Directionality Fields Generated by a Local Hilbert Transform in Optics

Conventional Hilbert transform in time (Kramers Kronig relation) is due to causality, in other words to time asymmetry, i.e. to the arrow of time. The same Hilbert transform applied in space breaks analogously the space symmetry and can introduce unidirectional invisibility, or unidirectional coupling or scattering of fields [1]. We propose a local Hilbert transform, in order to design non-Hermitian optical potentials, generating arbitrary directionality of scattering, with desired shapes and topologies [2], We derive a local Hilbert transform to systematically build such non-Hennitian potentials in two-dimensional space, by modifying the background potentials, the refraction index profiles, 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 (see Fig.1, left part), or for instance to generate the vortices or currents in the probe fields, as another example. Physically, the proposed directionality field generation provides a flexible new mechanism for dynamically shaping and precise control over probe fields leading to novel effects in wave dynamics [3].