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To create a FEL, a beam of electrons is accelerated to relativistic speeds. The beam passes through a periodic, transverse magnetic field. This field is produced by arranging magnets with alternating poles along the beam path. This array of magnets is sometimes called an undulator, or a "wiggler", because it forces the electrons in the beam to assume a sinusoidal path. The acceleration of the electrons along this path results in the release of a photon (bremsstrahlung or synchrotron radiation, but not in the most common sense of either term).
Viewed relativistically in the rest frame of the electron, the magnetic field can be treated as if it were a virtual photon. The collision of an electron with the virtual photon creates a freely propagating photon (Compton scattering). Mirrors capture the released photons to generate resonant gain. Adjusting either the beam energy (speed/energy of the electrons) or the field strength tunes the wavelength easily and rapidly over a wide range. Because Compton scattering is complicated in and of itself, it is easier to say that the undulator forces the electrons onto a sinusoidal trajectory along the length axis (longitudinal direction) of the undulator in the undulator's rest frame, then transform to a new rest frame in which the undulator moves along the longitudinal direction such that the electrons oscillate about a fixed point at rest. The radiation is then simple dipole radiation due to the oscillations of the electrons about that fixed point. In the rest frame of the undulator, this dipole radiation will be seen as radiation with a shorter wavelength, propagating in a forward direction along the length of the undulator.
Since the energy of an emitted photon (radiation) depends upon the electron velocity and and magnetic field strength, the FEL can be tuned, i.e. the frequency or color can be controlled.
What makes it a laser (light amplification by stimulated emission of radiation) is that the electron motion is in phase (coherent) with the field of the light already emitted, so that the fields add coherently. Since the intensity of light depends on the square of the field, this increases the light output. (Surprisingly, quantum mechanics is not required in this explanation.). In the rest frame moving along the undulator any radiation will still move with the speed of light and pass over the electrons and lets them communicate to get in synchronization. Often same light (that is radiation) is introduced from the outside. Depending on the position along the undulator the oscillation of an electrons is in phase or not in phase with this radiation. The light either tries to accelerate or decelerate these electrons. The electrons thereby gain or lose kinetic energy, and so move either faster or slower along the undulator. This causes the electrons to form bunches. Now they are synchronized, and will in turn emit synchronized (that is coherent) radiation.
