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

Abstract

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, $\ensuremath{\sigma}<60\ifmmode^\circ\else\textdegree\fi{}$). The analysis assumes a thin $(a,b\ensuremath{\ll}S)$ axially continuous beam, or very long charge bunch, propagating in the $z$ direction through a periodic focusing lattice with transverse focusing coefficients ${\ensuremath{\kappa}}_{x}(s+S)\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\kappa}}_{x}(s)$ and ${\ensuremath{\kappa}}_{y}(s+S)\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}{\ensuremath{\kappa}}_{y}(s)$, where $S\phantom{\rule{0ex}{0ex}}=\phantom{\rule{0ex}{0ex}}\mathrm{const}$ 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 ${\overline{f}}_{b}(\overline{x},\overline{y},{\overline{x}}^{\ensuremath{'}},{\overline{y}}^{\ensuremath{'}},s)$ and (normalized) self-field potential $\overline{\ensuremath{\psi}}(\overline{x},\overline{y},s)$ in the four-dimensional transverse phase space $(\overline{x},\overline{y},{\overline{x}}^{\ensuremath{'}},{\overline{y}}^{\ensuremath{'}})$. In the resulting kinetic equation for ${\overline{f}}_{b}(\overline{x},\overline{y},{\overline{x}}^{\ensuremath{'}},{\overline{y}}^{\ensuremath{'}},s)$, the average effects of the applied focusing field are incorporated in constant focusing coefficients ${\ensuremath{\kappa}}_{x\mathrm{sf}}>0$ and ${\ensuremath{\kappa}}_{y\mathrm{sf}}>0$, 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.