Abstract
A method is described to solve the Poisson problem for a three dimensional source distribution that is periodic into one direction. Perpendicular to the direction of periodicity a free space (or open) boundary condition is realized. In beam physics, this approach allows us to calculate the space charge field of a continualized charged particle distribution with periodic pattern. The method is based on a particle-mesh approach with equidistant grid and fast convolution with a Green's function. The periodic approach uses only one period of the source distribution, but a periodic extension of the Green's function. The approach is numerically efficient and allows the investigation of periodic- and pseudoperiodic structures with period lengths that are small compared to the source dimensions, for instance of laser modulated beams or of the evolution of micro bunch structures. Applications for laser modulated beams are given.
Highlights
The fundamental problem of the analysis of beam dynamics is to solve the equation of motion for a multiparticle system (N ∼ 103 Á Á Á 109) in the presence of external forces and self-forces
For bunches that are long compared to the transverse dimension Lt=γ, it can be estimated by the one-dimensional longitudinal space charge field
The electromagnetic field caused by a set S of discrete point particles with charge qi, position riðtÞ and velocity viðtÞ is estimated based on the assumption of collective uniform motion with the velocity vc
Summary
The fundamental problem of the analysis of beam dynamics is to solve the equation of motion for a multiparticle system (N ∼ 103 Á Á Á 109) in the presence of external forces (caused by magnets, cavities etc.) and self-forces. The problem is equivalent to an electrostatic problem, and in principle one has to add for each observer particle the one over distance contribution from all other sources This point-to-point interaction would allow the slightly better approach of individual uniform motion per particle, but its effort scales as N2. Sometimes the particle distribution has a fine substructure which has to be taken into account, for instance if the particles perform plasma oscillations [3,4,5], or a micro-modulation builds up (microbunch-instability) or they interact in an undulator with a laser Such high resolution computations need very fine meshes and high particle (or macroparticle) numbers so that they become numerically very expensive. Of one period Vp. The effort for numerical operations (per step of particle tracking) can be split into the effort for the computation of the Green’s function and the effort for all other operations (binning to mesh, convolution, field interpolation, and particle motion). For comparison of field calculation methods, we did tracking simulations with Astra [11] and QField
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