Guided modes in non-Hermitian optical waveguides

We study guided modes in non-Hermitian optical waveguides with dielectric layers having either gain or loss. For the case of a three-layer waveguide, we describe stationary regimes for guided modes when gain and loss compensate each other in the entire structure rather than in each layer. We demonstrate that, by adding a lossless dielectric layer to a double-layer waveguide with the property of parity-time ($\mathcal{PT}$) symmetry, we can control a ratio of gain and loss required to support propagating and nondecaying optical guided modes. This novel feature becomes possible due to the modification of the mode structure, and it can allow using materials with a lower gain to balance losses in various optical waveguiding structures. In addition, we find a non-$\mathcal{PT}$-symmetric regime when all guided modes of the system have their losses perfectly compensated.

Figure 1

Schematics of a three-layer non-Hermitian waveguide. Each layer can be either passive or exhibit gain or loss. For visual identification, we use red tint to denote gain layers, blue to denote loss layers, and grey to denote passive layers.

Figure 2

Location of the energy-conserving modes on the plane of parameters of $[\mathrm{Im}\left(n_2\right),\mathrm{Im}\left(n_3\right)]$ for three different values of layer thickness $d$: (a) $d=100$ nm, (b) $d=200$ nm, and (c) $d=300$ nm. In panel (b) points A, B, and C show the special cases, and points E and D correspond to general cases, which are discussed in the text.

Figure 3

Mode structure for three special cases; shown are the electric field amplitudes and phases for (a) a degenerate $\mathcal{PT}$-symmetric case with parameters corresponding to point A in Fig. 1 and (b,c) two cases corresponding to points B and C in Fig. 1, when one of the layers is passive. Parameters of the structures are shown in the corresponding figure panels.

Figure 4

Mode structure for two general cases. Shown are the electric field amplitudes and phases; (a) and (b) correspond to points E and F in Fig. 1, respectively.

Figure 5

Determinant of the matrix $M$ in the logarithmic scale on the plane of complex wave numbers. Shown are the points of the stationary propagating mode (marked by a red “x”) and the nonpropagating modes (marked by black circles).

Figure 6

Location of the energy conserving modes on the plane of parameters of $[\mathrm{Im}\left(n_2\right),\mathrm{Im}\left(n_3\right)]$ for the asymmetric case. Parameters are $d=200$ nm, $n_1=2+0.1i$, $\mathrm{Re}\left(n_2\right)=2$, and $\mathrm{Re}\left(n_3\right)=2.2$.