Abstract
The interaction of the pump wave, the scattered wave and the space-charge wave in a free-electron laser with an electromagnetic wave wiggler and an axial guide field is analysed, where both the scattered wave and pump wave are circularly polarised. The linear dispersion and non-linear growth of these three waves are derived in the laboratory frame. The possibility of explosive instabilities for a fast electromagnetic wave pump propagating counter to the electron beam is discussed and its critical condition is ( omega p+kp nu z0)- omega b/ gamma zo< omega c<(kp nu z0), where omega p and kp are the pump wave frequency and wave number, omega b, nu zO and gamma z0 are the plasma frequency, initial axial velocity and Lorentz factor of the relativistic electron beam, respectively, and omega c is the electron cyclotron frequency in the axial guide field. Group II orbits ( omega c> omega p+kp nu z0) may no longer give positive gain in the process of three-wave interaction.
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