Impedance of multilayer cylindrical structures with a material-filled beam region

Abstract

Beam-coupling impedances in material media are relevant for ionization-cooling channels of a future muon collider, where the beam propagates in matter rather than vacuum. We extend the cylindrical field-matching formalism for multilayer structures to the case of a material-filled beam region surrounded by external layers with arbitrary electromagnetic properties. Relative to the vacuum formulation, the usual factor $1/γ^2$ is replaced by the material-dependent factor $F=1/\varepsilon_1-μ_1β^2$, and the radial propagation constant is modified accordingly. Analytical expressions are obtained for the monopolar longitudinal and dipolar transverse impedances, with the surrounding structure encoded through reflection coefficients determined by field matching. The formalism reduces to the known vacuum and perfectly conducting limits in the appropriate cases. Representative calculations are presented for absorber-relevant configurations, illustrating the dependence on the material properties of both the beam region and the surrounding layers. In particular, a finite beam-region conductivity can generate a real impedance component and reverse the sign of the imaginary space-charge impedance, while sufficiently high permittivity can give rise to resonant structures associated with oscillatory radial fields. The effect of finite-conductivity surroundings is also examined, showing the approach towards the perfectly conducting limit as the external conductivity is increased. The applicability of the infinite-length approximation is also discussed by comparison with finite-length mode-matching results.