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We studied the topological properties of an extended Su&ndash, Schrieffer&ndash, Heeger (SSH) model composed of a binary waveguide array with alternating real and imaginary couplings. The topological invariant of the periodic structures remained quantized with chiral symmetry even though the system was non-Hermitian. The numerical results indicated that phase transition arose when the absolute values of the two couplings were equal. The system supported a topological zero mode at the boundary of nontrivial structures when chiral symmetry was preserved. By adding onsite gain and loss to break chiral symmetry, the topological modes dominated in all supermodes with maximum absolute value of imaginary energy. This study enriches research on the SSH model in non-Hermitian systems and may find applications in optical routers and switches.

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 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. © 2018 IEEE.