Degradation in gain for a free electron laser amplifier due to electron momentum spread

A finite spread in axial momentum for the electron beam in a free electron laser amplifier is shown to decrease the small-signal gain. For millimeter and sub-millimeter wave amplifiers, where exponential growth dominates the gain, it is shown that the gain is approximately 3 db below that for a cold beam if the relative momentum spread (Δu/u)_1/2 = (G_o/248)^1/2 (λ_o/L), where G_o≫1 is the gain in db for the cold-beam case, λ_o is the magnetic wiggler period, and L is the amplifier length. Exact numerical examples are given for representative FEL amplifiers at 35 and 550 GHz.