Temporal and spectral structure of an FEL oscillator during start-up

A previously developed Greenʹs function technique for analyzing the time evolution of low-gain FEL oscillators is applied to the study of the start-up of oscillators from random beam noise. These calculations describe the evolution of the total radiation power and the spectral power as a function of pass number. The theory suggests that an estimate of the linear gain of an oscillator during start-up, as well as the noise level on the electron beam, can be inferred from a measurement of the position of the spectral peak of the FEL output. Analytical results are obtained for electron beams much longer than the slippage distance. A computationally efficient method for the spectral analysis of finite pulse FEL oscillators is described.