On numerical errors to the fields surrounding a relativistically moving particle in PIC codes

Abstract

The particle-in-cell (PIC) method is widely used to model the self-consistent interaction between discrete particles and electromagnetic fields. It has been successfully applied to problems across plasma physics including plasma based acceleration, inertial confinement fusion, magnetically confined fusion, space physics, astrophysics, high energy density plasmas. In many cases the physics involves how relativistic particles (those with high relativistic γ factors) are generated and interact with plasmas. However, when relativistic particles stream across the grid both in vacuum and in plasma there are many numerical issues that may arise which can lead to incorrect physics. We present a detailed analysis of how discretized Maxwell solvers used in PIC codes can lead to numerical errors to the fields that surround particles that move at relativistic speeds across the grid. Expressions for the axial electric field as integrals in k space are presented. Two types of errors to these expressions are identified. The first arises from errors to the numer-ator of the integrand and leads to unphysical fields that are antisymmetric about the particle. The second arises from errors to the denominator of the integrand and lead to Cerenkov like radiation in "vacuum". These fields are not anti-symmetric, extend behind the particle, and cause the particle to accelerate or decelerate depending on the solver and parameters. The unphysical fields are studied in detail for two representative solvers -the Yee solver and the FFT based solver. Although the Cerenkov fields are absent, the space charge fields are still present in the fundamental Brillouin zone for the FFT based solvers. In addition, the Cerenkov fields are present in higher order zones for the FFT based solvers. Comparison between the analytical solutions and OSIRIS results are presented. A solution for eliminating these unphysical fields by modifying the k operator in the axial direction is also presented. Using a customized finite difference solver, this solution was successfully implemented into OSIRIS. Results from the customized solver are also presented. This solution will be useful for a beam of particles that all move in one direction with a small angular divergence.

A new field solver for modeling of relativistic particle-laser interactions using the particle-in-cell algorithm

“…5(c) are larger than their counterparts that have the time-stagger correction. As is well known, p 2 satisfies the canonical momentum conservation, p 2 − a L = const., (27) where a L is the normalized vector potential of the laser pulse. Since the test particle is initially stationary in the2-direction and placed where a L = 0, the subsequent evolution of p 2 is subject to p 2 = a L .…”

“…Many numerical issues can arise due to the discretization, requiring careful use to avoid subtle spurious effects. Examples of known issues include improper numerical dispersion, numerical Cerenkov radiation and the associated numerical Cerenkov instability (NCI) [19][20][21][22], finite-grid instability [23][24][25][26] and numerical errors in the fields that surround relativistic particles [27]. These errors do not always decrease proportionately with decreasing cell size and time step, making it important to deeply understand the cause of these effects in order to most efficiently remedy them.…”

“…Lorentz force remains, but in general the magnitude is smaller than those of the standard method. Note that choosing [k] 1 = [k] 1,t is exactly the solver proposed by Xu[27] to reduce field errors surrounding relativistic particles. This numerical-dispersionfree solver adopts identical operators, i.e.,[k] E1 = [k] B1 = [k] 1,t .Our proposed scheme can be simply viewed as a time-staggeredcorrection version of this solver.…”

Generation of topologically complex three-dimensional electron beams in a plasma photocathode

“…To illustrate how this injection scheme can be used to generate spiraling and other complex 3D structured electron beams, we consider a fully blown out wake generated by an electron beam driver. We simulate the ionization injection by an appropriately delayed but co-moving ultrashort laser using non-evolving forces characteristic of nonlinear wakefields using the 3D PIC code OSIRIS [59,60]. The forces for electrons with forward velocity…”

Ultra-Bright Electron Bunch Injection in a Plasma Wakefield Driven by a Superluminal Flying Focus Electron Beam

“…There are 16 macro-particles initialized in each cell to model a uniform plasma with T e ∼ 2 eV. We use customized finite-difference solvers [36,37] to eliminate the numerical Cerenkov instability [38] and spurious space-charge-like fields [39]. A bi-Gaussian drive beam consisting of 10 6 macroparticles was initialized at the plane f R = 10 4 k −1 p before the lens with γ R = 1000, h = −0.01, k p σ z = 2 and Λ = 4, where k p ≡ ω p /c and ω p are the plasma wavelength and frequency.…”