Leading order hard edge fringe fields effects exact in (1 + δ) and consistent with Maxwell's equations for rectilinear magnets

In a circular machine, where the linear lattice functions (α, β, γ) and a phase advance can be defined, one expects the fringe field effects to be negligible if the change in these functions is small through the element. However, this may not always be the case. In such situations, it is useful to have a leading order result which is adapted to tracking and analytical analysis. In this paper, we provide such a result for the quadrupole and we also provide a general formula for the effect of an arbitrary rectilinear multipole. Starting from the standard multipole expansion for the B field of a 2(n+1)-pole (n ≥ 1), we compute the missing terms in the vector potential expansion consistent with the pure 2(n+1)-pole symmetry. We then compute the leading effects of the fringing fields of a multipole on the dynamics. Finally, we apply this result to quadrupoles and reproduce the original results of Lee-Whiting, Matsuda, and Wollnik. For the quadrupole, we show how to write a symplectic (canonical) integrator for the dynamics which can be used in a standard circular machine kick code. For higher order multipoles, we display the implicit characteristic function solution as first proposed by Dragt. (orig.)