$\mathcal{PT}$-symmetric chiral metamaterials: Asymmetric effects and $\mathcal{PT}$-phase control

We investigate the influence of chirality on the $\mathcal{PT}$-symmetric and $\mathcal{PT}$-broken phase of $\mathcal{PT}$-symmetric chiral systems. Starting from the point that transverse magnetic (TM) and transverse electric (TE) waves have different exceptional points, we show that with circularly polarized waves (which are linear combinations of TM and TE waves) mixed $\mathcal{PT}$-symmetric phases can be realized and the extent of these phases can be highly controlled by either or both the chirality and the angle of incidence. Additionally, while the transmission of both TM and TE waves in nonchiral $\mathcal{PT}$-symmetric systems is the same for forward and backward propagation, we show that with chirality this symmetry can be broken. As a result, it is possible to realize asymmetric, i.e., side-dependent, rotation, and ellipticity in the polarization state of the transmitted wave. Our results constitute a simple example of a chiral $\mathcal{PT}$-symmetric optical system in which the various phases (full $\mathcal{PT}$, mixed, broken) and the asymmetric effects can be easily tuned by adjusting the chirality parameter and/or the angle of incidence.

Figure 1

The model $\mathcal{PT}$-symmetric chiral system studied in the present work. It consists of two layers characterized by complex permittivities, permeabilities, and chiralities. It is illuminated by obliquely incident either TM-polarized plane wave (blue) or TE-polarized plane wave (grey).

Figure 2

Schematic representation of the scattering of circularly polarized plane waves by a two-layer $\mathcal{PT}$-symmetric chiral optical system.

Figure 3

$\mathcal{PT}$-symmetric (white), mixed $\mathcal{PT}$-symmetric (light-gray) and broken $\mathcal{PT}$-symmetric (dark-gray) phases for obliquely incident waves for the system shown in Fig. 1 with chirality $\kappa =0$. The figure shows the eigenvalues of scattering matrices at ${\theta }_i={30}^{\circ }$: (a) for TM, (b) TE, and (c) for CP waves. The EPs versus frequency as the incidence angle varies are illustrated in (d). The relevant material parameters of the two media are: $\varepsilon \left(-z\right)=4-0.6i, \varepsilon \left(z\right)=4+0.6i, \mu \left(-z\right)=1.5-0.15i, \mu \left(z\right)=1.5+0.15i$.

Figure 4

$\mathcal{PT}$-symmetric, mixed $\mathcal{PT}$-symmetric, and broken $\mathcal{PT}$-symmetric phases in a chiral $\mathcal{PT}$ system under oblique incidence of CP waves. The relevant material parameter of the two media are: $\varepsilon \left(-z\right)=4-0.6i, \varepsilon \left(z\right)=4+0.6i, \mu \left(-z\right)=1.5-0.15i, \mu \left(z\right)=1.5+0.15i$ and $\kappa \left(-z\right)=-0.05+0.05i, \kappa \left(z\right)=0.05+0.05i$. Panel (a) shows the eigenvalues, $|{\sigma }_i|,i=1-4$, of $S$ matrix vs frequency for ${\theta }_i={30}^{\circ }$. Panels (b), (c), and (d) show these eigenvalues as a function of chirality parameter, while panels (e), (f), and (g) show them as a function of incidence angle for three different dimensionless frequencies, $\frac{\omega L}{c}=13.0, 14.0,$ and $15.5$, respectively.

Figure 5

Reflected and transmitted power coefficients for the chiral $\mathcal{PT}$ system shown in Figs. 1 and 2 under oblique incidence of CP waves. The relevant material parameters of the two media are the same as in Fig. 4. Panels (a) and (b) show the reflection and transmission vs chirality for RCP incident waves. Panels (c) and (d) show the reflection and transmission vs chirality for LCP incident waves. The angle of incidence is 30° and the dimensionless frequency is $\frac{\omega L}{c}=13$. The positions of the EPs [see Fig. 4] are marked by vertical dotted lines.

Figure 6

Transmission coefficients for TM and TE polarized plane waves incident from the left [(a), (c)] and right [(b), (d)] side of the $\mathcal{PT}$-symmetric system shown in Fig. 1. The relevant material parameter of the two media are $\varepsilon \left(-z\right)=4-0.6i, \varepsilon \left(z\right)=4+0.6i, \mu \left(-z\right)=1.5-0.15i, \mu \left(z\right)=1.5+0.15i$ and $\kappa \left(-z\right)=-0.05+0.05i, \kappa \left(z\right)=0.05+0.05i$. For (c) and (d) the angle of incidence is 30°, while for (a) and (b) ${\theta }_i=0^{\circ }$.

Figure 7

Ellipticity and orientation angle of polarization ellipse for TM [(c), (d)] and TE [(e), (f)] polarized plane waves incident from the left and right side of our $\mathcal{PT}$-symmetric system shown in Fig. 1. The material parameter of the two media are the same as in Fig. 6.

Figure 8

Copolarized transmission and reflection coefficients [(a), (b)] and cross-polarized transmission and reflection [(c), (d)] for TM and TE incident waves as a function of incidence angle, at dimensionless frequency $\frac{\omega L}{c}=18.3$. Panels (e)–(h) show the ellipticity and orientation angle of polarization ellipse for TM [(e), (g)] and TE [(f), (h)] polarized plane waves incident from the left and right side of our $\mathcal{PT}$-symmetric system. The material parameters of the two media are the same as in Fig. 6.

Figure 9

Problem geometry. A (1D) structure, consisting of two layers.

Figure 10

The polarization ellipse for an elliptically polarized plane wave with an ellipticity ($\chi$) and an orientation angle ($\psi$).

Figure 11

The eigenvalues of $S$ matrix as a function of (a) real part of the chirality parameter ($\kappa \left(-z\right)=-\mathrm{Re}\left[\kappa \right]+0.05i,\kappa \left(z\right)=\mathrm{Re}\left[\kappa \right]+0.05i$) and (b) imaginary part of the chirality ($\kappa \left(-z\right)=-0.05+\mathrm{Im}\left[\kappa \right]i,\kappa \left(z\right)=0.05+\mathrm{Im}\left[\kappa \right]i$), for obliquely incident waves at ${\theta }_i={30}^{\circ }$ and for dimensionless frequency $\frac{\omega L}{c}=13.0.$ The material parameter of the two media are: $\varepsilon \left(-z\right)=4-0.6i, \varepsilon \left(z\right)=4+0.6i, \mu \left(-z\right)=1.5-0.15i, \mu \left(z\right)=1.5+0.15i$.