  We study the tapering optimization scheme for a short period, less than two
cm, superconducting undulator, and show that it can generate 4 keV X-ray pulses
with peak power in excess of 1 terawatt, using LCLS electron beam parameters.
We study the effect of undulator module length relative to the FEL gain length
for continous and step-wise taper profiles. For the optimal section length of
1.5m we study the evolution of the FEL process for two different
superconducting technologies NbTi and Nb3Sn. We discuss the major factors
limiting the maximum output power, particle detrapping around the saturation
location and time dependent detrapping due to generation and amplification of
sideband modes.
