Periodically-focused solutions to the nonlinear Vlasov-Maxwell equations for intense beam propagation through an alternating-gradient quadrupole field

This paper considers an intense nonneutral ion beam propagating in the z-direction through a periodic focusing quadrupole field with transverse focusing force, E_foc=-κ_q(s)(xeˆ_x-yeˆ _y), on the beam ions. A third-order Hamiltonian averaging technique using a canonical transformation is employed to transform away the rapidly oscillating terms. This leads to a Hamiltonian, H(X˜,Y˜,X˜¹,Y˜¹ ,s)=(1/2)(X˜¹²+Y˜¹² )+(1/2)κ_fq(X˜ ²+Y˜² )+ψ(X˜,Y˜,s), in the transformed variables (X˜,Y˜,X˜¹,Y˜¹), where the focusing coefficient κ_fq is constant, and many solutions and properties of the Vlasov-Maxwell system are well known