Design formulas for short-wavelength FELs

Simple formulas for optimization of vacuum ultraviolet (VUV) and X-ray self-amplified spontaneous emission (SASE) free electron lasers (FELs) are presented. The FEL gain length and the optimal β-function are explicitly expressed in terms of the electron beam and undulator parameters. The FEL saturation length is estimated taking into account energy diffusion due to quantum fluctuations of the undulator radiation. Examples of the FEL optimization are given. Parameters of a SASE FEL, operating at the Compton wavelength, are suggested.