Bunching coefficients in Echo-Enabled Harmonic Generation

TL;DR

This paper models the impact of Coulombian diffusion on bunching coefficients in Echo-Enabled Harmonic Generation free electron lasers using heat and Liouville equations, employing analytical methods to deepen understanding.

Contribution

It introduces a novel analytical approach combining heat and Liouville equations with special functions to analyze bunching coefficient evolution.

Findings

Coulombian diffusion causes significant bunching coefficient dilution.

Analytical solutions provide insights into the physical mechanisms.

Method enhances understanding of phase space evolution in FELs.

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.