$\mathcal{PT}$-symmetric waveguide system with evidence of a third-order exceptional point

An experimental setup of three coupled $\mathcal{PT}$-symmetric waveguides showing the characteristics of a third-order exceptional point (EP3) has been investigated by Heiss and Wunner in an idealized model of three $\delta$-function waveguides [W. D. Heiss and G. Wunner, J. Phys. A 49, 495303 (2016)]. Here we extend these investigations to realistic, extended waveguide systems. We place major focus on the strong parameter sensitivity rendering the discovery of an EP3 a challenging task. We also investigate the vicinity of the EP3 for further branch points of either cubic- or square-root-type behavior.

Figure 1

$\mathcal{PT}$-symmetric directional coupling. The structure consists of three coupled slab waveguides on a background material with index $n_0=3.3$. The right-hand side displays the real and imaginary part of the refractive index, which only vary in the $x$ direction. Using the dimensionless parameters $s_m$ and $s_{1,2}$, the waveguides' width as well as the separation between them is adjusted. Here, a value of $a=2.5\mu \mathrm{m}$ is chosen. Also note that we allow for a larger real index difference between the middle waveguide and the background material as compared to the outer ones by adding an additional term $n_m$ to the fixed one, $\mathrm{\Delta }n=1.3\times {10}^{-3}$. The imaginary part of the index can be controlled by the gain-loss coefficient $\gamma$ with the vacuum wavelength taken to be $\lambda =1.55\mu \mathrm{m}$.

Figure 2

Propagation constants of the three guided modes of the waveguide system depicted in Fig. 1 as a function of the non-Hermiticity parameter $\gamma$. As $\gamma$ is increased, the outer eigenvalues approach each other, while the middle mode is mostly unaffected by this variation. For ${\gamma }_{\mathrm{EP}3}\approx 0.2568{\mathrm{cm}}^{-1}$, the eigenmodes coalesce in a third-order exceptional point. Beyond this branch point (gray area), the propagation constants become complex.

Figure 3

Verification of the EP3 in Fig. 2 by encircling it in the space of asymmetry parameters (breaking the system's $\mathcal{PT}$ symmetry in either the real or imaginary part of the refractive index) and distance (represented by $s_m$). The upper panels show the characteristic threefold permutation if one encircles the EP3 in the $s_m-a_r$ space (left) or in the $s_m-a_i(=-b_i)$ space (right). The starting points of the respective eigenvalues are depicted by specific symbols. They are the end points of the paths of another eigenvalue. Thus, one can see that the disturbed ground-state branch (leftmost starting point) ends at the starting point of one of the excited states. The path starting there ends at the starting point of the second of the excited states. Finally, the path of the latter ends at the starting point of the disturbed ground state. The corresponding curves shown on the bottom right are parametrized according to Eq. (11), where the loop is performed clockwise for both circles.

Figure 4

C-normalized stationary modes of the waveguide system for increasing non-Hermiticity parameter (a)–(c) $\gamma =0.0{\mathrm{cm}}^{-1}$, (d)–(f) $\gamma =0.1{\mathrm{cm}}^{-1}$, (g)–(i) $\gamma =0.256802{\mathrm{cm}}^{-1}$. The real part of the modes is illustrated by blue solid lines and the imaginary part by red dashed lines. From left to right are shown the ground-state mode and the two excited modes. The gray shaded areas represent the waveguides' positions. $x$ is given in $\mu \mathrm{m}$.

Figure 5

The power distribution for the propagating total field consisting of the three guided modes [see Eq. (15)] for three values of the non-Hermiticity $\gamma$. With increasing values of $\gamma$, the corresponding beat length also increases. In addition, an obvious rise in the intensity can be observed.

Figure 6

(a) Curves denoting the positions in the three-dimensional parameter space $\left(\gamma ,s_m,a_r\right)$ where EP2s are found for the waveguide system depicted in Fig. 1. The parameters $s_1^{\mathrm{EP}3}$, $s_2^{\mathrm{EP}3}$, $n_m^{\mathrm{EP}3}$, and $\mathrm{\Delta }n$ are held fixed and for every $\gamma$ the values of $\mathrm{Re}\left(\beta \right),\mathrm{Im}\left(\beta \right),s_m,a_r$ are determined in a four-dimensional root search such that Eq. (16) is fulfilled. Branches of EP2s connecting either ground-state and first-excited mode or first- and second-excited mode sprout out from the EP3. (b) Magnification of the space around the EP3 from which all lines originate. (c) The circle in the parameter space (solid green line) in Fig. 3 circumscribes the EP3 and two EP2s formed by the branches of the ground state and the first-excited state.

Figure 7

Projections of the two-dimensional curves of EP2s from Fig. 6 on the (a) $\gamma -s_m$, (b) $\gamma -a_r$, and (c) $s_m-a_r$ planes.

Figure 8

Curves denoting the positions in three-dimensional parameter space $\left(\gamma ,s_m,a_i=-b_i\right)$ where EP3s are found for the wave-guide system depicted in Fig. 1. For every $\gamma$, $\mathrm{Re}\left(\beta \right),\mathrm{Im}\left(\beta \right),s_m,a_i=-b_i$ are determined in a four-dimensional root search such that Eq. (16) is fulfilled while the other parameters of Eqs. (8) and (9) are held fixed. The figures contain the projections on the respective parameter planes. EP3s only exist in regions where $a_i=-b_i$ assumes nonzero values. The distributions differ in so far as they belong to different values of $a_i=-b_i$, which is also noticeable in mirrored imaginary parts of the propagation constants (not shown).