Analytical theory and control of the longitudinal dynamics of a storage-ring free-electron laser

A comprehensive analytical description is given of the longitudinal dynamics of a storage-ring free-electron laser in the presence of a finite light-electron beam temporal detuning. Closed analytical expressions for the main statistical parameters of the system (i.e., beam energy spread, intensity, centroid position, and r.m.s. value of the laser distribution) as a function of the detuning are provided. The transition between the stable “cw” regime and the unstable steady state is shown to be a Hopf bifurcation. This allows us to introduce a feedback procedure which suppresses the bifurcation and significantly improves the system stability. The critical value of the detuning above which the bifurcation occurs is analytically derived as a function of the electron energy and of the beam optics parameters. Results are compared to experiments and display good agreement. Comparisons with other theoretical models are also drawn.

Figure 1

Schematic layout of a SRFEL.

Figure 2

Schematic layout of the pass-to-pass laser-electron beam interaction. $\Delta T$ stands for the period between two successive interactions, ${\tau }^{*}$ is the position of the laser centroid with respect to the peak of the electron density, and $ϵ$ accounts for the laser-electron beam detuning at each pass.

Figure 3

${\sigma }_e∕{\sigma }_0$ is plotted versus the ratio $g_i∕P$. The circles represent the numerical solution of Eq. (5) while the solid line refers to the theoretical estimate (9). The difference keeps smaller than $2\%$.

Figure 4

Numerical simulation performed for the case of Super-ACO SRFEL. The results reproduce different “natural” regimes of the (normalized) laser intensity.

Figure 5

Phase-space portraits for $ϵ=0.1\text{fs}$. Top left panel: $\left(I_{n+1}-I_n\right)∕\Delta T$ versus $I_n$. Top right panel: $\left({\sigma }_{n+1}-{\sigma }_n\right)∕\Delta T$ versus ${\sigma }_n$. Bottom left panel: $\left({\tau }_{n+1}-{\tau }_n\right)∕\Delta T$ versus ${\tau }_n$. Bottom right panel: $\left({\sigma }_{l,n+1}-{\sigma }_{l,n}\right)∕\Delta T$ versus ${\sigma }_{l,n}$.

Figure 6

Phase-space portraits for $ϵ=2\text{fs}$. Top left panel: $\left(I_{n+1}-I_n\right)∕\Delta T$ versus $I_n$. Top right panel: $\left({\sigma }_{n+1}-{\sigma }_n\right)∕\Delta T$ versus ${\sigma }_n$. Bottom left panel: $\left({\tau }_{n+1}-{\tau }_n\right)∕\Delta T$ versus ${\tau }_n$. Bottom right panel: $\left({\sigma }_{l,n+1}-{\sigma }_{l,n}\right)∕\Delta T$ versus ${\sigma }_{l,n}$.

Figure 7

The fixed points are plotted as function of the detuning parameter $ϵ$. Top left panel: Normalized laser intensity. Top right panel: Normalized electron-beam energy spread. Bottom left panel: Laser centroid. Bottom right panel: r.m.s. value of the laser distribution. The symbols refer to the simulations, the solid line stands for the analytic approach based on the numerical solution of Eq. (44), while the long-dashed lines represent the closed analytical expressions.

Figure 8

Real (upper panel) and imaginary (lower panel) parts of the eigenvalues of the Jacobian matrix associated to the system (38) as a function of the detuning parameter $ϵ$. The circle in the picture for the real parts represents the transition from the stable to the pulsed regime, i.e., the Hopf bifurcation.

Figure 9

Behavior of the FEL intensity in absence (upper panel) and in presence (lower panel) of the derivative control system. The parameters utilized for the simulations are those of Super-ACO (see Table ). The value of ${ϵ}_0$ [see Eq. (60)] has been set to $1.3\text{fs}$, i.e., well inside the unstable region of the detuning curve. The stabilization has been achieved using $\beta =6\times {10}^{-3}$.

Figure 10

${ϵ}_c$ is plotted versus the ratio $g_i∕P$. The circles stand for the simulations. The diamonds represent the exact numerical study. The squares refer to the approximate analytical expression (66). Simulations refer to the case of Super-ACO.