Parametric instability of optical non-Hermitian system in the vicinity of a singular point

In contrast to Hermitian systems, eigenmodes of non-Hermitian systems are non-orthogonal. As a result, the system energy depends not only on the amplitude of eigenmodes, but also on the phase difference between them. We show that it is possible to amplify the amplitude of eigenmodes without the increase of the system energy. In addition, we show that whenever the system is in a singular point, any infinitesimal deviation of system parameters results in the amplification of eigenmode amplitudes. As a result, the system becomes non-stable with respect to perturbations. We demonstrate such an instability by the example of two coupled waveguides, where losses exceed gain and all eigenmodes are evanescent.