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Laser seeding technique have been envisioned to produce nearlytransform-limited pulses at soft X-ray FELs. Echo-Enabled Harmonic Generation(EEHG) is a promising, recent technique for harmonic generation with anexcellent up-conversion to very high harmonics, from the standpoint of electronbeam physics. This paper explores the constraints on seed laser performance forreaching wavelengths of 1 nm. We show that the main challenge in implementingthe EEHG scheme at extreme harmonic factors is the requirement for accuratecontrol 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 seedpulse, the chirp in the electron beam microbunch turns out to be roughlymultiplied by the harmonic factor. In the case of a Ti:Sa seed laser, suchfactor is about 800. For such large harmonic numbers, generation of nearlytransform-limited soft X-ray pulses results in challenging constraints on theTi:Sa laser. In fact, the relative discrepancy of the time-bandwidth product ofthe seed-laser pulse from the ideal transform-limited performance should be nomore than one in a million. The generated electron beam microbunching is alsovery sensitive to distortions of the seed laser wavefront, which are alsomultiplied by the harmonic factor. In order to have minimal reduction of theFEL input coupling factor, it is desirable that the size-angular bandwidthproduct 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 themodulator undulator.

Analytical studies of constraints on the performance for EEHG FEL seed lasers