An algorithm for simulating the geometric optics of charged particle instruments

Abstract

Abstract An algorithm is presented for characterizing an electromagnetic instrument such as a microscope, a spectrometer, an energy filter, or an accelerator for focusing and steering a beam of charged particles in a general geometry. The algorithm solves a system of second order, nonlinear, ordinary differential equations (ODE) for the evolution of a set of coupled, dependent variables. The relativistic Newton's equation falls into this category. Here, the variables represent the particle coordinates. The algorithm does not solve for the coordinates directly. The coordinates are expressed as a power series in their initial conditions, and the ODE is transformed to one for the series coefficients via the binomial and Taylor expansions. The solution to this ODE produces the series coefficients of the instrument to any order in an automated manner and thereby provides a map between the initial conditions of the particles and their coordinates along the axis of the instrument. This map gives the optical properties of the system in the neighborhood of the optical axis without a need for a solution to the individual trajectories. The algorithm is not limited to the solution to Newton's equation. It is extendable to a nonlinear ODE of higher order and therefore should have applications in disciplines outside of charged particle optics.