Simple formulae for the optimization of the FEL gain length including the effects of emittance, betatron oscillations and energy spread

Simple analytical formulae are presented for a quick optimization of the Free Electron Laser (FEL) gain length for given values of radiation wavelength, electron beam current, normalized transverse emittance and energy spread. The optimization parameters include the gap size of the wiggler, the wiggler period and the betatron wavelength (in the case of external focusing). The method is based on the handy formulae for the FEL gain of a Gaussian beam including the effects of energy spread, emittance, and betatron oscillations of the electron beam. We have found a simple relation between the minimum FEL gain length and the optimum betatron wavelength for given energy spread, emittance, and gap size of the wiggler. When the emittance is about the radiation wavelength divided by 4ρ and the energy spread is negligible, this relation shows that the gain length is optimized if the betatron wavelength is chosen so that the betatron phase advances by a half radian in the gain length