Low-gain free-electron laser near cyclotron resonance

The Einstein-coefficient method is used to calculate the growth rate of a low-gain free-electron laser for a tenuous relativistic electron beam propagating in the combined axial and transverse helical wiggler fields $B_0{\hat{e}}_z-(\delta B cos{k}_{0}z{\hat{e}}_x+\delta B sin{k}_{0}z{\hat{e}}_y)$. The analysis assumes that the system is close to resonance between the electron-cyclotron frequency in the guide field (${\omega }_c=\frac{e{B}_0}{\gamma \mathrm{mc}}$) and the wiggler frequency in the beam frame (${\omega }_0=k_0{V}_b$). It is shown that the gain near resonance can be substantially enhanced relative to the gain obtained in the region far from resonance.