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The effects of a correlated linear energy/velocity chirp in the electron beamin 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 theundulator, taking the limit of a `small' chirp. In the following, the physicsof compensating for chirp in the beam is revisited, including the effects ofvelocity dispersion, or beam compression or decompression, in the undulator. Itis found that the limit of negligible velocity dispersion in the undulator isdifferent from that previously identified as the small chirp limit, and is moresignificant than previously considered. The velocity dispersion requires ataper which is non-linear to properly compensate for the effects of thedetuning, and also results in a varying peak current (end thus a varying gainlength) over the length of the undulator. The results may be especiallysignificant for plasma driven FELs and low energy linac driven FEL testfacilities.

Velocity Dispersion of Correlated Energy Spread Electron Beams in the Free Electron Laser