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{\ensuremath{-}}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.