Generalized Form Factors for the Beam Coupling Impedances in a Flat Chamber

The exact formalism from B. Zotter to compute beam coupling impedances has been fully developed only in the case of an infinitely long circular beam pipe. For other two dimensional geometries, some form factors are known only in the ultrarelativistic case and under certain assumptions of conductivity and frequency of the pipe material. We present here a new and exact formalism to compute the beam coupling impedances in the case of a collimator-like geometry where the jaws are made of two infinite plates of any linear material. It is shown that the impedances can be computed theoretically without any assumptions on the beam speed, material conductivity or frequency range. The final formula involves coefficients in the form of integrals that can be calculated numerically. This way we obtain new generalized form factors between the circular and the flat chamber cases, which eventually reduce to the so-called Yokoya factors under certain conditions.