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 ($ensuremath{sim}10%$) provides effective suppression of the sideband instability in the postsaturation regime.
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