Due to start-up from shot noise, typical SASE XFEL pulses exhibit poor longitudinal coherence. Self-seeding
schemes can be use to improve it. Recently, a novel single-bunch self-seeding scheme was proposed, based on a
particular kind of monochromator, which relies on the use of a single crystal in Bragg-transmission geometry.
In its simplest configuration, the self-seeded XFEL consists of an input undulator and an output undulator
separated by such monochromator. However, in some experimental situations this simplest two-undulator configuration
is not optimal. The obvious and technically possible extension is to use a setup with three or more
undulators separated by monochromators. This amplification-monochromatization cascade scheme is distinguished,
in performance, by a small heat-loading of crystals and a high spectral purity of the output radiation,
and is particularly advantageous for the European XFEL. The power of the output signal can be further increased
by tapering the magnetic field of the undulator. Once the cascade self-seeding scheme is combined
with tapering in a tunable-gap baseline undulator at the European XFEL, a source of coherent radiation with
unprecedented characteristics can be obtained at hard X-ray wavelengths, promising complete longitudinal and
transverse coherence, and a peak brightness three orders of magnitude higher than what is presently available
at LCLS. Additionally, the new source will generate hard X-ray beams at extraordinary peak (TW) and average
(kW) power level. The proposed source can thus revolutionize fields like single biomolecule imaging, inelastic
scattering and nuclear resonant scattering. Our self-seeding scheme is extremely compact, and takes almost no
cost and time to be implemented. The upgrade proposed in this work could take place during the commissioning
stage of the European XFEL, opening a vast new range of applications from the very beginning of operations.
We present feasibility study and exemplifications for the SASE2 line of the European XFEL.