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U50 undulator: Electron beam transport simulations

The U50 undulator is used for the wavelength range between 15 and 150 um. We investigated the electron beam transport and the single pass gain in this undulator as a function of electron bunch parameters and undulator strength which is adjusted with the width of the gap between the two arrays of magnets. Here we report a peculiar transport simulation result.

Defocusing

In a planar undulator with infinitely broad magnets the electron beam wiggles in one plane and is focused in the other plane. The finite width of the magnets gives rise to focusing or defocusing in the wiggle plane, depending on the height of the magnets. A Halbach configuration gives a stronger field but more defocusing in the wiggle plane. The U50 undulator is of the Halbach type. We investigated the defocusing of this undulator with GPT using the model for individual magnet. Magnet parameters are chosen such that the magnetic field corresponds to the measured field. Below we give a result of an investigation of this problem. An array of electrons, equidistantly spaced in the wiggle plane (x-plane) is injected into the undulator at 20 MeV. The x-position at the exit of the undulator is plotted as a function of the width of the gap of the undulator. It shows that the defocusing effect is strongest at a gap width of 9 mm.

Loci of x-position (wiggle plane) of electrons at the end of the U50 undulator when the undulator gap is varied. At large gap this position approaches the x-positions of the electrons at the entrance of the undulator.

Chicane between two U27 undulators

The wavelength range from 5 to 15 um is covered by a series of two U27 undulators. We investigated the effect of the chicane, in the drift space between the two undulators, on the free electron laser gain. (ref: R. Wuensch, E. Grosse, U. Lehnert, C.A.J.van der Geer, Numerical study of an Optical Klystron FEL for ELBE, 23tn FEL conference, www.fel2001.de). Here we report on the possibility of phase compensation with the chicane placed in the drift space between the two undulators.
The micro bunching of the electron beam, induced in the first undulator loses its synchronism with the ponderomotive wave in the drift gap. To restore the synchronism, and thereby the FEL mechanism, one should either tune the length of the drift gap or only the pathlength of the electrons in the gap with a chicane. The later method is more convenient as it requires no moving parts.
We found that almost complete compensation is obtained when the chicane is activated proportional to the undulator strength. This is not too strange if one considers that the chicane can be seen as a very short undulator. Its K-value should be equal to the K-value of the real undulators. In our model the chicane has three bending magnets, equally spaced in the drift gap. The result of this compensation can be judged from the spontaneous emission spectrum shown below, as obtained with "G00mf", the GPT model for a set of Gaussian waves in free space.

Spontaneous emission generated by one macroparticle of electrons shot through two undulators ''U27'' with a chicane in the drift gap of 300~mm. The chicane is activated proportional to K. The resonance wavelengt varies from 8.5 to 17 over the range of K.

Our current research for FZR

The FEL mentioned before is being constructed first with only one U27 undulator. Here we report on the detailed analysis of this FEL from start-up to saturation. This analysis is now possible with GPT thanks to the development of a model for multiple Gaussian modes, G00mf. The simulations are time consuming but give very detailed results on the electron beam near saturation. The model is tested with a few simple cases and then applied to this FEL with one U27 undulator. The corresponding parameters are given in the table at right.

One of the model tests

For one pass of one particle through the undulator without initial radiation, the inverse Fourier transform of the output spectrum is a wave train, which has the structure of the static undulator field. The output wavetrain has as many periods as the number of periods of the undulator. This is consistent with dipole radiation from all the periods of the undulator. In practice, the time domain pulse train is used to verify whether the spectral range and the number of modes used in the simulation is sufficient.

When electrons are injected repeatedly, while the radiation remains captured between the mirrors, the amplitude of the whole wave train grows, provided that the new electrons are injected at exactly the right time. If the new electrons are injected a little later, then the wave train gets a triangular envelop. This case is shown in the figure below, for one particle after 10 passes, for exact synchronism and for one period delay.

Upper plot: Spectrum from one macroparticle of 1 pC (electrons) after 10 passes for exact synchronism between electron source and radiation (green) and for one period desynchronism (red). Lower plot: Corresponding pulse trains. The dot is the position of the macroparticle.

Start-up of FEL with one U27 undulator

The result of a simulation with GPT for the FEL from start up to saturation is shown in the figure below. For every new pass a new bunch of 10000 macroparticles, distributed according to the parameters in the table, is sent through the undulator. 50 Modes are assumed to represent the radiation.

Spectral evolution during 100 passes of a new electron bunch through the undulator.

Electron bunch: Evolution of energy distribution

One reason for these simulations is to determine the distribution of the exit electron bunch in saturation in order to design the beam transport system behind the FEL. We show a part of the evolution of the energy distribution up to saturation in the figure below:


Energy distribution after indicated number of passes

Collaboration: This project is commissioned by the Forschungszentrum Rossendorf.