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By S. Varro

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33. a) Gaussian mode and b) guided mode The Figure also illustrates the gain guiding mechanism, which becomes relevant as soon as the Rayleigh length becomes longer than the gain length. It is obvious that even in this case the  parameter plays a fundamental role. Indeed, the condition for the guided modes can be written as follows:   u (75) 4 2 3 w02 Without optical cavity, the waist is essentially related to the transverse section of the electronic beam. Before concluding this section, we note that the effects of inhomogeneous broadening and relative reduction of the gain in SASE can be treated using the same procedure given in the case of a FEL oscillator.

Free Electron Lasers The conventional sources use electron beams accelerated in a storage ring, while SASE FEL use an electron beam accelerated in a LINAC (linear accelerator) of high energy. The transverse dimensions of the emitting source (the electron beam) is therefore 100 times smaller in the case of the SASE FEL that in the case of the synchrotron sources. According to what has been said in (1) and given that the brightness is inversely proportional to the transverse section of the beam, the magnification factor reaches ten orders of magnitude.

Typically, γ ≈ 1 mm mrad for λ in the nm range. Note that the local emittance, referred to as “slice”, can vary significantly along the bunch to give hot-spots where lasing can occur. In fact, in contrast to linear colliders where particle collisions effectively integrate over the entire bunch length, X-ray FELs usually concern only very short fractions of the electron bunch length. The integration length is given by the electron-to-photon longitudinal slippage over the length of the undulator, prior to FEL power saturation.

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