General analysis of the longitudinal coupling impedance of a rectangular slot in a thin coaxial liner

As previously discussed [Fedotov and Gluckstern, Phys. Rev. E 54, 1930 (1996)], for a long narrow slot whose length may be comparable with the wavelength, the usual static approximation for the polarizability and susceptibility which enter into the impedance is a poor one. Therefore, finding semi-analytic expressions for the impedance of a rectangular slot in a broad frequency range is highly desirable. We develop a general analysis based on a variational formulation, which includes both the realistic coaxial structure of the beam-pipe and the effect of finite wavelength in the calculation of the coupling impedance of a rectangular slot in a liner wall of zero thickness. We then present a numerical study of the frequency dependence of the coupling impedance of a transverse rectangular slot.