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 $\frac{k_{0}v}{{\omega }_p}>{\gamma }^{\frac{-3\mathrm{}}{2}}+{\gamma }^{\frac{-1\mathrm{}}{2}}$, the free-electron laser is unstable at all angles with the low-frequency waves being absolutely unstable.

• Received 8 October 1979

DOI:https://doi.org/10.1103/PhysRevA.22.2088