Directionally 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-Hermitian 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]. The presentation consists of the general part on the derived local Hilbert transform, and on different modifications of it. For instance, we generalize the theory to multicolour local Hilbert transform to manipulate separately the different frequency components of the field, also consider the discrete local Hilbert transform to unidirectionally couple the arrays of micro-lasers. However we will mainly focus on the applications to manage the fields in spatially extended, nonlinear, laser-like optical systems. In particular we show the concentration and the regularisation of the fields in active micro-laser system, for instance micro-modificated VCSELs (see Fig.1.right part) following the proposed local Hilbert transform of the potentials (of the surface of the substrate)