Visualization of Branch Points in $\mathcal{P}\mathcal{T}$-Symmetric Waveguides

The visualization of an exceptional point in a $\mathcal{P}\mathcal{T}$-symmetric directional coupler (DC) is demonstrated. In such a system the exceptional point can be probed by varying only a single parameter. Using the Rayleigh-Schrödinger perturbation theory we prove that the spectrum of a $\mathcal{P}\mathcal{T}$-symmetric Hamiltonian is real as long as the radius of convergence has not been reached. We also show how one can use a $\mathcal{P}\mathcal{T}$-symmetric directional coupler to measure the radius of convergence for non-$\mathcal{P}\mathcal{T}$-symmetric structures. For such systems the physical meaning of the rather mathematical term radius of convergence is exemplified.

Figure 1

A $\mathcal{P}\mathcal{T}$-symmetric directional coupler. The structure consists of two coupled slab waveguides. The real and imaginary part of the refractive index is also portrayed. The refractive index only varies in the $x$ direction.

Figure 2

The two trapped modes of the waveguide depicted in Fig. 1 as a function of the non-Hermiticity parameter strength. The eigenmodes approach each other on the real axis as $\Delta \alpha$ increases until a critical value of $\Delta {\alpha }_c\sim 8.4$ is reached. At the critical value one finds a branch (exceptional) point where the two modes coalesce. Beyond the branch point the directional coupler sustains one gain-guiding mode and one loss-guiding mode.

Figure 3

The power distribution for a propagating sum field consisting of the two guided modes, see Eq. (7), for three values of $\Delta \alpha$. As can be readily observed, the beat length (analogous to the beat time period in quantum mechanics) increases as the value of $\Delta \alpha$ approaches the critical value.