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Mathematical Modeling of Fringe Fields in Beam Line Control Systems

Yury Tereshonkov
Usually beam dynamic modeling in beam lines is implemented using a
piecewise approximation for of steering magnet fields. Real magnet
fields have a bell-shaped form, which describes a field distribution
along the electrical axis of a steering element. Influence of fringe
fields in beam lines is taken into account only on tuning stage,
practically on final steps before usage. This paper deals with
methodology, which allows including fringe field effects during an
initial stage of a modeling process. Also wide spectrum of modeling
functions for fringe field distributions is presented. It is known,
that fringe field effects are intrinsic and unremovable effects and
could heavily impact on the beam dynamic and corresponding beam
characteristics. Mathematical and computer models for fringe field
help to estimate their influence and include information of real
magnet field into the designing model. Research is based on the
matrix formalism for Lie algebraic tools. This approach gives large
flexibility, because it could admit the usage of computer algebra
methods and technologies. Computer algebra methods can be easily
paralleling, which gives line benefit with increasing number of
processors (cores).
Besides, this approach can be extended for
nonlinear aberrations without loss of generality.
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