A free-electron laser (FEL)  transforms the kinetic energy of a relativistic electron beam, produced by a particle accelerator like a microtron, storage ring, or radio frequency (RF) linac, into electromagnetic (EM) radiation. The transformation occurs when the beam goes through an alternating magnetic field, produced by a magnet called an undulator , that forces the electrons to move in an oscillatory trajectory about the axis of the system, as shown in Fig.1. An electromagnetic wave propagates together with the electron beam along the undulator axis, and interacts with the electrons.
Fig.1: Schematic representation of an FEL oscillator, showing its main component: the electron beam, undulator magnet, and optical cavity. The undulator shown is a permanent-magnet planar undulator. The arrows indicate the magnetic field direction. In a FEL amplifier of SASE FEL set-up the optical cavity is omitted. The FEL amplifier is seeded by an external radiation field.
The undulator magnet is a periodic structure in which the field alternates between positive and negative values and has zero average value. It produces an electron trajectory having a transverse velocity component perpendicular to the axis and parallel to the electric field of the wave, thus allowing an energy exchange between the two to take place. One can either transfer energy from the beam to the wave, in which case the device is an FEL, or from the wave to the beam, in which case it is an inverse FEL (IFEL) . In the second case the system is acting like a particle accelerator and can be used to accelerate the electron beam to higher energies.
The energy transfer can take place only if a condition of synchronism between the wave and the beam oscillations is satisfied. This condition gives a relationship between the radiation wavelength , the electron beam velocity z along the undulator axis (measured in units of the light velocity c), and the undulator period 0:
The dimensionless quantity aw=eB00/2 mc is the undulator vector potential normalized to the electron rest mass mc2; e is the electron charge and B0 is the undulator peak magnetic field. The quantity aw is called the undulator parameter, which is the normalized vector potential of the undulator field. Undulator can be of two types: helical and planar. The resonance equation is valid for a helical undulator. In the case of a planar undulator the resonance equation is still valid if we devide the undulator parameter with the square root of 2.
Because of the dependence of the radiation wavelength on the undulator period, magnetic field, and electron-beam energy - quantities that can be easily and continuously changed - the FEL is a tunable device that can be operated over a very large frequency range. At present the range extends from the microwave to the UV . A new FEL is now under construction in the USA  to reach the x-ray region, about 0.1 nm. A similar program is being developed in Germany , and other FEL to cover the intermediate region between 0.1 nm and the UV  are also being considered by several countries.
The efficiency of the energy transfer from the beam kinetic energy to the em wave is between 0.1 to a few percent for most FELs, but it can be quite large, up to about 40%, for specially designed systems. The beam energy not transferred to the em wave remains in the beam and can be easily taken out of the system, to be disposed of, or recovered elsewhere. This fact suggests that high-average power FELs can be designed without the problem, common in atomic and molecular lasers, of heating the lasing medium.
The time structure of the laser beam mirrors that of the electron beam. Depending on the accelerator used, one can design systems that are continuous-wave (cw) or with pulses as short as picoseconds or subpicoseconds.
Tunability, high efficiency, and time structure make the FEL a very attractive source of coherent em power. In some wavelength regions, like the x-ray, the FEL is unique. Its applications range from purely scientific research in physics, chemistry, and biology to military, medical, and industrial applications.
FELs originate in the work carried out in the 1950s and 1960s on the generation of coherent em radiation from electron beams in the microwave region [2,9]. As scientists tried to push power sources to shorter and shorter wavelengths, it became apparent that the efficiency of the microwave tubes, and the power they produced, dropped rapidly in the millimeter region. It was then realized that this problem could be overcome by using an undulator magnet to
modify the beam trajectory , making it possible for the beam to interact with a wave, away from any metallic boundary. Two pioneering experiments at the Stanford University [10,11] proved that the FEL is a useful source of coherent radiation.
The current disadvantage of FELs is the greater complexity and cost associated with the use of a particle accelerator. For this reason the use and development of FELs are mainly oriented to the following:
An order-of-magnitude comparison of the peak brightness obtained or expected for the FEL, compared to other sources of electromagnetic radiation, is given in Fig.2, pointing out again the interest for FELs in the soft to hard x-ray regions.
Fig.2: Peak brightness of Free-electron lasers in the VUV and X-ray regime, compared to 3rd generation light sources .