Small-Signal Gain
The first FEL experiments had an enhancement of the seeding radiation field of a few percent. In this low-gain regime the undulator length is shorter than the gain length. All modes - growing, decaying, and oscillating - have approximately the same amplitude and the radiation power at the end of undulator is a result of the interference of these modes. If the wavelength of the seeding radiation field fulfills the synchronism condition, the interference is completely destructive and
the gain - the relative change in the radiation field intensity - is zero. The growth rate of each mode depends differently on the deviation from the synchronism condition and a slight deviation 
results in an enhancement of the electromagnetic field.
The small-signal gain was first evaluated by Madey [1]. Although his original theoretical discussion of the FEL used a quantum-mechanical description, it was soon realized that under most conditions a classical description is sufficient to describe the system, since the most relevant quantities, such as gain are independent of the Planck constant.
When the initial em field amplitude is small - and the relative change in the intensity is also small - the gain is called the small-signal gain. For an initially mono-energetic electron beam the small-signal gain, calculated by Madey, is
where 
, Nw is the number of undulator periods, Ip is the beam current, IA=ec/re is the Alfven current (about 17 kA), re is the classical electron radius (re = 2.8.10-15 m), and 
02 is the transverse cross section of the electromagnetic wave (assumed to be larger than that of the electron beam).
The gain curve is proportional to the derivative of the spectrum of the spontaneous radiation. The gain is zero for x=0, at synchronism, and is maximum for x ~ 2.6. The linewidth is of the order of 1/Nw, the inverse of the number of periods in the undulator.
Since the electromagnetic radiation frequency and the electron energy are related by the synchronism condition we can also redefine x, for a fixed frequency, as 
. The gain linewidth then corresponds to an relative energy change of the order of 1/2Nw.
When the radiation field increases the beam kinetic energy decreases, and when the change is about 1/2Nw the gain becomes 0. Hence the maximum FEL efficiency, defined as ratio of the laser intensity to the initial beam kinetic energy, is of the order of 1/2Nw.