The operation of free-electron lasers can be severely limited by the a
xial velocity spread of the beam electrons. In this Letter we propose
methods for reducing the axial velocity spread in electron beams by re
distributing the electron energy via interaction with an axially symme
tric, slow, TM waveguide mode. In the first method, the energy redistr
ibution is correlated with the electrons' betatron amplitude, while in
the second method it is correlated with the electrons' synchrotron am
plitude. Reductions by more than a factor of 40 in the rms axial veloc
ity spread have been obtained in simulations.