Sideband control in a millimeter-wave free-electron laser

The frequency offset of the sideband instability in a free-electron laser (FEL) should depend on (1-$v_{?}$/$v_g$ ${)}^{\mathrm{-}1}$, where $v_{?}$ is the average longitudinal velocity of the electrons and $v_g$ is the group velocity of the electromagnetic waves. We have tested the $v_{?}$/$v_g$ dependence of the sideband shift in a 2-mm, Raman-regime FEL oscillator. A change of $v_{?}$/$v_g$ from 0.93 to 0.98, accomplished by an increase in the undulator period, resulted in the measured sideband shift increasing from 6% to 40%, in approximate agreement with theory.