Inclusion of Space-Charge Effects with Maxwell's Equations in the Single-Particle Analysis of Free-Electron Lasers

In free-electron lase:rs, 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 Ute electron beam. In this analytical study, we consider the space-charge effect as well as tdgh interaction strength in the small· signal limi t so that the result can be compared directly with the traveling· wave tube theozy. It is found that the weU-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 onJ.y th ree controlling parantete:rs: 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 limi ted or in an unstable regime where .it grows e.xponenManuscript received January 6, 198 1; revised March 26, 1981. This work was supported by the U.S. Air Force Office of Scientific Re· search. C.-C. Shih was with the Steele Laboratory, California Institute of Technology, Pasadena, CA 91125. He is now with TRW, Redondo Beach, CA 90278. A. Yariv is with Steele Laboratozy, California Institute of Tecllnology, Pasadena, CA 91125. tially. The boundary between these two regimes is defined quandta· lively. The effect of the plasma resonance is observed at high electron densities as a natutal result of maximum single-pass gains. The traveling· wave tube is then analyzed as a special e.xample.