Non-Hermiticity-induced wave confinement and guiding in loss-gain-loss three-layer systems

Following up on previous studies on parity-time-symmetric gain-loss bilayers, and inspired by formal analogies with plasmonic waveguides, we study non-Hermiticity-induced wave confinement and guiding phenomena that can occur in loss-gain-loss three-layers. By revisiting previous well-established “gain-guiding” concepts, we investigate analytically and numerically the dispersion and confinement properties of guided modes that can be supported by this type of structures, by assuming realistic dispersion models and parameters for the material constituents. As key outcomes, we identify certain modes with specific polarization and symmetry that exhibit particularly desirable characteristics, in terms of quasireal propagation constant and subwavelength confinement. Moreover, we elucidate the effects of material dispersion and parameters and highlight the potential advantages by comparison with the previously studied gain-loss bilayer configurations. Our results provide additional perspectives on light control in non-Hermitian optical systems and may find potentially intriguing applicability to reconfigurable nanophotonic platforms.

Figure 1

Problem schematic. (a) LGL three-layer configuration, consisting of a layer of gain medium sandwiched between two lossy half-spaces, with relative permittivity distribution as in (1). (b) GL bilayer configuration, with relative permittivity distribution as in (2). All field quantities are assumed as $y$-independent, and propagation is studied along the $z$ direction.

Figure 2

(a), (b) Real and imaginary parts, respectively, of the relative permittivity of the lossy (orange-dashed curves) and gain (purple-solid curves) media, as a function of frequency, assuming the Lorentz-type dispersion models in (3) with ${\varepsilon }_0^{\prime }=5.887, {\varepsilon }_0^{\prime \prime }=2.110$, and $\mathrm{\Gamma }=4.523\times {10}^{-3}{\omega }_0$.

Figure 3

Parameters as in Fig. 2. Imaginary part of the propagation constant as a function of the gain-layer electrical thickness, for the four lowest-order modes at the center frequency ${\omega }_0$. Red squares and diamonds identify TE odd and even modes, respectively. Blue up- and down-triangles identify TM odd and even modes, respectively.

Figure 4

Parameters as in Fig. 2. (a–c) Real and imaginary part of the propagation constant $k_z$, and decay length $L_d$, respectively, as a function of frequency, for the TE ($d=0.168{\lambda }_0$, red-dashed curves) and TM ($d=0.211{\lambda }_0$, blue-solid curves) odd modes supported by the LGL three-layer. The gain-layer thickness $d$ is chosen so as to attain a purely real propagation constant at the center frequency ${\omega }_0$ (cf. Fig. 3). Also shown (magenta-dotted curves) are the results pertaining to the TM mode supported by the GL bilayer.

Figure 5

Parameters as in Fig. 4. (a) Finite-element-computed electric-field magnitude map ($|E_y|$) at the center frequency ${\omega }_0$ pertaining to the TE odd mode in an LGL three-layer of $10{\lambda }_0$ size along the $z$ direction (the gain layer is delimited by a black-dashed rectangle). The structure is excited by an electric line source located at $x=z=0$, and the false-color scale is suitably saturated for better visualization. (b) Transverse cut (black-solid curve) at $z=2{\lambda }_0$, compared with analytical prediction (cyan-dashed curve). Fields are normalized with respect to their maxima. (c–d) Corresponding results for TM odd mode. In this case, the magnetic field ($|H_y|$) is displayed, and a magnetic line source located at $x=z=0$ is considered in the finite-element simulations.

Figure 6

(a, b) As in Figs. 5 and 5, respectively, but at $\omega =0.995{\omega }_0$.

Figure 7

(a, c) As in Figs. 5 and 5, respectively, but with gain-layer thickness reduced by 20% ($d=0.134{\lambda }_0$ and $d=0.169{\lambda }_0$, respectively). (b, d) Corresponding longitudinal cuts (black-solid curves) at the gain-loss interface $x=d/2$, normalized with respect to their maxima. Also shown (cyan-dashed curves) as references are the exponentially decay trends pertaining to the theoretical propagation constants $k_z=\pm (2.059+0.163i)k_0$ and $k_z=\pm (2.323+0.079i)k_0$, respectively.

Figure 8

As in Fig. 4, but assuming $\mathrm{\Gamma }=1.809·{10}^{-2}{\omega }_0$ in (3), with all other parameters unchanged.

Figure 9

As in Fig. 3, but assuming ${\varepsilon }_0^{\prime \prime }=1.055$ in (3), with all other parameters unchanged.

Figure 10

As in Fig. 4, but assuming ${\varepsilon }_0^{\prime \prime }=1.055$ in (3), with all other parameters unchanged. The gain-layer thicknesses in the LGL three-layer are now $d=0.237{\lambda }_0$ for the TE mode, and $0.268{\lambda }_0$ for the TM mode (cf. Fig. 9).