Symplectic scaling of transfer maps including fringe fields

A method is introduced that provides an accurate and fast approximation of high-order maps of fringe fields and other fields that change along the reference trajectory. While the effect of main fields of optical elements can be determined very efficiently with differential algebraic (DA) methods via exponentiation of the respective propagator, the computation of high-order maps of nonstationary fields in general requires time-consuming DA integration. The method of symplectic scaling presented in this paper provides a very fast approximation of such maps by relating an arbitrary map to a specific previously computed map. This is achieved by a combination of geometric scaling and scaling with rigidity performed, in a canonically perturbative treatment of a strength parameter. The method is useful for detailed analysis of nonlinear motion in particle optics, which in many cases is strongly influenced or even dominated by the presence of fringe fields. The use of the symplectic scaling method typically speeds up the computation of fringe-field effects by around two orders of magnitude and thus approaches speeds similar to that of the main-field calculation. The method has been implemented in the code COSY INFINITY; several examples from various subfields of beam physics are given to illustrate the accuracy and speed of the method.