Analytical studies of constraints on the performance for EEHG FEL seed   lasers
  Laser seeding technique have been envisioned to produce nearly transform-limited pulses at soft X-ray FELs. Echo-Enabled Harmonic Generation (EEHG) is a promising, recent technique for harmonic generation with an excellent up-conversion to very high harmonics, from the standpoint of electron beam physics. This paper explores the constraints on seed laser performance for reaching wavelengths of 1 nm. We show that the main challenge in implementing the EEHG scheme at extreme harmonic factors is the requirement for accurate control of temporal and spatial quality of the seed laser pulse. For example, if the phase of the laser pulse is chirped before conversion to an UV seed pulse, the chirp in the electron beam microbunch turns out to be roughly multiplied by the harmonic factor. In the case of a Ti:Sa seed laser, such factor is about 800. For such large harmonic numbers, generation of nearly transform-limited soft X-ray pulses results in challenging constraints on the Ti:Sa laser. In fact, the relative discrepancy of the time-bandwidth product of the seed-laser pulse from the ideal transform-limited performance should be no more than one in a million. The generated electron beam microbunching is also very sensitive to distortions of the seed laser wavefront, which are also multiplied by the harmonic factor. In order to have minimal reduction of the FEL input coupling factor, it is desirable that the size-angular bandwidth product of the UV seed laser beam be very close to the ideal i.e. diffraction-limited performance in the waist plane at the middle of the modulator undulator. 
Gianluca Geloni, Vitali Kocharyan and Evgeni Saldin
physics.acc-ph