  In some options for circular polarization control at X-ray FELs, a helical
radiator is placed a few ten meters distance behind the baseline undulator. If
the microbunch structure induced in the baseline (planar) undulator can be
preserved, intense coherent radiation is emitted in the helical radiator. The
effects of betatron motion on the preservation of micro bunching in such
in-line schemes should be accounting for. In this paper we present a
comprehensive study of these effects. It is shown that one can work out an
analytical expression for the debunching of an electron beam moving in a FODO
lattice, strictly valid in the asymptote for a FODO cell much shorter than the
betatron function. Further on, numerical studies can be used to demonstrate
that the validity of such analytical expression goes beyond the above-mentioned
asymptote, and can be used in much more a general context. Finally, a
comparison with Genesis simulations is given.
