Coupling impedance in modern accelerators

A systematic review of theoretical results for the longitudinal and transverse impedances obtained by different methods is presented. Definitions, general theorems, modal analysis, diffraction model, and analytical results comprise the content of the paper. Several new results are included. In particular, necessary and sufficient conditions for the independence of the impedance on the beam longitudinal direction are given. The impedances of two basic simple structures—that of a cavity and that of a step—are studied in detail. The transition from the regime of a cavity to the regime of a step is explained, an approximate formula describing this transition is given, and the criterion for determining the applicability of each regime is established. The asymptotic behavior of the impedance for a finite number $M$ of periodically arranged cavities as a function of $M$ is studied. The difference in the behavior of the impedance for a single cavity and that for an infinite number of cavities is explained as the result of the interference of the diffracted waves. A criterion for determining the transition in the impedance behavior from small $M$ to large $M$ is presented.