Optimized setups for detection of Megatesla-level magnetic fields through Faraday rotation of XFEL beams

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.