  We show that radiation pulses from an X-ray Free-Electron Laser (XFEL) with a
planar undulator, which are mainly polarized in the horizontal direction,
exhibit a suppression of the vertical polarization component of the power at
least by a factor $\lambda_w^2/(4 \pi L_g)^2$, where $\lambda_w$ is the length
of the undulator period and $L_g$ is the FEL field gain length. We illustrate
this fact by examining the XFEL operation under the steady state assumption. In
our calculations we considered only resonance terms: in fact, non resonance
terms are suppressed by a factor $\lambda_w^3/(4 \pi L_g)^3$ and can be
neglected. While finding a situation for making quantitative comparison between
analytical and experimental results may not be straightforward, the qualitative
aspects of the suppression of the vertical polarization rate at XFELs should be
easy to observe. We remark that our exact results can potentially be useful to
developers of new generation FEL codes for cross-checking their results.
