Guiding-center Vlasov-Maxwell description of intense beam propagation through a periodic focusing field

This paper provides a systematic derivation of a guiding-center kinetic model that describes intense beam propagation through a periodic focusing lattice with axial periodicity length S, valid for sufficiently small phase advance (say, σ<60°). The analysis assumes a thin (a,b≪S) axially continuous beam, or very long charge bunch, propagating in the z direction through a periodic focusing lattice with transverse focusing coefficients κ_x(s+S)  =  κ_x(s) and κ_y(s+S)  =  κ_y(s), where S  =  const is the lattice period. By averaging over the (fast) oscillations occurring on the length scale of a lattice period S, the analysis leads to smooth-focusing Vlasov-Maxwell equations that describe the slow evolution of the guiding-center distribution function f̅ _b(x̅ ,y̅ ,x̅ ^′,y̅ ^′,s) and (normalized) self-field potential ψ̅ (x̅ ,y̅ ,s) in the four-dimensional transverse phase space (x̅ ,y̅ ,x̅ ^′,y̅ ^′). In the resulting kinetic equation for f̅ _b(x̅ ,y̅ ,x̅ ^′,y̅ ^′,s), the average effects of the applied focusing field are incorporated in constant focusing coefficients κ_{x sf}>0 and κ_{y sf}>0, and the model is readily accessible to direct analytical investigation. Similar smooth-focusing Vlasov-Maxwell descriptions are widely used in the accelerator physics literature, often without a systematic justification, and the present analysis is intended to place these models on a rigorous, yet physically intuitive, foundation.