Symplectic Expression for Chromatic Aberrations

Chromatic aberrations characterize how linear optics parameters, including tunes, beta functions, and x-y coupling parameters, depend on momentum deviation. The definition of chromatic aberration is based on static or slowly varying momentum deviations. Linear optics parameters are expanded in terms of momentum deviation; their coefficients are called chromatic aberrations in this paper. The Hamiltonian and a six-dimensional symplectic map, which are taken into account in both betatron and synchrotron motions, are reconstructed using the measured chromatic aberrations. This map can be used to study the effects of chromatic aberrations in beam-beam, space charge, and electron cloud phenomena.