  We propose a new approach based on a local Hilbert transform to design
non-Hermitian potentials generating arbitrary vector fields of directionality,
p(r), 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). In particular, 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 new mechanism for dynamically
shaping and precise control over probe fields leading to novel effects in wave
dynamics.
