The passage of a charged particle through the field B of a magnetic element may be described by a symplectic transfer map M. This map is generated by the Hamiltonian H specifying charged-particle motion. M may be written as a product of Lie transformations, and the generators for these transformations can be found by integrating a set of differential equations whose driving terms are the Taylor coefficients arising in the Taylor expansion of H about a design orbit. The Hamiltonian formulation of charged particle motion, required to exploit symplectic structure, involves the use of a vector potential A. Therefore expanding H in a Taylor series requires a Taylor expansion of A. For realistic elements B is known (using codes such as those available from Vector Fields) only on a three-dimensional grid. The challenge is to reliably compute high-order Taylor coefficients for A based on this grid data.
At first sight this appears to be an impossible task. Numerical differentiation of grid data is well known to be very sensitive to numerical noise: numerical differentiation amplifies noise. And high derivatives are required to compute M to high order. This problem can be overcome by employing surface methods which involve the use of inverse Laplacian kernels. Such kernels are smoothing, and this smoothing overcomes the noise associated with numerical differentiation. Moreover, the Maxwell equations are satisfied exactly, and analyticity is assured.
One final point is that the optimal termination of fringe fields requires the use of a gauge at element ends for which A is as small as possible. This minimum gauge is found to be the Poincare-Coulomb gauge.
In summary, with the use of surface methods, and the use of the minimum gauge at element ends, it is now possible for the first time to compute realistic symplectic transfer maps to high order including all multipole-error and fringe-field effects. These maps can then be used to realistically predict/evaluate the expected performance of both linear and
circular machines.
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