  A solid density target irradiated by a high-intensity laser pulse can become
relativistically transparent, which then allows it to sustain an extremely
strong laser-driven longitudinal electron current. The current generates a
filament with a slowly-varying MT-level azimuthal magnetic field that has been
shown to prompt efficient emission of multi-MeV photons in the form of a
collimated beam required for multiple applications. This work examines the
feasibility of using an x-ray beam from the European XFEL for the detection of
the magnetic field via the Faraday rotation. Post-processed 3D particle-in-cell
simulations show that, even though the relativistic transparency dramatically
reduces the rotation in a uniform target, the detrimental effect can be
successfully reversed by employing a structured target containing a channel to
achieve a rotation angle of $10^{-4}$ rad. The channel must be relativistically
transparent with an electron density that is lower than the near-solid density
in the bulk. The detection setup has been optimized by varying the channel
radius and the focusing of the laser pulse driving the magnetic field. We
predict that the Faraday rotation can produce $10^3$ photons with polarization
orthogonal to the polarization of the incoming 100 fs long probe beam with $5
\times 10^{12}$ x-ray photons. Based on the calculated rotation angle, the
polarization purity must be much better than $10^{-8}$ in order to detect the
signal above the noise level.
