Wake of a Rough Beam Wall Surface

We have shown that the two models of roughness impedance investigated in this report have some similarities and some differences, and can be thought of as being complementary. They both are applicable only when the frequencies of interest are low compared to c/r, with r the typical size of the surface discontinuities, and both yield an approximation to the impedance that is purely inductive. The first model approximates a rough surface by a random collection of non-interacting bumps. It finds the impedance of a single, small bump on a beam tube surface, and then uses averaging to estimate the impedance of a rough surface. The second model, through use of the spectral function of the rough surface analytically finds the impedance, though it is limited to surfaces with slowly varying discontinuities. In the specific case of a surface with non-interacting, smooth bumps the two models will give the same result. The micro-geometry of a metallic surface--for example, the beam tube in the LCLS undulator--depends on the manufacturing and polishing process that had been applied to that surface. For either of the models discussed in this report to accurately estimate the impedance of a surface requires a specific characterization of themore » micro-geometry. Once such a characterization is performed, through measurement, one can begin to apply these models to obtain a realistic estimate of the surface impedance and derive conclusions about the effect of the impedance on beam dynamics.« less