Undulators are widely used in storage rings as brilliant sources of synchrotron radiation. In this case the superposition of spontaneous radiation of many electrons has no phase correlation, so that the total intensity is proportional to the electron number Ne. In a FEL, as in an atomic laser, there is a phase correlation between emitting electrons. This correlation is obtained by modulating the longitudinal beam density on the scale of the radiation wavelength, a process called bunching. When electrons are bunched together in a distance that is short compared to the wavelength of the radiation, they emit in phase, thus increasing the intensity in the emitted line and simultaneously reducing its width.
The formation of bunching is caused by the interaction between the electrons and the radiation field. It can be summarized by three actions, fundamental to any FEL process:
Each point will be explained in detail below. The FEL amplification couples these basic processes and exploits the inherent
instability of this system. A stronger modulation in the electron density increases the emission level of the radiation, which, in return, enhances the energy modulation within the electron bunch. With stronger energy modulation the formation of the bunching (modulation in the electron positions) becomes even faster. The FEL amplification is stopped, when the electron density modulation has reached a maximum. At this point all electrons have the csame phase of emission and the radiation is fully coherent. The Free-Electron Laser has reached saturation.
The FEL process is either initiated by a seeding radiation field or by the inherent fluctuation in the electron position at the undulator entrance. The first classifies the Free-Electron Laser as an FEL amplifier, while the latter is called a Self-Amplifying Spontaneous Radiation (SASE) FEL, where the initial radiation level is produced by the spontaneous radiation.
In the following we describe the basic FEL process. We start with a simplified model, where transverse effects such as the diffraction of the radiation field due to its finite size is ignore. The discussion of these 3D effects is postponed for now.
be the electric field of a plane wave propagating along the undulator axis. The initial phase of the radiation field at t,z=0 is . This field is transverse to the undulator axis, and thus has components parallel to the transverse electron velocity. An energy transfer between the em field and the electron beam can take place with
where the phase is
This quantity is called the ponderomotive force phase  and plays a key role in FEL physics. The ponderomotive force phase has terms changing at the "fast'' em field frequency and terms changing with the ``slow'' undulator period. The time-average value of the electron energy change, averaged over a few radiation periods, is 0 unless we satisfy a "synchronism condition'' d/dt=0. This condition can be written as with as the "resonant'' velocity for a given undulator period and radiation field wavelength. In the case of relativistic particles, using the relationship and assuming that the energy is much larger than its rest mass and that the normalized transverse velocities are much smaller than one, this condition can be approximately written as =(0/22)(1+aw2). If the synchronism condition is satisfied, the ponderomotive force phase is constant, and the electron beam, oscillating at the slow undulator frequency, can exchange energy with the fast oscillating em wave.
If the electron moves with the resonant velocity, its ponderomotive force phase is constant according to the synchronism condition. Particles with higher energy moves forward with respect to the initial phase 0 , while particles with lower energy fall back.
The distribution of the initial position of the electrons within the bunch is determined by the electron source and the acceleration process. If the electron beam is produced by an accelerator using rf cavities to accelerate the beam, the distribution is characterized by two scale lengths. One is the rf wavelength; the other is the radiation wavelength. In a FEL the first is much larger than the second. The wavelengths used in rf cavities vary from minimum of about 10 cm for an rf linac to infinity for an electrostatic accelerator. At the scale length defined by the rf system the electrons are distributed longitudinally into bunches separated by one rf wavelength. The length of these bunches is a fraction of the wavelength of the accelerating rf and varies from about 1 mm for the case of a 3 GHz rf linac to a continuous beam for the electrostatic case.
At the scale length of the radiation wavelength the electron distribution is in good approximation uniform; there is no correlation between the longitudinal electron positions at the scale of the radiation wavelength except for a residual, but small bunching due to the finite number of electrons per radiation wavelength. It represents the spontaneous radiation level at the given wavelength , which is amplified in a SASE FEL.
If the electrons have the resonant velocity or are close to it, the interaction with the radiation field results in a sinusoidal modulation of the electron energy. Electrons, lying within the phase interval [,+], gains energy and become faster (d/dt>0), while electrons in the interval [+,+2] are slowed down (d/dt<0). As a result all electrons tends to drift towards the phase + and become bunched with the periodicity of the radiation field. The electrons emit in phase and the radiation is coherent.
The radiation emitted by the electrons adds up to the interacting electromagnetic wave. The amplitude E0 and the phase of the electromagnetic field change with the ongoing interaction between radiation field and electron beam. This is apparent because the electrons tends to bunch at a phase different to the radiation field. The radiation phase is slowly adapting to the new phase defined by the bunching phase of the electrons. For a self-consistent description of the FEL interaction, the Maxwell equation for the electromagnetic wave has to be solved as well. If the fast oscillation of the electromagnetic wave is split from the amplitude and phase information, the complex amplitude changes slower than the "slow'', transverse oscillation of the electrons. The change is given by :
with 0 as the magnetic permeability and the sum over all electrons. Note that the right hand side is determined by the degree of bunching in the electron density and is at maximum when all electron phases are identical. It is also clear from the same equation that if more electrons are contributing to the FEL process, the growth in the radiation field is stronger. The strength of this coupling scales with the FEL parameter 
where is the beam plasma frequency, n0 is the electron-beam density, and . Typical values of vary from 10-2 for a millimeter FEL to 10-3 or less for a visible or UV FEL.
The solutions for the FEL equations (change in energy, phase and field amplitude) are of type , where the growth rate depends on the electron distribution in energy and the deviation of the mean energy from the resonance energy . For the simplest case of a mono-energetic beam with <>=R, is the solution of the dispersion equation 3=-1 (). Although energy deviation or finite energy spread add additional terms, the resulting dispersion equation is always cubic for this 1D model.
Of particular interest for the FEL process is the solution of the dispersion equation, which has a negative imaginary part. It corresponds to an exponentially growing mode. The maximum growth rate in this 1D model is . Inserted into the ansatz for the radiation field amplitude E, the e-folding length of the amplification is
which is also referred to as the gain length . Besides the exponentially growing mode, the other modes are either oscillating or decaying. In the beginning of the Free-Electron Laser all modes are of the same magnitude and it requires a few gain lengths before the exponentially growing mode dominates. This retardation in the amplification of the seeding field is called lethargy regime .
The growth in the radiation field amplitude stops at saturation, when the modulation in the electron distribution has reached a maximum. At this point the bunch is broken up into multiple microbunches. The spacing of the microbunches is the radiation wavelength. Because the wavelength is tunable with the electron beam energy the Free-Electron Laser is also a device to modulate the current of a long pulse with free control on the periodicity. This property is useful for some experiments on plasma-beam interaction