Determination of the characteristics of toroidal electrostatic analyzers. Application to the optimization of analyzers used in space physics

A numerical method for determining the characteristics of electrostatic particle analyzers is presented, applicable in the general case of toroidal electrodes. The electrostatic field is expressed as a Taylor series expansion in terms of reduced coordinates, defining the position of a point relative to the average equipotential. The coefficients of this expansion are evaluated up to an arbitrary given order, and the trajectory of a particle is determined by numerical integration of the equations of motion. By exploring different incidence cases, the characteristic quantities of the analyzer are obtained. The results are compared with those obtained through a rheographic tank simulation. Two concrete examples of the use of the developed numerical program are presented, addressing measurement problems in space environments: an analyzer with high resolution in energy and angle, and an analyzer with a large input range and wide energy bandwidth. The toroidal shape significantly broadens the performance framework achieved with traditional cylindrical or spherical analyzers.