In this report, I present results of a theoretical 1-D model discussed by G. Dattoli, A. Reniere and myself. The validity of a steady-state analysis is discussed and some estimates are given of the effects introduced by the ''lethargy'' of the laser pulse due to the finite length of the e/sup /minus// pulse. Also, I present analytical expressions for the laser pulse in terms of supermodes (wave-packets of cavity modes), their evolution, physical width as well as the associated frequency spectrum. Next, I present results obtained with a 3-D code for the single pass, small-signal gain. In the appendix, I summarize the symbols used in this report. 2 refs., 6 figs., 3 tabs.

We present a novel approach to the 3-dimensional high-gain free- electron laser amplifier problem. The method allows us to write the laser field as an integral equation which can be efficiently and accurately evaluated on a small computer. The model is general enough to allow the inclusion of various initial electron beam distributions to study the gain reduction mechanism and its dependence on the physical parameters. 16 refs., 8 figs., 1 tab.