Higher harmonic emission by a relativistic electron beam in a longitudinal magnetic wiggler

The classical limit of the Einstein-coefficient method is used in the low-gain regime to calculate the stimulated emission from a tenuous relativistic electron beam propagating in the combined solenoidal and longitudinal wiggler fields $(B_0+\delta Bsin{k}_{0}z){\hat{e}}_z$ produced near the axis of a multiple-mirror (undulator) field configuration. Emission is found to occur at all harmonics of the wiggler wave number $k_0$ with Doppler upshifted output frequency given by $\omega =\frac{(l{k}_0{V}_b+{\omega }_{\mathrm{cb}})(1+\frac{V_b}{c}){\gamma }_b^2}{(1+\frac{{\gamma }_b^2{V}_{\perp }^2}{c^2})}$, where $l\ge 1$. The emission is compared to the low-gain cyclotron maser with $\delta B=0\mathrm{}$ and to the low-gain free-electron laser (operating at higher harmonics) utilizing a transverse linearly polarized wiggler field.