Sideband instability analysis based on a one-dimensional high-gain free electron laser model

When an untapered high-gain free electron laser (FEL) reaches saturation, the exponential growth ceases and the radiation power starts to oscillate about an equilibrium. The FEL radiation power or efficiency can be increased by undulator tapering. For a high-gain tapered FEL, although the power is enhanced after the first saturation, it is known that there is a so-called second saturation where the FEL power growth stops even with a tapered undulator system. The sideband instability is one of the primary reasons leading to this second saturation. In this paper, we provide a quantitative analysis on how the gradient of undulator tapering can mitigate the sideband growth. The study is carried out semianalytically and compared with one-dimensional numerical simulations. The physical parameters are taken from Linac Coherent Light Source-like electron bunch and undulator systems. The sideband field gain and the evolution of the radiation spectra for different gradients of undulator tapering are examined. It is found that a strong undulator tapering ($\sim 10\%$) provides effective suppression of the sideband instability in the postsaturation regime.

Figure 1

The growth rate $|\mathrm{Im}k|$ as a function of $\kappa$. The dispersion curve is obtained by solving Eq. (20) with ${\mathrm{\Omega }}_{\mathrm{syn},0}\approx 4.4$, $|{\mathcal{E}}_0|\approx 10$, and $f_B=f_R=1$. The green dashed lines indicate the analytical approximate solutions of the maximum growth rate $\max |\mathrm{Im}k|$ [Eq. (21)] at $\kappa \approx {\mathrm{\Omega }}_{\mathrm{syn},0}$.

Figure 2

Sideband growth rate as a function of $z$ for two cases: the gentle and strong undulator tapering. The red solid line refers to the gentle tapering and is obtained by numerically solving Eq. (20) and taking the maximum value from the dispersion curve at each $z$ location; the red dotted line is calculated from Eq. (22) for $\mathrm{\Delta }=0.1\%$. The blue solid and black dotted lines refer to the strong tapering, in which the former is obtained from Eq. (20) and the latter from Eq. (23). In the black dotted line, $\mathrm{\Xi }\approx 4.7$ at $z\approx 50\mathrm{m}$. Here $\rho \approx 1.57\times {10}^{-3}$ and $|{\mathcal{E}}_0^{(0)}|\approx 2.52$.

Figure 3

FEL peak power as a function of $z$ for untapered (red), 0.8%-tapered (green), and 10%-tapered (blue) cases.

Figure 4

Sideband field gain $\ln G(z)$ as a function of $z$ for (a) untapered, (b) 0.8%-tapered, and (c) 10%-tapered cases. The numerical simulations are averaged results from 50 independent runs. The solid lines are obtained from solving Eq. (20) and the dashed lines are evaluated by Eqs. (22) and (28). The shaded error bars, denoted as statistical fluctuations ($\pm 1\sigma$) from each independent 1-D FEL numerical simulations, are marked with red and blue and correspond to the lower and upper sideband signals, respectively.

Figure 5

FEL output spectra for (a) untapered, (b) 0.8%-tapered, and (c) 10%-tapered cases. Each of the spectra are overlapped at several equal-distance $z$ locations. The thin red and blue lines denote the lower and upper sampling synchrotron sideband frequencies, respectively. The thick red and blue lines indicate the final synchrotron sideband frequency sampled at the undulator exit.

Figure 6

The main signal (in logarithmic scale) and the ratio $R$ as a function of the taper ratio $\mathrm{\Delta }$. The curves in this figure are obtained from the averaged results of 50 independent runs for each taper ratio.