Fig. 9. Electron motions in a free electron laser. periodic magnetic field. The mechanism by which electron energy is converted into wave energy is described below and in Figs. 9, 10, and 11. The description is based on a paper by Colson (15). In Fig. 9, (1) and (2) show the path of an electron in they-z plane, whereas Fig. 8 shows the x-z plane. Because the magnetic field points alternately up and down in the x direction (see Fig. 8), the path of an electron is as shown in (1) of Fig. 9 when viewed from the laboratory frame of reference. Viewed in the frame moving to the right along the laser axis with the electron's z-direction velocity, the electron simply oscillates as shown in (2) of Fig. 9. Figures 10 and 11 show what can happen, viewed in frame (2) of Fig. 9, when an EM wave propagating to the right interacts with the electron. The top, middle, and bottom pictures in these two figures show successive instants in time, with time increasing from top to bottom. In Fig. 10, the electron velocity and the electric field of the EM wave are in opposite directions. The field does (positive) work on the electron: the wave loses energy as shown by the diminution in amplitude of the electric field, and the electron gains energy. In Fig. 11, the electron velocity and the electric field of the EM wave are in the same direction. The electron does (positive) work on the electric field: the wave gains energy, and the electron loses energy. Initially, that is, on the left side of the periodic magnet in Fig. 8, as many electrons are in the Fig. 10 situation as in the Fig. 11 situation, and there is no net transfer of energy from the electrons to the wave. However, a wave that is slightly off- ■■esonance (one whose crests pass up an electron at a rate that differs slightly from
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