# 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**for the undulators of ELBE (first harmonic). Additionally, the corresponding wavelength λ

_{e}^{kin}_{1}(2) of the radiation is shown.

The gain was calculated for 3 different bunch lengths

**σ**by means of the formula in ref.[1] which describes the medium-gain regime including terms up to third order in the gain parameter

_{t}**g**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.

_{0}The beam and resonator parameters used in the calculation are given in the table below.

U27 |
||
---|---|---|

σ_{t}=0.7ps |
σ_{t}=1.5ps |
σ_{t}=3.0ps |

2007/03/13 |
||

U100 |
||

σ_{t}=1ps |
σ_{t}=2ps |
σ_{t}=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].

The following effects have been taken into account:

- the electron energy spread
**σ**_{E} - the undulator magnetic field inhomogeneity
**σ**(variance of field amplitudes)_{B} - the transverse electron beam emittance (normalized, rms)
**ε**.^{n}_{trans}

Their contributions to the line broadening δλ/λ are given in the following table. They have to be compared with the natural line width arising from the finite interaction time of the electrons traversing the undulator.

Quantity | Parametric description [2] | Estimated value [%] | |
---|---|---|---|

U27 |
U100 |
||

Energy spread at 20 (40) MeV | 2σ_{E}/E |
0.5 (0.25) | |

Field inhomogeneity (at max. field) | 2K_{rms}/(1+K_{rms}^{2}) *σ_{B}/B |
0.4 | |

Transverse emittance (at max. field) | 2K_{rms}/(1+K_{rms}^{2}) *ε^{n}_{trans>}/λ_{u} |
0.05 | 0.01 |

Natural line width (σ) | 1/(2N_{U}) |
0.74 | 1.3 |

[1] S.V.Benson, CEBAF TN#94-065 and private communication

[2] G.Dattoli, A.Renieri and A.Torre, Lectures on the Free-Electron Laser Theory and Related Topics (World Scientific, Singapore, 1993)