Directionality fields generated by a local Hilbert transform

We propose an approach based on a local Hilbert transform to design non-Hermitian potentials generating arbitrary vector fields of directionality, (p) over right arrow((r) over right arrow), with desired shapes and topologies. We derive a local Hilbert transform to systematically build such potentials by modifying background potentials (being either regular or random, extended or localized). We explore particular directionality fields, for instance in the form of a focus to create sinks for probe fields (which could help to increase absorption at the sink), or to generate vortices in the probe fields. Physically, the proposed directionality fields provide a flexible mechanism for dynamical shaping and precise control over probe fields leading to novel effects in wave dynamics.