Longitudinal Effects, Starting from Noise

The radiation field propagates faster then the electron beam and advances one radiation wavelength per undulator period. Thus, information cannot propagate further than the slippage length N_w where N_w is the total number of undulator periods. If the bunch is longer than the slippage length, parts of the bunch amplifies the radiation independently. While a FEL amplifier is seeded by a longitudinal coherent signal, this is not the case for a FEL, starting from the spontaneous emission (SASE FEL). SASE FELs with electron bunches longer than the slippage length generate longitudinally incoherent signals. Further processing, succeeding the FEL, is required for fully longitudinal coherence (e.g. it can be achieved with a monochromator with a bandwidth smaller than the Fourier limit of the electron bunch length).

A FEL amplifier has a coherent seeding signal and the electron beam can be more or less in resonance with it. The output power level depends on the deviation of the mean energy of the electron beam from the resonant energy _R. A SASE FEL, on the other hand, uses the broadband signal of the spontaneous emission to start the FEL amplification. Independent on the beam energy the resonance condition is always fulfilled, but the SASE FEL amplifies all frequency components within the acceptance bandwidth of the FEL as well. The relative width is /= and is typically much larger than the observed width of an FEL amplifier.

The initial emission level of the spontaneous radiation depends on the fluctuation in the electron density. Because there are only a finite number of electrons per radiation wavelength and the initial ponderomotive force phase is random, electrons can cluster together at random. The radiation from parts of the bunch with a larger clustering will have larger intensity than that of other parts. The variation in these clusters and, thus, the fluctuation in the beam current, depends on the total number of electrons. Due to the random nature of the electron positions, more electrons results in a smoother distribution. The chances for large clusters are reduced.

The effective power level, emitted due to spontaneous radiation and then further amplified by the FEL interaction, is called the shotnoise power [25]. A FEL amplifier, seeded with a power signal smaller than the shotnoise power level, does not operate as a FEL amplifier, but as a SASE FEL instead. Typical values are a few watts for Free-Electron Lasers in the IR regime to a few kilowatts for FELs in the VUV and X-ray regime.

When the electron bunch length is longer than the slippage length the radiation profile contains many spikes. The spikes are also present in the frequency spectrum. The origin is the random fluctuation in the beam density. The shot to shot fluctuation in the radiation pulse energy follows a Gamma distribution [26]\. The only free parameter of this distribution, M, can be interpretated as the number of the spikes in the radiation pulse. The relative width of the distribution is the inverse square root of M. The length of the spikes is approximately (/₀)L_g [27] \cite{cooplength}. A shorter pulse would result in a larger fluctuation of the radiation energy. The fluctuation of the instantaneous power is given by a negative exponential distribution.

From the optical point of view, the electron beam acts as a dielectric, dispersive medium for the radiation field. The field amplitude E evolves as . The imaginary part of results in the growth of the radiation field and, thus, gain guiding, while the real part adds a phase advance to the fast oscillating wave . The effective phase velocity differs from c, the speed of light. The electron beam is dielectric and the radiation field is focused due to the same principle of fiber optic cables. In addition, depends on the deviation of the radiation frequency from the frequency which fulfills the synchronism condition. This dispersion reduces the group velocity below the speed of light [28] and spikes advance less than one radiation wavelength per undulator period. Amplification stops at saturation and the "dielectric'' electron beam becomes non-dispersive.