Effect of an angular trajectory kick in a high-gain free-electron laser

In a free-electron laser, a transverse momentum offset (or “kick”) results in an oscillation of the centroid of the electron beam about the undulator axis. Studying the influence of this effect on the free-electron laser (FEL) interaction is important both from a tolerance point of view and for its potential diagnostic applications. In this paper, we present a self-consistent theoretical analysis of a high-gain FEL driven by such a “kicked” beam. In particular, we derive a solution to the three-dimensional, linearized initial value problem of the FEL through an orthogonal expansion technique and also describe a variational method for calculating the average FEL growth rate. Our results are benchmarked with genesis simulations and provide a robust theoretical background for a comparison with previous analytical results.

Figure 1

Electron beam centroid, FEL gain and bunching factor as functions of $z$, for a kick angle of $p_0=0/0.77/1.54/2.31\mu \mathrm{rad}$ (blue/red/green/magenta). The solid and dashed lines represent the linearized solution and genesis simulation results, respectively.

Figure 2

Radiation beam size versus $z$ for a kick angle of $1.54\mu \mathrm{rad}$ (theory and genesis simulation results).

Figure 3

Electron and radiation horizontal centroids for a kick angle of $1.54\mu \mathrm{rad}$ (analytical results).

Figure 4

Comparison between the scaled, local power growth rate $-{\widehat{\mu }}_{zi}=-{\mu }_{zi}/(2\rho k_u)$ and the results of the periodic analysis (PE). Also shown is the growth rate in the absence of a kick (results for ${\beta }_{ex}=30\mathrm{m}$ and $\widehat{\nu }=-0.5$, as in Figs. 1, 2, 3).

Figure 5

Comparison of the periodic analysis (PE) results with the estimates derived from the Tanaka formula [Eq. (1)] for ${\beta }_{ex}=30\mathrm{m}$. A stronger-than-anticipated reduction of the growth rate is observed.