  This paper describes an effective frequency doubler scheme for SASE free
electron lasers. It consists of an undulator tuned to the first harmonic, a
dispersion section, and a tapered undulator tuned to the second harmonic. The
first stage is a conventional soft X-ray SASE FEL. Its gain is controlled in
such a way that the maximum energy modulation of the electron beam at the exit
is about equal to the local energy spread, but still far away from saturation.
When the electron bunch passes through the dispersion section this energy
modulation leads to effective compression of the particles. Then the bunched
electron beam enters the tapered undulator and produces strong radiation in the
process of coherent deceleration. We demonstrate that a frequency doubler
scheme can be integrated into the SASE FEL at the TESLA Test Facility at DESY,
and will allow to reach 3 nm wavelength with GW-level of output peak power.
This would extend the operating range of the FEL into the so-called water
window and significantly expand the capabilities of the TTF FEL user facility.
