The quantum-kinetic equation for free-electron lasing in a longitudinal electric wiggler is expressed by two methods: a method expanding the momentum distribution function up to the second order and the transition probability up to the next order in h/2pi in the Taylor series forms and a method utilizing only the first-order derivative of the momentum distribution function and the lowest order in 'h' of the transition probability by means of the corresponding principle, the momentum and energy-conservation laws, and the time-reversal invariance of the transition probability. The gain of a free-electron laser using a longitudinal electric wiggler is determined, and emphasis is placed on the single-pass gain with consideration of the increase in the momentum spread.
Shang Hoon Kim, Shang Hoon Kim,
"Quantum-kinetic theory of free-electron lasing in a longitudinal electric wiggler", Proc. SPIE 1227, Free-Electron Lasers and Applications, (1 May 1990); doi: 10.1117/12.18605; https://doi.org/10.1117/12.18605