# FisMat2017 - Submission - View

**Abstract's title**: Simulation of Longitudinal and Transverse Phase Space of Electron Beam in SASE FEL Using Lienard-Wiechert Fields

**Affiliation**: University of Tehran

**Affiliation Address**: Tehran 14399, Iran

**Country**: Iran

Operation of self-amplified spontaneous emission free electron lasers (SASE FELs) is based on micro-bunching which is an instability that converts electrons spontaneous radiation to a coherent emission. This instability leads to the collection of electrons in bunches and conversion of their random distribution to a periodic (bunched) distribution where the period length is the fundamental wavelength of undulator. Micro-bunching also enhances the radiation power of FEL and induces a modulation in longitudinal phase space. The present article is devoted to the simulation of the phase space of electron beam in SASE FEL and the study of micro-bunching. The evolution of 6-D phase space of electron beam is considered through Lienard-Wiechert fields and transfer matrices of charged particles. Six equations describing the evolution of electrons phase space are derived. These coupled equations are solved numerically for a sample of electrons to investigate the temporal evolution of the phase space of electron beam. The results of simulations show that as electrons move in the undulator, the modulation will become more pronounced. Furthermore, electrons are collected in specific positions forming electron bunches. This process continues up to the point of saturation where SASE enters in post-saturation regime. In this regime, micro-bunching is reduced and some electrons do not participate in micro-bunching process. In transverse phase space, it is observed that beam emittance is almost conserved by the presence of quadrupoles.