Guided modes in a PT -symmetric waveguide: Real dispersion curves

A $\mathcal{P}\mathcal{T}$-symmetric optical waveguide has a dielectric function with symmetric and antisymmetric real and imaginary parts, respectively. It is well known that if the amplitude $\ensuremath{\sigma}$ of the imaginary part of the dielectric function (representing the balanced gain and loss) is small, the $\mathcal{P}\mathcal{T}$-symmetric waveguide can have real guided modes that are confined around the waveguide core and have a real frequency and a real propagation constant. Many studies on $\mathcal{P}\mathcal{T}$-symmetric waveguides are concerned with the dependence of guided modes on $\ensuremath{\sigma}$ for a fixed frequency. In particular, $\mathcal{P}\mathcal{T}$-symmetric waveguides have exceptional points (EPs) where typically two guided modes coalesce to a single degenerate mode. We analyze the dispersion curves of the real guided modes in a $\mathcal{P}\mathcal{T}$-symmetric waveguide, show that for any positive $\ensuremath{\sigma}$ the number of dispersion curves is always finite and, when $\ensuremath{\sigma}$ is increased, each dispersion curve goes through two transitions and finally disappears on the light line. The EPs in $\mathcal{P}\mathcal{T}$-symmetric waveguides are normally studied for a fixed frequency with $\ensuremath{\sigma}$ being the parameter. We show that the dispersion curves of the real guided modes have local extremum points that are EPs with alternative parameters. The evolution of real dispersion curves (as $\ensuremath{\sigma}$ is increased) is accompanied by the appearance and disappearance of different types of EPs. Although our study is based on a simple $\mathcal{P}\mathcal{T}$-symmetric slab waveguide, similar results are expected for other $\mathcal{P}\mathcal{T}$-symmetric waveguides.