Single-pass gain of the ELBE FELs

The single-pass gain is another important quantity characterizing an FEL. It is determined by the parameters of the electron beam (energy, current, emittance, see the tables below), of the undulator (undulator period and parameter), and by the overlap between the electron and the optical beam. The Figures below show an estimate of the single-pass gain G as a function of undulator parameter K_rms (1) and kinetic electron energy E_e^kin for the undulators of ELBE (first harmonic). Additionally, the corresponding wavelength λ₁ (2) of the radiation is shown.
The gain was calculated for 3 different bunch lengths σ_t by means of the formula in ref.[1] which describes the medium-gain regime including terms up to third order in the gain parameter g₀ using a filamentary filling factor and an improved description of gain degradation due to beam quality parameters. Optimum desynchronisation of the optical cavity has been assumed.
The beam and resonator parameters used in the calculation are given in the table below.

U27
σt=0.7ps	σt=1.5ps	σt=3.0ps
		2007/03/13
U100
σt=1ps	σt=2ps	σt=4ps
		2007/03/13

Factors degrading the gain

Some properties of the actual electron beam and of the undulator field may diminish the energy transfer from the electrons to the optical field. The resulting gain degradation can be described by their contributions to the inhomogeneous line broadening in the spontaneous undulator radiation [2].