Computation of Charged-Particle Transfer Maps for General Fields and Geometries Using Electromagnetic Boundary-Value Data

TL;DR

This paper introduces a boundary-value based method to accurately compute charged-particle transfer maps in complex electromagnetic fields, overcoming noise issues from direct differentiation of field data.

Contribution

The authors develop a novel boundary integral approach using surface source distributions and Helmholtz's theorem to generate high-order transfer maps for arbitrary static fields.

Findings

Method reduces noise in derivative calculations

Enables high-order map generation for complex geometries

Integrates with existing electromagnetic solvers

Abstract

Three-dimensional field distributions from realistic beamline elements can be obtained only by measurement or by numerical solution of a boundary-value problem. In numerical charged-particle map generation, fields along a reference trajectory are differentiated multiple times. Any attempt to differentiate directly such field data multiple times is soon dominated by "noise" due to finite meshing and/or measurement errors. This problem can be overcome by the use of field data on a surface outside of the reference trajectory to reconstruct the fields along and around the reference trajectory. The integral kernels for Laplace's equation that provide interior fields in terms of boundary data or boundary sources are smoothing: interior fields will be analytic even if the boundary data or source distributions fail to be differentiable or are even discontinuous. In our approach, we employ all three components of the field on the surface to find a superposition of single-layer and double-layer surface source distributions that can be used together with simple, surface-shape-independent kernels for computing vector potentials and their multiple derivatives (required for a Hamiltonian map integration) at interior points. These distributions and kernels are found by the aid of Helmholtz's theorem (or equivalently, by Green's theorem). A novel application of the Dirac-monopole vector potential is used to find a kernel for the single-layer distribution. These methods are the basis for map-generating modules that can be added to existing numerical electromagnetic field-solving codes and would produce transfer maps to any order for arbitrary static charged-particle beamline elements.