# Scattering Properties of PT-Symmetric Chiral Metamaterials

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## Abstract

**:**

## 1. Introduction

## 2. Physical Model and Main Equations

#### 2.1. PT-Symmetry Conditions in Chiral Systems

**H**,

**B**, and

**D**). Nevertheless, despite the lack of PT-symmetry in the eigenwaves, there is still the possibility for real eigenvalues (PT-symmetric phase). This possibility is ensured by the degeneracy of RCP/LCP eigenwaves with respect to the frequency, as can be easily seen by applying the condition (3) into Equation (1) [26]. Hence, our analysis demonstrates the possibility to obtain real eigenvalues in PT-symmetric chiral systems and provides the associated conditions. However, as numerical results in scattering systems have shown, the conditions (3) for the chirality parameter are not the only ones that lead to fully PT-symmetric phase, at least in scattering configurations [26]. In such configurations there are cases beyond condition (3) for κ which yield similar features and similarly useful properties [27].

#### 2.2. Chiral Systems in Scattering Configurations under Normal Incidence: The Scattering Matrix

_{0}matrix as is simplified in our case. In Equation (8), ${t}_{++}\equiv {t}_{++}^{\left(L\right)}={t}_{++}^{\left(R\right)}$ and ${t}_{--}\equiv {t}_{--}^{\left(L\right)}={t}_{--}^{\left(R\right)}$ are the transmission coefficients for RCP($+$)/LCP($-$) light, while ${r}^{\left(L\right)}\equiv {r}_{+-}^{\left(L\right)}={r}_{-+}^{\left(L\right)}$ and ${r}^{\left(R\right)}\equiv {r}_{+-}^{\left(R\right)}={r}_{-+}^{\left(R\right)}$ are the reflection coefficients for LCP($-$)/RCP($+$) light (the superscript (L) or (R) indicates incidence from left or right, respectively). The remaining eight scattering coefficients (${t}_{+-}^{\left(L\right)},{t}_{-+}^{\left(L\right)},{t}_{+-}^{\left(R\right)},{t}_{-+}^{\left(R\right)}$ and ${r}_{++}^{\left(L\right)},{r}_{--}^{\left(L\right)},{r}_{++}^{\left(R\right)},{r}_{--}^{\left(R\right)}$) are zero in our case. (Note that the first subscript in t and r indicates the scattered wave polarization and the second the incident wave indicates polarization.)

#### 2.3. Generalized Unitarity Relations for Chiral PT-Symmetric Systems

## 3. Results and Discussion

#### 3.1. PT-Symmetric Chiral Bilayer under Normal Incidence: Different PT Phases and Scattering Characteristics

_{0}) eigenvalues (see Equation (9)) as a function of frequency are plotted in Figure 2e, demonstrating the existence of the two different phases; the PT-symmetric one characterized by unimodular eigenvalues and the PT-broken one characterized by eigenvalues of inverse moduli. At the exceptional point (at $\omega L/c=22.25$) all eigenvalues coincide.

_{A}+κ

_{Β}); see Figure 1). Such a dependence, although having strong influence on the transmission magnitude, does not affect the frequency position of the zeros and the resonances of the transmission, something observable also from Figure 2.

#### 3.2. PT-Symmetric Chiral Bilayer under Oblique Incidence: Controlling the PT-Symmetry Phase

_{0}(see Equation (8)), should be considered [27]. By numerically calculating the eigenvalues of this scattering matrix, the attainable PT-related phases of the bilayer can be identified.

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Our model one-dimensional chiral bilayer. It is infinite along the x- and y-directions and of thickness L = 2d along the z-direction. For parity-time (PT)-symmetry the material parameters of the two media (A and B) should obey ${\epsilon}_{A}={\epsilon}_{B}^{*}$, ${\mu}_{A}={\mu}_{B}^{*}$, ${\kappa}_{A}=-{\kappa}_{B}^{*}$, with ε, μ, κ being the relative permittivity, permeability, and chirality parameter, respectively. The amplitudes of the incident $\left({a}_{\pm},{d}_{\pm}\right)$ and scattered $\left({b}_{\pm},{c}_{\pm}\right)$ waves are also shown, where the subscripts +/− account for RCP/LCP waves.

**Figure 2.**(

**a**–

**d**) Left column: PT-symmetric bilayer (see Figure 1) of length $L$ without chirality. (

**e**–

**h**) Middle column: Chiral PT-symmetric bilayer. (

**i**–

**l**) Right column: Chiral bilayer beyond PT-symmetry (for the material parameters of the systems see text). First row: Eigenvalues of the scattering matrix ${S}_{0}$ (see Equation (9)). Second row: Eigenvectors of the scattering matrix ${S}_{0}$ (see Equations (10) and (11)). Third row: Reflection, R, and transmission, T, (power) coefficients for LCP($-$) and RCP($+$) waves impinging on both left (see superscript (L)) and right (see superscript (R)) sides of the systems. The left hand side (l.h.s.) of the generalized unitarity relation ${R}^{\left(L\right)}{R}^{\left(R\right)}+2\sqrt{{T}_{++}{T}_{--}}-{T}_{++}{T}_{--}=1$ (see Equation (24)) is also plotted (dashed line). Bottom row: Ellipticity, $\eta $, and optical activity, $\theta ,$ of a wave transmitted through the three systems studied. All features are plotted as a function of the dimensionless frequency $\omega L/c$, with c the vacuum speed of light and $L=2d$ the total system length/thickness.

