  In a previous paper we discussed the physics of a microbunched electron beam
kicked by the dipole field of a corrector magnet by describing the kinematics
of coherent undulator radiation after the kick. We demonstrated that the effect
of aberration of light supplies the basis for understanding phenomena like the
deflection of coherent undulator radiation by a dipole magnet. We illustrated
this fact by examining the operation of an XFEL under the steady state
assumption, that is a harmonic time dependence. We argued that in this
particular case the microbunch front tilt has no objective meaning; in other
words, there is no experiment that can discriminate whether an electron beam is
endowed with a microbunch front tilt of not. In this paper we extend our
considerations to time-dependent phenomena related with a finite electron bunch
duration, or SASE mode of operation. We focus our attention on the
spatiotemporal distortions of an X-ray pulse. Spatiotemporal coupling arises
naturally in coherent undulator radiation behind the kick, because the
deflection process involves the introduction of a tilt of the bunch profile.
This tilt of the bunch profile leads to radiation pulse front tilt, which is
equivalent to angular dispersion of the output radiation. We remark that our
exact results can potentially be useful to developers of new generation XFEL
codes for cross-checking their results.
