Arbitrary order description of arbitrary particle optical systems

Abstract

Abstract The differential algebraic approach for the design and analysis of particle optical systems and accelerators is presented. It allows the computation of transfer maps to arbitrary orders for arbitrary arrangements of electromagnetic fields, including the dependence on system parameters. The resulting maps can be cast into different forms. In the case of a Hamiltonian system, they can be used to determine the generating function or Eikonal representation. Also various factored Lie operator representations can be determined directly. These representations for Hamiltonian systems cannot be determined with any other method beyond relatively low orders. In the case of repetitive systems, a combination of the power series representation and the Lie operator representation allows a nonlinear change of variables such that the motion is very simple and its long term behaviour can be studied very efficiently. Furthermore, it is now possible to compute quantities relevant to the study of circular machines like tune shifts and chromaticities much more efficiently. Besides these aspects, the ability to compute maps depending on parameters provides analytical insight into the system. In addition, this approach allows very efficient optimization, to the extent that in many cases it is almost completely analytic.