Velocity dispersion of correlated energy spread electron beams in the free electron laser

The effects of a correlated linear energy/velocity chirp in the electron beam in the FEL, and how to compensate for its effects by using an appropriate taper (or reverse-taper) of the undulator magnetic field, is well known. The theory, as described thus far, ignores velocity dispersion from the chirp in the undulator, taking the limit of a ‘small’ chirp. In the following, the physics of compensating for chirp in the beam is revisited, including the effects of velocity dispersion, or beam compression or decompression, in the undulator. It is found that the limit of negligible velocity dispersion in the undulator is different from that previously identified as the small chirp limit, and is more significant than previously considered. The velocity dispersion requires a taper which is non-linear to properly compensate for the effects of the detuning, and also results in a varying peak current (end thus a varying gain length) over the length of the undulator. The results may be especially significant for plasma driven FELs and low energy linac driven FEL test facilities.