Differential algebra based magnetic field computations and accurate fringe field maps

For the purpose of precision studies of transfer maps of particle motion in complex magnetic fields, we develop a method for Differential Algebra based $3D$ field computation and multipole decomposition. It can be applied whenever a model of a magnet is given which consist of line wire currents, and the wires are utilized to represent both the coils and the iron parts via the so-called image current method. Such a model exists for most modern superconducting magnets and a large variety of others as well. It is stressed that it is the only practically possible way to extract the multipoles and its derivatives, and hence the transfer map of the particle motion, analytically to high order. We also study various related topics like aspects of computational complexity of the problem, Maxwellification of fields, importance of vanishing curl, etc., and its applications to very accurate computation of magnetic fields including fringe fields. Refs 19. Figs 4. Tables 6.