Nonlocal homogenization of $\mathcal{PT}$-symmetric multilayered structures

Unique and highly tunable optical properties of $\mathcal{PT}$-symmetric systems and metamaterials enable a plenty of entirely new linear and nonlinear optical phenomena with numerous applications, e.g., for designing subdiffraction lenses, nonreciprocal devices, etc. Therefore, the artificial media with the $\mathcal{PT}$ symmetry attract ever-increasing attention and are now a subject for intensive investigations. One of the commonly used methods providing information about the optical response of artificial nanostructural media is a so-called effective medium theory. Here we examine the possibility of utilizing the effective medium theory for a comprehensive analysis of $\mathcal{PT}$-symmetric multilayered systems composed of alternating loss and gain slabs. We show that applicability of local effective material parameters (or Maxwell Garnett approximation) is very limited and cannot be exploited for a prediction of exceptional points marking a $\mathcal{PT}$ symmetry breaking. On the other hand, nonlocal bianisotropic effective medium parameters can be reliably used, if the thickness of a unit cell is much smaller than the radiation wavelength. In the case of obliquely incident plane waves, we reveal the limitation on the loss-gain coefficient, which should not be too large compared with the real part of the permittivity. We believe that our findings can improve the fundamental understanding of physics behind $\mathcal{PT}$-symmetric systems and advance the development of auxiliary tools for analyzing their peculiar optical response.

Figure 1

Geometry of the $\mathcal{PT}$-symmetric system as an $N$-unit-cell multilayer composed of alternating loss and gain slabs of equal thickness $d/2$ and permittivities ${\varepsilon }_L={\varepsilon }^{\prime }+i{\varepsilon }^{″}$ and ${\varepsilon }_G={\varepsilon }^{\prime }-i{\varepsilon }^{″}$, respectively. A homogeneous bianisotropic slab with material tensors ${\widehat{\varepsilon }}_{\mathrm{eff}}$, ${\widehat{\mu }}_{\mathrm{eff}}$, and ${\widehat{\alpha }}_{\mathrm{eff}}$ calculated using the OEMA is shown in the bottom part of the sketch.

Figure 2

Transmission of the $\mathcal{PT}$-symmetric multilayer as a function of ${\varepsilon }^{″}$ calculated within the full TMM (indicated with “multilayer”) and OEMA of different orders for the period thickness (a) $d=100$ nm and (b) $d=200$ nm. The case of a normally incident wave with $\lambda =1.55\mu \mathrm{m}$ is considered; the structure consists of $N=20$ slabs; ${\varepsilon }^{\prime }=2$.

Figure 3

Dependence of the scattering-matrix eigenvalues on ${\varepsilon }^{″}$ calculated within the full TMM (indicated by “multilayer”) and OEMA of different orders for the period thickness (a) $d=100$ nm and (b) $d=200$ nm. Other parameters are the same as in Fig. 2. The first and the second eigenvalues are designated with solid and dashed curves, respectively.

Figure 4

Angular dependence of transmission for the $\mathcal{PT}$-symmetric multilayer calculated within the full TMM (indicated by “multilayer”) and the second-order OEMA. Upper panel shows the results for ${\varepsilon }^{″}=1$, lower panel, for ${\varepsilon }^{″}=5$. Behavior of reflection is shown only for ${\varepsilon }^{″}=1$ and is omitted in the lower panel not to encumber the figure. The period thickness is $d=100$ nm, and the other parameters are the same as in Fig. 2.

Figure 5

Angular dependence of the scattering-matrix eigenvalues calculated within the full TMM (indicated by “multilayer”) and the second-order OEMA. The upper panel shows the results for ${\varepsilon }^{″}=10$, the lower one, ${\varepsilon }^{″}=20$. The other parameters are the same as in Fig. 4. The first and the second eigenvalues are designated with solid and dashed curves, respectively.

Figure 6

Angular dependence of transmission and reflection calculated within the full TMM (indicated by “multilayer”) and the second-order OEMA for ${\varepsilon }^{\prime }=10$ and ${\varepsilon }^{″}=1$. The other parameters are the same as in Fig. 4.

Figure 7

Square unit cell of the two-dimensional $\mathcal{PT}$-symmetric structure comprising four $z$-oriented circular waveguides.