Variational Self-Consistent Theory for Beam-Loaded Cavities

A variational theory is presented for beam loading in microwave cavities. The beam-field interaction formulated as a dynamical whose stationarity according to Hamilton's principle will naturally lead steady-state solutions that indicate how cavity's resonant frequency, $Q$, and optimal coupling coefficient detune result of the loading. driven cavity Lagrangian derived from first principles, including effects cavity-wall losses, input power, interaction. general formulation applied typical klystron predict appropriate detuning parameters required maximize gain (or modulation depth) average Lorentz factor boost, $⟨\mathrm{\ensuremath{\Delta}}\ensuremath{\gamma}⟩$. Numerical examples are presented, showing agreement with trends previously observed literature. developed carries several advantages analysis design beam-loaded structures. It provides self-consistent model (nonlinear) interaction, procedure maximizing under beam-loading conditions, useful set guide cavity-shape optimization during systems. Enhanced clarity physical picture underlying problem seems be gained using this approach, allowing straightforward inclusion or exclusion different field configurations calculation expressing final results terms measurable quantities. Two discussed cavity, finite magnetic confinement no at all. Formulating language directly accessible powerful techniques found Hamiltonian dynamics canonical transformations may potentially carry an additional advantage analytical computational gains, suitable conditions.