Analytical treatment of free-electron laser in the long-pulse limit

Abstract
We discuss a method to obtain analytical solutions of the evolution equation of the optical signal in long-pulse free-electron-laser oscillators in the low-gain and small-signal regime. Supermodes are identified with the eigenstates, harmonic-oscillator orthonormal functions, of a non-Hermitian Hamiltonian. Inhomogeneous broadening effects due to energy spread and emittance are included. The space-time characteristics of the laser field are obtained with a significant reduction in the computer time.