On the influence of electron velocity spread on Compton FEL operation

For the field amplitude, a nonlinear integro-differential equation is derived that describes the operation of a Compton FEL in the presence of electron velocity spread typical for modern facilities. Numerical solutions of the equation are in good agreement with particle simulations for the bunching factor less than 0.6, reproduce the frequency detuning spectrum near its maximum, and describe the amplification process up to saturation.