Theoretical and experimental studies of energy modulation to demodulation in seeded free-electron lasers

Laser manipulation plays a critical role in precisely tailoring relativistic electron beams through energy modulation, enabling the generation of coherent, intense, and ultrashort radiation in accelerator-based light sources such as synchrotron radiation facilities and free-electron lasers (FELs). However, laser-induced energy modulation inevitably degrades electron beam quality by increasing the energy spread, thereby limiting high-repetition-rate operation. Here, we investigate energy modulation and demodulation in a seeded FEL using two modulators separated by a tunable phase shifter. Analytical analysis and three-dimensional simulations show that a $\pi$ phase delay can nearly reverse the laser-beam interaction and substantially suppress the residual modulation. Diagnostics based on coherent undulator radiation and time-resolved measurements are established to characterize weak residual modulation, and a dedicated demodulation undulator is designed for controlled studies. Preliminary experiments performed at the Shanghai soft X-ray FEL facility using the existing seeding beamline demonstrate laser-induced energy-modulation suppression. Together with the analytical and numerical studies, these results establish a practical framework for investigating the transition from energy modulation to demodulation in seeded FELs, with potential applications in high-repetition-rate, fully coherent X-ray sources with improved preservation of electron beam quality.