Axisymmetric charge-conservative electromagnetic particle simulation algorithm on unstructured grids: Application to microwave vacuum electronic devices

Abstract

We present a charge-conservative electromagnetic particle-in-cell (EM-PIC) algorithm optimized for the analysis of vacuum electronic devices (VEDs) with cylindrical symmetry (axisymmetry). We exploit the axisymmetry present in the device geometry, fields, and sources to reduce the dimensionality of the problem from 3D to 2D. Further, we employ 'transformation optics' principles to map the original problem in polar coordinates with metric tensor diag(1, ρ 2 , 1) to an equivalent problem on a Cartesian metric tensor diag(1, 1, 1) with an effective (artificial) inhomogeneous medium introduced. The resulting problem in the meridian (ρz) plane is discretized using an unstructured 2D mesh considering TE φ -polarized fields. Electromagnetic field and source (node-based charges and edge-based currents) variables are expressed as differential forms of various degrees, and discretized using Whitney forms. Using leapfrog time integration, we obtain a mixed E − B finite-element time-domain scheme for the full-discrete Maxwell's equations. We achieve a local and explicit time update for the field equations by employing the sparse approximate inverse (SPAI) algorithm. Interpolating field values to particles' positions for solving Newton-Lorentz equations of motion is also done via Whitney forms. Particles are advanced using the Boris algorithm with relativistic correction. A recently introduced charge-conserving scatter scheme tailored for 2D unstructured grids is used in the scatter step. The algorithm is validated considering cylindrical cavity and space-charge-limited cylindrical diode problems. We use the algorithm to investigate the physical performance of VEDs designed to harness particle bunching effects arising from the coherent (resonance) Cerenkov electron beam interactions within micro-machined slow wave structures.

Finite element time-domain body-of-revolution Maxwell solver based on discrete exterior calculus

“…In this section, we consider a backward-wave oscillator (BWO) driven by energetic electron beams in the relativistic regime designed to produce a high-power microwave signal [65], as depicted in Fig. 14. The proposed FETD-BOR solver is incorporated into a PIC algorithm [66,67,68] to simulate the wave-plasma interaction in the device [9]. The PIC algorithm is based on an unstructured grid and explained in detail in [9,24,25].…”

“…14. The proposed FETD-BOR solver is incorporated into a PIC algorithm [66,67,68] to simulate the wave-plasma interaction in the device [9]. The PIC algorithm is based on an unstructured grid and explained in detail in [9,24,25]. For this problem it suffices to consider the TE φ polarized field with m = 0.…”

Compatible Maxwell solvers with particles I: conforming and non-conforming 2D schemes with a strong Ampere law

“…Electromagnetic PIC schemes are a popular approach to model the propagation of charged particle beams in vacuum devices, see for instance [44,67,26] demonstrating the numerical efficiency of DG-PIC schemes using divergence cleaning techniques, or [60] where a conforming FEM scheme using low order Nédélec elements (Whitney forms) is coupled with particles using a charge-conserving deposition method.…”

Quasi-Helmholtz Decomposition, Gauss' Laws and Charge Conservation for Finite Element Particle-in-Cell

“…Of particular note in seminal work in 26 , wherein rubrics for charge conservation for FEM-PIC where developed. These have been refined and developed further in a series [27][28][29] . By and large, these papers follow a similar rubric, use a leap-frog scheme to step through a solution for the field, update position and velocity of the particle and solve for the fields again.…”