Interplay of the chirps and chirped pulse compression in a high-gain seeded free-electron laser

In a seeded high-gain free-electron laser (FEL), where a coherent laser pulse interacts with an ultrarelativistic electron beam, the seed laser pulse can be frequency chirped, and the electron beam can be energy chirped. Besides these two chirps, the FEL interaction introduces an intrinsic frequency chirp in the FEL even if the above-mentioned two chirps are absent. We examine the interplay of these three chirps. The problem is formulated as an initial value problem and solved via a Green function approach. Besides the chirp evolution, we also give analytical expressions for the pulse duration and bandwidth of the FEL, which remains fully longitudinally coherent in the high-gain exponential growth regime. Because the chirps are normally introduced for a final compression of the FEL pulse, some conceptual issues are discussed. We show that to get a short pulse duration, an energy chirp in the electron beam is important.