Finite pulse effects in self-amplified-spontaneous-emission

We study the effects of the electron density profile on self-amplified-spontaneous-emission (SASE). A general formalism in the linear regime is developed by deriving the coupled Maxwell-Klimontovich equations for an arbitrary density profile including the effects of the energy spread, diffraction, and the betatron oscillation. An explicit solution is obtained for the one-dimensional (1-D) case. The temporal and the spectral intensity profiles of SASE depend linearly on the initial electron correlation function. The correlation function consists of two terms, a term giving rise to the usual spontaneous radiation and its amplification to SASE, and a term representing the coherent bunched beam effect. The latter term has been neglected so far in the treatments of SASE, but it could be significant when there is a variation in the electron density at a length scale comparable to the wavelength. The theory reproduces the well-known results when the electron density is uniform. It also reproduces a recent theory for a finite top-hat density profile and a vanishing energy spread.