Directionality Fields generated by a Local Hilbert Transform

TL;DR

This paper introduces a novel method using a local Hilbert transform to design non-Hermitian potentials that can generate complex vector fields with specific shapes and topologies, enabling advanced control over wave dynamics.

Contribution

It presents a systematic way to construct non-Hermitian potentials for arbitrary vector field generation using a local Hilbert transform, expanding capabilities in wave control.

Findings

Successfully designed potentials for sinks and vortices

Demonstrated control over wave field topology

Enhanced wave manipulation techniques

Abstract

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.