Coupling impedance of a long slot and an array of slots in a circular vacuum chamber

A method is presented for calculating the real part of the longitudinal impedance Z at a frequency ω for both a small hole and a long slot in a beam vacuum chamber with a circular cross section. The length of the slot can be arbitrarily large; the only requirement of the dimensions of the slots is that its width be much smaller than c/ω. Regular arrays of N slots periodically distributed along the pipe are also considered. The equation relating the real part of the impedance to the energy radiated by the slot into waveguide eigenmodes is the basis of this method. We show that in addition to broadband impedance, there are narrow peaks in ReZ. For a single slot, the peaks are located slightly above the cutoff frequencies of various waveguide modes. For a periodic array, additional peaks are generated at frequencies determined by the array period. These peaks can be partially suppressed by randomization of the positions of the slots.