Chiral Metamaterials with $PT$ Symmetry and Beyond

Optical systems with gain and loss that respect parity-time ($PT$) symmetry can have real eigenvalues despite their non-Hermitian character. Chiral systems impose circularly polarized waves which do not preserve their handedness under the combined space- and time-reversal operations and, as a result, seem to be incompatible with systems possessing $PT$ symmetry. Nevertheless, in this work we show that in certain configurations, $PT$ symmetric permittivity, permeability, and chirality is possible; in addition, real eigenvalues are maintained even if the chirality goes well beyond $PT$ symmetry. By obtaining all three constitutive parameters in realistic chiral metamaterials through simulations and retrieval, we show that the chirality can be tailored independently of permittivity and permeability; thus, in such systems, a wide control of new optical properties including advanced polarization control is achieved.

Figure 1

A system of two homogeneous chiral slabs for the realization of a chiral $PT$-symmetric structure. The complex material parameters ${ϵ}_i$, ${\mu }_i$, and ${\kappa }_i$ are the relative permittivity, the relative permeability, and the chirality parameter, respectively, and the subscript $i=\{g,l\}$ denotes whether they are located in the “gain” or the “loss” region. For $PT$ symmetry the shown parameters should satisfy ${ϵ}_g={ϵ}_l^{*}$, ${\mu }_g={\mu }_l^{*}$, ${\kappa }_g=-{\kappa }_l^{*}$, and $L_g=L_l$. The amplitudes of the incident ($b,c$) and scattered ($a,d$) waves are shown, where the subscript $+$ ($-$) accounts for right- (left-) circularly polarized [RCP (LCP)] waves.

Figure 2

$PT$-chiral system of length $L$ with $n=2\pm 0.2i$. The chirality is (a) $\kappa =\pm 0.165+0.165i$ and (b) $\kappa =0.165\pm 0.165i$. Top row: Eigenvalues $\lambda$ of scattering matrix. Middle row: Transmittance $T_{++}$, $T_{--}$ and reflectance $R_L$, $R_R$. Bottom rows: Optical activity $\theta$ and ellipticity $\eta$. The generalized unitarity relation Eq. (5) is plotted as $R_L{R}_R+2\sqrt{T_{++}{T}_{--}}-T_{++}{T}_{--}=1$ (dashed horizontal line). The shaded area denotes the broken $PT$ phase and the dashed vertical line marks an ATR with strong chiral features.

Figure 3

(a) Unit cell of the basic CMM block (top) and its $PT$ counterpart (bottom) satisfying the conditions Eq. (1) at 220 THz. The unit cell is periodically repeated on the $xy$ plane. The length of the CMM along $z$ direction is $w$. (b) Retrieved effective permittivity ${ϵ}_{\text{eff}}$, permeability ${\mu }_{\text{eff}}$, and chirality ${\kappa }_{\text{eff}}$ as a function of frequency for the structures shown in (a). The results of the top (bottom) row correspond to the CMM ($PT$-CMM) system.

Figure 4

Demonstration of $PT$ transition (exceptional point) with a realistic $PT$-chiral metamaterial pair in terms of ${ϵ}^{\prime \prime }$, the imaginary part of the permittivity of the auxiliary gain or loss slab pair (the real part ${ϵ}^{\prime }=2.1$ is constant). The shaded area denotes the broken $PT$ phase. (a) Numerical calculations at 220 THz for the CMM and $PT$-CMM blocks of Fig. 3 in the configuration shown in the schematic on top. (b) Results with the same CMM and the modified $PT$ CMM, which has chirality of inverted sign. Panels: Scattering matrix eigenvalues $\lambda$ (top left), transmittance $T_{++}$, $T_{--}$ and reflectance $R_L$, $R_R$ (bottom left), optical activity $\theta$ (top right), and ellipticity $\eta$ (bottom right). The dashed line in the bottom left-hand panel is the quantity $R_L{R}_R+2\sqrt{T_{++}{T}_{--}}-T_{++}{T}_{--}$. The relaxed $PT$ condition in (b) does not affect the exceptional point, allowing for the $PT$ aspect to be tuned independently from the chiral properties.