A model for the simulation of the time dependent behavior and the analysis of the equilibrium of the coupled system of storage ring and Free Electron Laser (FEL) is presented. The analysis comprises both amplifier and oscillator FEL. Bunch lengthening and energy widening due to wake forces are taken into account in a self-consistent way. The method is based on a mapping algorithm for means and correlations of the electron distribution function, pioneered by K. Hirata. The evolution of the laser field in the oscillator FEL is described by supermodes. The model is used to simulate an FEL in a small 500 MeV storage ring with 100 m circumference. Typical values for the output power, spatial, and spectral characteristics of the emitted radiation are presented.In this paper we present a method to investigate the interaction of a storage ring and a small signal-small gain FEL. We address the time-dependent transient behavior and the equilibrium. Special attention is paid to the temporal and spectral characteristics of the laser output in the oscillator configuration.In the model we will consider the longitudinal phase space of the storage ring, only. Damping and quantum excitation due to synchrotron radiation lead to Gaussian bunches, which are perturbed by two processes. First, discontinuities in the vacuum chamber lead to wake fields by which the leading particles in a bunch perturb the trailing particles. Second, in an amplifier FEL the energy exchange between the transversely undulating electrons and a copropagating laser wave depends crucially on the relative phases and predominantly acts as a noise source for the electron beam. Only the non-linear character of the FEL process leads to an overall energy loss of the electrons and amplification of the laser wave. In an oscillator the situation is complicated due to the presence of a multitude of resonator modes. Furthermore, the FEL process leads to mode coupling and the formation of light pulses, called supermodes, typically having a length of a few percent of the electron bunch length. The supermodes consist of thousands of coupled modes which then act back on the electron beam and perturb it locally where the light pulse is situated, usually slightly ahead of the peak of the electron distribution.
THE STORAGE FLINGSince both the wake field and the FEL depend on collective properties of the electron distribution, a statistical approach is in order. To avoid the need of huge * Work supported in part by Department of Energy contract and by Bundesministerium fiir Forschung und Technologie under contract 05 AX 334 B2. computers we adopt K. Hirata's approach [I] and construct mappings for the means Xi = (zi), and correlations bij = ((zi -Xi)(zj -Xj)), of a Gaussian distribution function in the variables ~1 = wsv,r/ac and zz = (E -Es)/,?&. Here z and E are the longitudinal position and energy of an electron, and ws, v,,a are the revolution frequency, the synchrotron tune, and the momentum compaction factor, respectively. EO is the design energy. In these variab...
