Effect of space charge on free-electron-laser gain

The first-order correction due to space-charge effects to the small-signal gain of a free-electron laser is calculated classically. For a current density of 10² A/cm², an interaction length of 1 m and an electron beam energy of 20 MeV, the maximum small-signal gain is reduced by about 0.5% while for 10³ Amp/cm², it is reduced by about 5%. The reduction is proportional to (ωₚLc)²γ₀³≡S, where ωₚ is the electron plasma frequency, L is the interaction length, c is the velocity of light, and mc²γ₀ is the initial electron energy. $S=0.092$ in the former case and 0.92 in the latter.