538.566We study the eigenmode structures with large excesses over the generation threshold in the electron-beam -backward-wave system in terms of the linear theory. It is shown that with increase in the length of the interaction space, the field maximum of both the fundamental and the higher-order modes shifts toward the collector end of the system. Asymptotic expressions for the growth rates, frequencies and spatial structures of the eigenmodes are obtained in the specified limiting case. When a superradiance pulse is generated in the system, a spatial field structure coinciding with the structure of the fundamental modes with a characteristic maximum near the collector end of the system is formed at the initial stage of interaction.1. The studies of the superradiance effects of short electron bunches [1-3] has recently generated interest in analysis of the nonstationary processes in the electron-beam -backward-wave system for large reduced lengths of the interaction space. It was noted in the numerical simulation that the field distribution at the initial linear stage of interaction has a characteristic maximum near the collector end of the system. As the wave propagates further upstream of the electron beam, the mentioned field distribution gives rise to a superradiance pulse at the cathode end of the system. In this relation, it is interesting to study the linear eigenmodes of this system with large excesses over the generation threshold. It should be mentioned that in most papers devoted to the theory of backward-wave oscillators (BWOs), the eigenmodes are considered in terms of stationary nonlinear theory [4][5][6]. For such modes, the field maximum is near the cathode end of the system for any length of the interaction space. It will be shown in the present paper that for the modes calculated within the framework of nonstationary linear theory, a structure with the maximum near the cathode end takes place only with small excesses over the generation threshold. As the length of the interaction space increases, the field maximum of both the fundamental and the higher-order modes shifts toward the collector end of the system. We succeed in analytically calculating the growth rates and eigenfrequencies of the modes, which determine the initial field distribution, in the specified limit.2. Consider interaction between an electron beam and an electromagnetic wave, whose group velocity is opposite to the electron direction, within the framework of the simplest one-dimensional nonstationary model [7]. In the laboratory reference frame, the linear stage of interaction in such a system can be described on the basis of the following system of equations:where z is the dimensionless (normalized to the Pierce parameter) longitudinal coordinate, t is the dimensionless time, and a and J are slowly varied amplitudes of the electromagnetic wave and the high-frequency