Abstract
I present an analytical solution for the phase space evolution of electrons in a self-amplified spontaneous emission (SASE) free-electron laser (FEL) operating in the linear regime before saturation in the resonant case by solving the one dimensional FEL equation together with the solution of the cubic equation, which represents the evolution of the SASE FEL field. The electrons are shown to be bunched around $\ensuremath{\pi}/6$ ahead of a resonant electron every resonant FEL wavelength in the high gain regime. The phase relation is similar to that in a low gain FEL where an electron beam above resonance is injected, explaining the positive FEL gain. The analytical solutions agree well with numerical simulations and are applied to obtain the coherent optical transition radiation (OTR) intensity produced from electron microbunching at FEL wavelength. The coherent OTR intensity is shown to be proportional to FEL intensity.
Highlights
A self-amplified spontaneous emission (SASE) freeelectron laser (FEL) has been developed worldwide as an intense coherent x-ray radiation source [1–3]
I present an analytical solution for the phase space evolution of electrons in a self-amplified spontaneous emission (SASE) free-electron laser (FEL) operating in the linear regime before saturation in the resonant case by solving the one dimensional FEL equation together with the solution of the cubic equation, which represents the evolution of the SASE FEL field
The analytical solutions agree well with numerical simulations and are applied to obtain the coherent optical transition radiation (OTR) intensity produced from electron microbunching at FEL wavelength
Summary
A self-amplified spontaneous emission (SASE) freeelectron laser (FEL) has been developed worldwide as an intense coherent x-ray radiation source [1–3]. The development has been supported by extensive theoretical studies [4 –7], which can account for various types of experimental results such as the exponential increase of SASE power with the undulator length [1] This exponential growth of SASE FEL power can be represented by the solution of the cubic equation, which is derived from one dimensional (1D) FEL equations representing both the electron dynamics in the laser field and the dynamics of the FEL field [4,6,8]. Evolutions of the field envelope a; , the energy i and phase i of the ith electron during FEL interaction are, respectively, given by [8]. The angular bracket indicates the average of all the electrons in the volume V around
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