COSY INFINITY and its use for singlepass and multipass systems

We discuss the computational treatment of both singlepass systems like those used in microscopy and spectroscopy where careful consideration and control of aberrations is important, as well as multipass systems where other priorities arise. This is largely due to the fact that many of the aberrations cancel out when traversing a system repeatedly, which is greatly helped by the adjustment of the linear transfer matrix to be non-resonant. However, some other effects have a tendency to build up over time either linearly or exponentially, and a specific form of analysis is necessary to understand and control this behavior. This is achieved using normal form methods which allow a clear separation of multipass effects that are transient and those that are persistent, as well as the use of symplectic integration. The Differential Algebraic (DA) methods employed in COSY INFINITY allow for the computation of aberrations of arbitrary order and also the relevant normal forms. The tools allow the automatic computation of fully Maxwellian 3D fields if only midplane or on-axis field information is available, which for example allows recovering all nonlinear effects arising from increasing or decreasing fields in the fringes of particle optical elements. They also allow the computation of such fields from surface or volume field measurements, leading to a fully Maxwellian representation even in the presence of noise in the data. Utilizing metrics on symplectic spaces, it is possible to construct minimally invasive symplectification schemes for study of multipass systems based on transfer maps.