**Figure 3.**Eigenvalues (σ) of the scattering matrix ${S}_{0}$ for the chiral bilayer shown in Figure 1, for incidence angle ${\theta}_{in}={45}^{\xb0},$ as a function of the dimensionless frequency $\omega L/c$, where L is the system length and c the vacuum speed of light. The permittivity and permeability of the two media are as in Figure 2. Panel (

**a**): Simple PT-symmetric system without chirality. Panel (

**b**): Chiral PT-symmetric system. Panel (

**c**): Chiral system beyond PT-symmetry $\kappa \left(-z\right)\ne -{\kappa}^{*}\left(z\right)$ (i.e., ${\kappa}_{A}\ne -{\kappa}_{{\rm B}}^{*})$.

**Figure 4.**Eigenvalues (σ) of the scattering matrix ${S}_{0}$ at incidence angle ${\theta}_{in}={45}^{\xb0}$ as a function of the chirality parameter, κ, for chiral PT-symmetric systems, as the one shown in Figure 1, and for dimensionless frequency $\omega L/c=15.5$. The permittivity and permeability of the two media are as in Figure 2. Panel (

**a**): Scan of the chirality parameter both real and imaginary parts; Panel (

**b**): Scan of the real part of the chirality for $\mathrm{Im}\left(\kappa \right)=-0.04$, i.e., $\mathrm{Im}\left(\kappa \right)\equiv \mathrm{Im}\left({\kappa}_{A}\right)=\mathrm{Im}\left({\kappa}_{{\rm B}}\right)$. Panel (

**c**): Scan of the imaginary part of the chirality for $\mathrm{Re}\left(\kappa \right)=\pm 0.04$, i.e., $\mathrm{Re}\left({\kappa}_{A}\right)=-0.04,$ and $\mathrm{Re}\left({\kappa}_{{\rm B}}\right)=0.04$.

**Figure 5.**PT-symmetric system, as the one of Figure 1 (of length $L$), without chirality (left column) and with chirality (right column), with permittivity and permeability as in Figure 2. (

**a**,

**c**): Eigenvalues (σ) of the scattering matrix ${S}_{0}$ at incidence angle ${\theta}_{in}={39}^{\xb0}$, as a function of the dimensionless frequency ωL/c, with c being the vacuum speed of light. (

**b**,

**d**): Phase diagrams showing the different attainable phases for different incidence angles. Red and green lines indicate the positions of the exceptional points. The horizontal dashed lines in (

**b**,

**d**) indicate the incidence angle considered in plots (

**a**,

**c**), respectively, and their crossings with the vertical dashed lines correspond with the exceptional points of (

**a**,

**c**).

**Figure 6.**Reflection (R), transmission (T), optical activity (θ), and ellipticity (η) for a linearly polarized plane wave incident at both sides of a chiral bilayer, as the one of Figure 1, with permittivity and permeability values as in Figure 2, at $\omega L/c=15.5$ as a function of the chirality parameter $\kappa $ (left two columns) and as a function of incident angle (right two columns). Note that $\kappa $ here denotes the absolute value of both real and imaginary part of the two layers $A$, $B$ (i.e., ${\kappa}_{A}=\kappa -i\kappa $ and ${\kappa}_{B}=-\kappa -i\kappa )$. The subscripts ‖ and ⊥ in the transmission and reflection indicate the components parallel and perpendicular to the plane of incidence, respectively, while the first (second) component indicates the output (input) wave polarization. The coefficients R and T are the squared magnitude of the corresponding coefficients (r and t) of Equations (29) and (30).

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**MDPI and ACS Style**

Katsantonis, I.; Droulias, S.; M. Soukoulis, C.; N. Economou, E.; Kafesaki, M.
Scattering Properties of PT-Symmetric Chiral Metamaterials. *Photonics* **2020**, *7*, 43.
https://doi.org/10.3390/photonics7020043

**AMA Style**

Katsantonis I, Droulias S, M. Soukoulis C, N. Economou E, Kafesaki M.
Scattering Properties of PT-Symmetric Chiral Metamaterials. *Photonics*. 2020; 7(2):43.
https://doi.org/10.3390/photonics7020043

**Chicago/Turabian Style**

Katsantonis, Ioannis, Sotiris Droulias, Costas M. Soukoulis, Eleftherios N. Economou, and Maria Kafesaki.
2020. "Scattering Properties of PT-Symmetric Chiral Metamaterials" *Photonics* 7, no. 2: 43.
https://doi.org/10.3390/photonics7020043