similar way, but can reach infrared, optical, UV, or even x-ray wavelengths. The bunched beam then radiates coherently and amplifies the existing radiation.

There is always an energy spread imposed on the electron beam in order to move the electrons together in a bunch. When the beam has an initial energy, or velocity spread, the bunching process can be impaired, reducing gain. In the FEL oscillator working in steady state at high power, the electrons become over-bunched at the end of the undulator and actually take energy back from the radiation field. This is the normal saturation process, and typically limits the fraction of the electron-beam power converted to radiation in a single pass to 1/N.

In an FEL amplifier (as opposed to an oscillator), the high current of the electron beam increases the rate of the electron bunching process along the undulator. For sufficiently high beam current entering a sufficiently long undulator, the same electron bunching process that occurs over many passes in the FEL oscillator can be completed in a single pass along the FEL amplifier. The amplified wavelength can be selected by an external oscillator laser. In the superradiant FEL, there is no external oscillator, so that radiation must first grow from spontaneous emission. As the optical power grows, the spectrum narrows to establish coherence. This process is called self-amplified spontaneous emission (SASE).

The relativistic electrons “see” each rapidly advancing undulator period Lorentz contracted to a shorter wavelength, λ0′ = λ0/γ. Also, the electrons “see” the radiation field passing over them as Doppler shifted to longer wavelengths, λ′ = (1 + βz) γλ ≈ 2 γλ. The condition of resonance between the undulator and radiation forces in the beam frame, λ0′ = λ′, gives the FEL resonance condition in the laboratory frame, λ = λ0/2 γ2.

Two interesting reviews of free electron lasers can be found in Brau (1988) and Pellegrini (1994).


Many of the properties of the FEL can be inferred from the above discussion. These properties are summarized below.

  1. Tunability. Because the FEL uses a single gain medium, the relativistic electron beam, and because the resonant condition can be easily tuned by changing either the electron beam energy or the magnetic field strength, FELs are broadly and easily tuned. A factor-of-10 tunable frequency range has already been demonstrated with the same accelerator and undulator.

  2. High peak power. Because waste energy is carried away at nearly the speed of light and because the lasing medium cannot be damaged by high optical fields, FELs can produce very high peak powers. Gigawatt peak powers have been demonstrated.

  3. Flexible pulse structure. Because the pulse structure of the radiation follows the pulse structure of the electron beam, the mature RF technology of linear

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