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.
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Dattoli et al. (1987) studied this question.
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