Management of Light Patterns Based on Local Hilbert Transform

We propose a new approach of optical field management based on a local Hilbert transform, where the non-Hermitian potentials generating arbitrary vector fields of directionality $\vec p(\vec r)$, with desired shapes and topologies are designed. We derive a local Hilbert transform to build systematically 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, to generate vortices in the probe fields, and others. 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.