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We discuss a phenomenological one-dimensional, self-consistent, multimode (longitudinal) theory for a free-electron-laser oscillator. A strong fundamental mode and a weak sideband mode are followed through many passes in the optical cavity as the energy of the electron beam decreases monotonically. The time evolution of the laser power exhibits oscillations produced by the sequential growth to saturation and subsequent decay of longitudinal modes evenly distributed in frequency.

We propose an analytical solution to the propagation equation of the optical pulse both in free-electron lasers and in optical klystrons. Our results include in a natural way the lethargy and wave-packet spreading.

In this paper we use a recently suggested analytical solution of the pulse propagation equation in free-electron lasers (FEL) to get further information on the lethargy and on the velocity of propagation of the optical packet. We also introduce a refractive index for the FEL interaction. Finally, the supermodes relevant to the regime of long electron bunch are found analytically and a comparison with earlier numerical results is carried out.