  Recently, the self-modulation scheme of a weakly pre-bunched electron beam
has been proposed [Yan et al., Physical Review Letters 126, 084801 (2021)],
which is of great promise for high-repetition-rate seeded free-electron lasers
(FELs), such as high-gain harmonic generation (HGHG). In this paper, the
self-modulation scheme is systematically analyzed and optimized, and further
experiments in which the self-modulator is resonant at the second harmonic of
the seed laser are conducted. The three-dimensional numerical simulations show
that the required seed laser intensity in the self-modulation scheme is around
three orders of magnitude lower than that of the standard HGHG through the
optimization of the beam size or the peak current. More importantly, by
reasonably setting the initial energy modulation and the resonance of the
self-modulator, a more prominent bunching factor and lase at the 30th harmonic
of the seed laser can be achieved in a single-stage HGHG. Moreover, the
experiment results confirm that varying the resonance conditions of the
self-modulator can still amplify the laser-induced energy modulation, where
coherent radiation generated even at the 12th harmonic can be observed. These
results indicate that the self-modulation scheme can remarkably reduce the
requirements of the seed laser system while improving the harmonic
up-conversion efficiency, which paves the way for realizing
high-repetition-rate and fully coherent soft x-ray FELs.
