Low-Gain FEL with an Optical Cavity

Principle of a low-gain FEL with an optical resonator

Principle of a low-gain FEL with an optical resonator

Principle of a free-electron laser

Principle of a free-electron laser: (a) For visible or infrared light an optical resonator can be used. A gain of a few percent for each passage of a short undulator magnet is sufficient to achieve laser saturation within many round trips through the undulator. (b) In the ultraviolet and X-ray region one can apply the mechanism of self-amplified spontaneous emission where the laser gain is achieved in a single passage of a very long undulator.

The main components of low-gain FELs operating at infrared and optical wavelengths are an electron storage ring or a recirculating linear accelerator in which a train of relativistic electron bunches makes many revolutions, a short undulator magnet, and an optical cavity.

We assume the presence of an initial light wave which may be provided either by an external source such as an optical laser, or by the spontaneously emitted undulator radiation which is captured into an optical eigenmode of the cavity. The bunches take many turns through the undulator. Upon each turn the light intensity grows by only a few percent, which is the reason why such a device is called a low-gain FEL. The small gain per undulator passage, however, does not prevent the FEL from reaching very high output powers in the order of Gigawatts, if the electron beam makes a sufficiently large number of turns.

Special Relativity and Undulator Radiation

As the electrons are accelerated to high energies and relativistic speeds, the ratio between their moving and rest mass (resp. the ratio between total energy W and rest energy W0 ) is called the Lorentz factor:


The relativistic length contraction of the undulator is determined by the factor 1/. The radiation that is emitted by electrons moving with high speed towards an observer is blue-shifted by a factor of about 1/(2) by the relativistic Doppler effect. We can then estimate the wavelength of undulator radiation by applying special relativity twice. We call λu the period of the magnet arrangement, which is the distance between two identical poles. A typical value is λu = 30 mm. As the electrons move straight through the undulator with an average speed close to the speed of light its period appears length-contracted to λ* = λu/ = 30 μm for = = 1000. The electrons oscillate at a high frequency and emit radiation with the wavelength λ*. The observer in the laboratory sees a source that approaches him with high speed. The relativistic Doppler effect then reduces the light wavelength to Formel