High-Gain FEL by Microbunching

Numerical simulation of microbunching

Numerical simulation of microbunching

The exponential growth of the FEL pulse energy E

The exponential growth of the FEL pulse energy E as a function of the length z traveled in the undulator. The data (open red circles) were obtained at the first stage of the SASE FEL at DESY, the electron energy was 245 MeV. The solid curve shows the theoretical prediction. The progress of microbunching is indicated schematically. Laser saturation sets in for z ≥ 12 m. Here the microbunches are fully developed and no further increase in laser power can be expected.

The essential advantage of FEL radiation as compared to undulator radiation is its much higher intensity because a large number of electrons radiate coherently. If it were possible to concentrate all electrons of a bunch into a region far smaller than the light wavelength, then all these particles would radiate like a “point macroparticle”. The problem is, however, that the concentration of some 109 electrons into such a tiny volume is totally unfeasible, even the shortest particle bunches are much longer than the wavelength of an X-ray FEL.

The way out of this dilemma is given by the process of microbunching, which is based on the following principle: Electrons losing energy to the light wave travel on a wavelike trajectory of larger amplitude than electrons gaining energy from the light wave. The result is a modulation of the longitudinal velocity which eventually leads to a concentration of the electrons in slices that are shorter than the wavelength. These microbunches are close to the positions where maximum energy transfer to the light wave can happen, and the particles within a microbunch radiate like a single particle of high charge. This increase in the radiation field enhances the microbunching even further and leads to an exponential growth of the energy of the radiation pulse as a function of the length of the undulator.

The FEL power is almost constant in the first section of the undulator, an exponential growth starts only after a certain distance. The exponential regime ends when the microbunches are fully developed and no more electrons are available for increasing the periodic density modulation. Then the FEL power levels off. In this saturation regime it may even happen that energy is pumped back from the light wave into the electron beam.

A key quantity for the technical realization of a high-gain FEL is the gain length, that is the length in which the FEL power grows by a factor e = 2.718. At FLASH the gain length is about 1 meter. The gain length depends critically on the electron beam parameters. To obtain a short gain length, the peak current in the bunch must be very high, in the order of several 1000 A, and the electron beam diameter must be less than 100 μm throughout a long undulator magnet. With increasing electron energy, and correspondingly decreasing FEL wavelength, the gain length grows, and longer undulators are needed. At FLASH the undulator is 27m long, but FELs operating in the hard X-ray regime will be technically even more demanding. In order to achieve laser saturation at a sub-nanometer wavelength the undulator must be more than 100 m long.

Another key quantity is the FEL parameter, which is in the order of a few per mille at FLASH. This parameter characterizes two important properties of the FEL: The bandwidth in the saturation regime is equal to the FEL parameter, and the laser power at saturation is the FEL parameter times the power of the electron beam. Unfortunately the FEL parameter drops by an order of magnitude when going into the X-ray region.