Preserving Simplecticity in the Numerical Integration of Linear Beam Optics

Abstract

Presented are mathematical tools and methods for the development of numerical integration techniques that preserve the symplectic condition inherent to mechanics. The intended audience is for beam physicists with backgrounds in numerical modeling and simulation with particular attention to beam optics applications. The paper focuses on Lie methods that are inherently symplectic regardless of the integration accuracy order. Section 2 provides the mathematically tools used in the sequel and necessary for the reader to extend the covered techniques. Section 3 places those tools in the context of charged-particle beam optics; in particular linear beam optics is presented in terms of a Lie algebraic matrix representation. Section 4 presents numerical stepping techniques with particular emphasis on a third-order leapfrog method. Section 5 discusses the modeling of field imperfections with particular attention to the fringe fields of quadrupole focusing magnets. The direct computation of a third order transfer matrix for a fringe field is shown.