Periodically-focused solutions to the nonlinear Vlasov-Maxwell equations for intense charged particle beams

An intense nonneutral ion beam propagates in the z-direction through a periodic focusing quadrupole field with transverse focusing force, F_foc= - κ_q(s)(xê_x - yê_y), on the beam ions. Here, the oscillatory lattice coefficient satisfies κ_q(s + S) = κ_q(s), where S = const, is the axial periodicity length. The model employs the Vlasov-Maxwell equations to describe the nonlinear evolution of the distribution function f_b(x, y, x′, y′, s) and the normalized self-field potential ψ(x, y, s) in the transverse laboratory-frame phase space (x, y, x′, y′). Using a third-order Hamiltonian averaging technique, a canonical transformation is employed with an expanded generating function which transforms away the rapidly oscillating terms, and leads to a Hamiltonian in the 'slow' transformed variables (X̃,Ỹ,X̃′,Ỹ′), with constant focusing coefficient κ_fq = const.