Large lisplacement and livergence analytic transfer maps through quadrupoles

Linear-field non-scaling FFAGs are proposed for multi-GeV acceleration of muons and order hundreds MeV/u proton or carbon for medical applications. The large momentum acceptance lattices employ alternating focusing and defocusing combined-function magnets. In one implementation, rectangular quadrupole magnets are used, with the dipole component generated by off-setting the magnet centre relative to the reference orbit. This feature, coupled with the large radial aperture for momentum variation, gives rise to large amplitude orbits. The angles are so large that there is a partial interchange of the longitudinal and transverse momentum relative to the fixed coordinate system of the quadrupole. We examine methods to devise non-linear transfer maps to high order. The map is constrained by the fact that its first partial derivative produces the linear transfer matrix which must have unity determinant.