respectively, in the weak and strong magnetic field cases. Our simulation results show that in the case of weak magnetic field and high ion velocity, the wakes exhibit typical V-shaped cone structures and the opening cone angles decrease with the increasing ion velocity. As the magnetic field becomes strong, the wakes lose their typical V-shaped structures and become highly asymmetrical. Similar results can be obtained in the case of low ion velocity and strong magnetic field. In addition, stopping power is calculated and compared with previous one-dimensional and full three-dimensional PIC results.

Detailed particle-in-cell simulations on the transport of a relativistic electron beam in plasmas

We present comprehensive two-dimensional (2D) particle-in-cell (PIC) simulations on the transport of a relativistic electron beam in a plasma in the context of fast ignition fusion. The 2D PIC simulations are performed by constructing two different simulation planes and have shown the complete stabilization and destabilization of the Weibel instability due to the beam temperature and background plasma collisions, respectively. Some three-dimensional PIC simulation results on the filamentary structures are also shown thereby further shedding light on the filamentation of the electron beam in plasmas. The linear growth rates of fastest growing mode in the beam-plasma system are compared with a theoretical model developed and are found in good agreement with each other.

Nonlinear turbulent transport in magnetic fusion plasmas

For more than a decade, the study of microturbulence driven by ion temperature gradient (ITG) drift instabilities in tokamak devices has been an active area of research in magnetic fusion science for both experimentalists and theorists alike. An important impetus for this avenue of research was the discovery of the radial streamers associated with the ITG modes in the early 1990s using the particle-in-cell (PIC) simulation method. Subsequently, ITG simulations using codes with increasing realism have been made possible by the dramatic increase in computing power. Notable examples were the demonstration of the importance of nonlinearly generated zonal flows in regulating ion thermal transport and the transition from Bohm to gyroBohm scaling with increased device size. In this paper, we will describe an interesting nonlinear physical process, as well as the resulting turbulent transport, that is associated with the interactions between the nonlinear parallel acceleration of the ions and the zonal flow modes. This study was carried out by utilizing a fully parallelized three-dimensional PIC code in global toroidal geometry on the most advanced, modern, massively parallel supercomputers.

The Role of the Guide Field in Relativistic Pair Plasma Reconnection

We study the role of the guide field in relativistic magnetic reconnection in a Harris current sheet of pair (e±) plasmas, using linear theories and particle-in-cell (PIC) simulations. Two-dimensional PIC simulations exhibit the guide field dependence to the linear instabilities; the tearing or reconnection modes are relatively insensitive, while the relativistic drift-kink instability (RDKI), the fastest mode in a relativistic current sheet, is stabilized by the guide field. Particle acceleration in the nonlinear stage is also investigated. A three-dimensional PIC simulation demonstrates that the current sheet is unstable to the RDKI, although a small reconnection occurs in the deformed current sheet. Another three-dimensional PIC simulation with a guide field demonstrates a completely different scenario. Secondary magnetic reconnection is triggered by nonlinear coupling of oblique instabilities, which we call the relativistic drift-sausage tearing instability. Therefore, particle acceleration by relativistic guide field reconnection occurs in three-dimensional configuration. Based on the plasma theories, we discuss an important role of the guide field: to enable nonthermal particle acceleration by magnetic reconnection.