Energy spectra from electromagnetic fields generated by ultra-relativistic charged bunches in a perfectly conducting cylindrical beam pipe

The spectrum of electromagnetic fields satisfying perfectly conducting boundary conditions in a segment of a straight beam pipe with a circular cross-section is discussed as a function of various source models. These include charged bunches that move along the axis of the pipe with constant speed for which an exact solution to the initial-boundary value problem for Maxwell's equations in the beam pipe is derived. In the ultra-relativistic limit all longitudinal components of the fields tend to zero and the spectral content of the transverse fields and average total electromagnetic energy crossing any section of the beam pipe are directly related to certain properties of the ultra-relativistic source. It is shown that for axially symmetric ultra-relativistic bunches interference effects occur analogous to those that occur due to coherent synchrotron radiation in cyclic machines despite the fact that in this limit the source is not accelerating. The results offer an analytic description of the fields showing how enhanced spectral behaviour depends on the geometry of the source, its location in the beam pipe and the details of the stochastic distribution of the source structure. The results are illustrated for different situations associated with the motion of on-axis ultra-relativistic bunches. The field energy spectra associated with a source containing N identically charged ultra-relativistic pulses, each with individual longitudinal Gaussian profiles distributed according to a uniform probability distribution with compact support, is compared with that generated by charged bunches containing a distribution with 2n + 1 peaks in a region with compact support (modelling micro-bunches). These results are of relevance for the experimental determination of properties of the longitudinal charge distribution of short relativistic electron bunches with micro-structure in straight segments of a beam pipe, from observation of the associated electromagnetic energy spectra.