Symplectic scaling, a DA based tool

Maps of magnetic particle optical elements written in geometric coordinates have two scaling properties. These are connected to the fact that the maps depend only on the ratio of field strength to magnetic rigidity and on the product of field strength to the size of the element. Once the map of an element is known for a given type of beam particles as a function of the magnetic field strength at the pole tip, the first scaling property can be used to compute the map for any particle type. With the second scaling property, the map can be computed for any similar element which differs in size. Usually the map is not known as a function of the magnetic field. With DA based programs, however, one can obtain the Taylor expansion of that function. The expansion can serve to approximate maps which could otherwise only be calculated by very time consuming numerical integration. To make this method applicable to cases where the symplectic structure is important, canonical maps have to be approximated. The approximated maps still have to be completely symplectic up to their expansion order. To meet this requirement, we have examined how the scaling properties can be used in connection with the symplectic representations of Lie transformations and generating functions. Useful examples of the resulting symplectic scaling method include maps of fringe fields as well as solenoids. Speed and accuracy of the method, which was implemented into version 6 of COSY INFINITY, will be demonstrated and a guide given how to apply this method most efficiently