Non-Hermitian propagation in equally-spaced waveguide arrays

A non-unitary transformation leading to a Hatano-Nelson problem is performed on an array of equally-spaced optical waveguides. Such transformation produces a non-reciprocal system of waveguides, as the corresponding Hamiltonian becomes non-Hermitian. This may be achieved by judiciously choosing an attenuation (amplification) of the injected (or exciting) field. The non-Hermitian transport induced by such transformation is studied for several cases and closed analytical solutions, not present in the available literature, are straightforwardly obtained. The corresponding non-Hermitian Hamiltonian may represent an open system that interacts with the environment, either loosing to or being provided with energy from the exterior.