Eigenmodes in a PT -symmetric waveguide: Complex dispersion curves

Many novel wave phenomena have been observed in parity-time (PT)-symmetric optical waveguides with a balanced gain and loss. These phenomena are highly related to the eigenmodes which are solutions of Maxwell's equation without sources. An eigenmode is associated with a propagation constant and an angular frequency. Most existing studies are concerned with guided modes at a real frequency. In this paper, we analyze various eigenmodes in a PT-symmetric slab waveguide, including guided modes with a complex frequency or a complex propagation constant, leaky modes, resonant modes, lasing-threshold modes, and perfect-absorption modes. We study the dispersion curves of eigenmodes and focus on the connection between the real and complex eigenmodes. It is observed that as the amplitude of balanced gain and loss is varied, there exist a few transition points where the connections between real and complex eigenmodes exhibit abrupt changes. In particular, there exists a transition point corresponding to the emergence of lasing-threshold and perfect-absorption modes. It is expected that the wave phenomena revealed in this paper are generic and can be found in other PT-symmetric waveguides. Our work improves theoretical understanding on PT-symmetric waveguides, and may have potential applications in non-Hermitian photonics.