Electromagnetic instability of the ion-focused regime

A relativistic electron beam propagating through a wide plasma channel in the ion-focused regime exhibits an electromagnetic instability coupled to the ‘‘betatron’’ motion, with peak growth rate near a resonant angular frequency ω∼2γ2ωβ, where γ is the Lorentz factor for the beam, and ωβ is the angular frequency of transverse oscillations. An eikonal formalism is derived analogous to that for the free-electron laser and the dispersion relation is determined. Constraints due to energy spread and other sources of detuning are noted. Analytic scaling laws are compared to the results of numerical simulation for a practical example corresponding to a microwave amplifier. Constraints due to competing instabilities are noted and the extension to short wavelengths is discussed.