Faraday synthesis in direction-dependent imaging

Context. Modern radio interferometers enable high-resolution polarization imaging, providing insights into cosmic magnetism through rotation measure (RM) synthesis. Traditional 2+1D RM synthesis treats the 2D spatial transform and the 1D transform in frequency space separately. A fully three-dimensional approach transforms the data directly from two spatial frequencies and one wave frequency (u, υ, v) to sky-Faraday depth space, using a 3D Fourier transform. Faraday synthesis uses the entire dataset for improved reconstruction, but also requires a 3D deconvolution algorithm to subtract artifacts from the residual image. However, applying this method to modern interferometers requires corrections for direction-dependent effects (DDEs). Aims. We extend Faraday synthesis by incorporating direction-dependent corrections, allowing for accurate polarized imaging in the presence of instrumental and ionospheric effects. Methods. We implemented this method within <ns0:sc>DDFACET</ns0:sc>, introducing a direction-dependent deconvolution algorithm (DDFSCLEAN) that applies DDE corrections in a faceted framework. Additionally, we parametrized the CLEAN components and evaluated the model on a larger subset of frequency channels, naturally correcting for bandwidth depolarization. We tested our method on both synthetic and real interferometric data. Results. Our results show that Faraday synthesis enables deeper deconvolution, reducing artifacts and increasing the dynamic range. The bandwidth depolarization correction improves the recovery of polarized flux, allowing coarser frequency resolution without losing sensitivity at high Faraday depths. From the 3D reconstruction, we also identified a polarized source in a LOFAR survey pointing that was not detected by previous RM surveys. Faraday synthesis is memory-intensive due to the large transforms between the visibility domain and the Faraday cube, and thus is only now becoming practical. Nevertheless, our implementation achieves comparable or faster runtimes than the 2+1D approach, making it a competitive alternative for polarization imaging.