Abstract
The two-dimensional dispersion relation for the free-electron laser is derived in the cold-beam limit. It is found that the forward wave has the largest growth rate, but the backward wave has the greatest spatial amplification rate. For $\frac{{k}_{0}v}{{\ensuremath{\omega}}_{p}}g{\ensuremath{\gamma}}^{\frac{\ensuremath{-}3}{2}}+{\ensuremath{\gamma}}^{\frac{\ensuremath{-}1}{2}}$, the free-electron laser is unstable at all angles with the low-frequency waves being absolutely unstable.
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