Inclusion of space-charge effects with Maxwell´s equations in the single-particle analysis of free-electron lasers

In free-electron lasers, the motion of electrons is governed by the Lorentz force equations in the presence of a radiation field. The variation of the radiation field then follows from Maxwell equations in which the modulated electron current acts as a source. A self-consistent analysis, which completely incorporates these two concepts, is presented to describe the behavior of the radiation field and the modulation of the electron beam. In this analytical study, we consider the space-charge effect as well as high interaction strength in the small-signal limit so that the result can be compared directly with the traveling-wave tube theory. It is found that the well-known three-wave solution is essentially applicable to the electron beam only. The variation of the radiation field is much more complicated. According to the analysis, there are only three controlling parameters: the pumping strength, the electron density, and the electron energy detuning. For different choices of those parameters, the field can be in a stable regime where its growth is limited or in an unstable regime where it grows exponentially. The boundary between these two regimes is defined quantitatively. The effect of the plasma resonance is observed at high electron densities as a natural result of maximum single-pass gains. The traveling-wave tube is then analyzed as a special example.