The use of harmonics in 3-D magnetic fields

Motivated by the need for new means for specification and determination of 3-D fields that are produced by electromagnetic lens elements in the region interior to coil windings and seeking to obtain techniques that will be convenient for accurate conductor placement and dynamical study of particle motion, the authors have generalized the representation of a 2-D magnetic field to 3-D. They have shown that the 3-D magnetic field components of a multipole magnet in the curl-free divergence-free region near the axis r = 0 can be derived from one dimensional functions An(z) and their derivatives. In this report they apply both methods to the ''end'' region of a 40 mm bore SSC quadrupole, calculating first the field harmonics and then reconstructing the field comparing both results with direct Biot-Savart calculation