Abstract
Coulombian diffusion determines a dilution of bunching coefficients in free electron laser seeded devices. From the mathematical point of view the effect can be modeled through a heat-type equation, which can be merged with the ordinary Liouville equation, ruling the evolution of the longitudinal phase space beam distribution. We will show that the use of analytical tools like the generalized Bessel functions and algebraic techniques for the solution of evolution problems may provide a useful method of analysis and shine further light on the physical aspects of the underlying mechanisms.
Highlights
In this paper we will pursue some technical details concerning the computation and the physical understanding of the recent analysis by Stupakov [1] on the effect of Coulomb diffusion on bunching coefficients in echoenabled harmonic generation (EEHG) free electron laser (FEL) seeded devices [2]
The considerations developed in this paper should be understood as a complement to Refs. [1,2], with the aim of providing a more general computational framework, benefiting from the formalism of beam transport employing algebraic means [3]. In this treatment the transport through magnetic elements is treated in terms of an exponential operator acting on an initial phase space distribution and usually does not contain ‘‘transport’’ elements provided by a heat-type diffusion
The effect of the Coulomb diffusion on the e-beam longitudinal phase space distribution is ruled, according to [1,2], by the equation
Summary
In this paper we will pursue some technical details concerning the computation and the physical understanding of the recent analysis by Stupakov [1] on the effect of Coulomb diffusion on bunching coefficients in echoenabled harmonic generation (EEHG) free electron laser (FEL) seeded devices [2]. [1,2], with the aim of providing a more general computational framework, benefiting from the formalism of beam transport employing algebraic means [3]. In this treatment the transport through magnetic elements is treated in terms of an exponential operator acting on an initial phase space distribution and usually does not contain ‘‘transport’’ elements provided by a heat-type diffusion. It is evident that the Coulomb term smears out the oscillations, associated with the bunching coefficients, which tend to disappear or to be strongly suppressed, as we will further discuss in the following. The suppression factor can be written as eÀðm2=LÞfs=1⁄21þð2=B21LÞsg, where L 1⁄4 1=B21D; it coincides with the analogous expression given in Ref. [2], provided that
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