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

In non-Hermitian systems the amplitude of the scattering matrix eigenvalues is either below (decay) or above (growth) unity. In the case of PT-symmetry, where gain and loss are in balance, the eigenvalues of the scattering matrix can be unimodular and are in fact unimodular below the exceptional point; thus their calculation offers the possibility to identify the exceptional point and the different PT-related phases. To investigate this possibility and its associated effects in the case of chiral systems, we investigated the system of

Figure 1 with some representative material parameters satisfying the PT-symmetry conditions (Equation (3)):

${\epsilon}_{A}=3.0-0.4i,{\epsilon}_{{\rm B}}=3.0+0.4i,{\mu}_{A}=1.1-0.1i,{\mu}_{{\rm B}}=1.1+0.1i$ and

${\kappa}_{A}=-0.04-0.04i,{\kappa}_{{\rm B}}=0.04-0.04i$. For those parameters the scattering matrix (

S_{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.

As has been discussed already in

Section 2, the eigenvalues in our case are independent of chirality, since both the reflection coefficients and the transmission product

${t}_{++}{t}_{--}$ are chirality independent. This is also verified numerically in

Figure 2, where, besides the eigenvalues of the chiral PT system, we have also plotted the eigenvalues of a system with the above PT-symmetric permittivity and permeability values, but with

κ = 0 in both media (see

Figure 2a) and for a system without any symmetry in

κ; i.e.,

${\kappa}_{A}=-0.06-0.02i,{\kappa}_{{\rm B}}=-0.04-0.01i$ (see

Figure 2i). For all three systems the scattering matrix eigenvalues shown in

Figure 2 are identical.

For the above-mentioned three systems we have calculated also the amplitudes of the scattering matrix eigenvectors (second row of

Figure 2) and the transmission (

T) and reflection (

R) power coefficients for circularly polarized incident waves (third row of

Figure 2). From the eigenvector plots (depicting the non-zero, non-unity components of Equations (10)−(11)) it can be observed that, while in the case of non-chiral PT-symmetric systems (

Figure 2b) the ratio of the two non-vanishing components for each eigenvector is unimodular below the exceptional point, in chiral PT-symmetric structures (

Figure 2f) it is not. Considering a system excitation configuration of the form of an eigenvector, the corresponding scattered waves, which are pure circularly polarized waves of opposite handedness at the left and right sides of the slab, above the exceptional point will be either exponentially growing or attenuating (depending on the incident wave configuration). At particular frequencies above EP, these two modes (growing-attenuating) are expected to give simultaneous

CPA and lasing of circularly polarized waves, analogous with the CPA-lasing modes of the non-chiral PT-symmetric systems [

18,

19]. The particular circular polarization favored in our foreseen CPA-laser modes depends mainly on the sign of the Im(

κ) of the two chiral slabs. We have to note here that the above-described eigenvector behavior seem to survive qualitatively, even if the chiral system is beyond PT-symmetry (see

Figure 2j).

Regarding the transmission and reflection coefficients for our isotropic chiral medium, as expected

${T}_{++}$ and

${T}_{--}$ do not depend on the side of incidence (since the system is reciprocal), while a similar condition does not hold for the reflection coefficients, as can be seen in

Figure 2g. Comparing

Figure 2g with

Figure 2c,k (i.e., the case of non-chiral PT-symmetric structure and the case beyond PT-symmetry in

κ, respectively), we can see that the chirality parameter has strong influence on the transmission coefficient and none on the reflection. Note that the chirality independence of the reflection coefficient was also observed in the corresponding analytical expressions (even for non-PT systems; see

Section 2), while from the same expressions (see Equations (12)–(21) and the paragraph after them) one can conclude that the transmission (power) coefficient depends on the chirality parameter exponentially (in particular

T depends on the Im(

κ_{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.

In the rest of the current subsection we investigate the impact of our system on the polarization of an incoming wave. For that we examine first the optical activity,

θ, of our structure, which for absence of cross-polarized transmission terms (e.g.,

${t}_{+-}$) is given by

$\theta =0.5\left[\mathrm{Arg}\left({t}_{++}\right)-\mathrm{Arg}\left({t}_{--}\right)\right]$. Taking into account Equations (12) and (13), we can find that for a general chiral bilayer as the one of

Figure 1 (i.e., prior application of PT-symmetry conditions)

Due to the symmetries imposed by applying PT (Equation (3)), the real part of the chirality

$\kappa $ changes sign across the bilayer. Hence, the polarization rotation occurring in the gain slab is subsequently canceled out when the wave passes through the loss slab, resulting in zero optical rotation, as is demonstrated also in

Figure 2h. Next we consider the other important polarization-related property, which is the circular dichroism (CD). A quantity directly related with the CD response is the transmitted wave ellipticity,

η, for a linearly polarized incident wave, which is given by

The second right hand side (r.h.s.) of Equation (26) has been obtained employing Equations (12) and (13), concerning general material parameters in the two slabs. In a chiral PT-symmetric system, the imaginary part of chirality preserves its sign across the system implying a very strong circular dichroism (CD), resulting in a large degree of the transmitted wave ellipticity, as shown/confirmed also in

Figure 2h.

Summarizing, as shown in

Figure 2 and discussed also in connection with analytical calculations, the PT-symmetric features of a chiral PT-symmetric bilayer (i.e., different PT-phases, exceptional point, structure resonances) are totally independent of chirality. On the other hand, the chirality strongly affects the polarization state of the wave passing through the bilayer (through its effect to its ellipticity). This shows that if one has the ability to control separately the system permittivity/permeability and the system chirality (as is to a large extent possible in chiral metamaterials), one can combine or superimpose almost at will the PT-related and the chirality-related features, achieving fascinating or important in applications effects (e.g., CPA-lasing for circularly polarized waves).

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

As has been already discussed, in the case of waves normally incident on the structure of

Figure 1, the position of EP which characterizes the transition from the PT-symmetric to the PT-broken phase, is totally independent of chirality. As we discuss in this section, the situation changes for oblique incidence, where chirality strongly affects the different PT-related phases and the EPs.

Since, as was showed recently [

40], transverse electric (TE) and transverse magnetic (TM) polarized waves are associated with different exceptional points, it follows as a result that for circularly polarized light under oblique incidence a mixed phase is realizable. Therefore, it is interesting to examine the different PT-related phases and the phase transitions for CP light obliquely incident on our chiral bilayer structure. In this case, the transmission and reflection coefficients

${t}_{+-}^{\left(L\right)}$,

${t}_{-+}^{\left(L\right)}$,

${r}_{++}^{\left(L\right)}$,

${r}_{--}^{\left(L\right)}$ and

${t}_{+-}^{\left(R\right)}$,

${t}_{-+}^{\left(R\right)}$ ${r}_{++}^{\left(R\right)}$,

${r}_{--}^{\left(R\right)}$ are not zero anymore, and, hence, the most general scattering matrix,

S_{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.

As an example, we investigate here a system with permittivity and permeability values the same as in

Figure 2 (i.e.,

${\epsilon}_{A}=3.0-0.4i,{\epsilon}_{{\rm B}}=3.0+0.4i,{\mu}_{A}=1.1-0.1i,{\mu}_{{\rm B}}=1.1+0.1i)$. In

Figure 3 we plot the eigenvalues of the scattering matrix

${S}_{0}$ as a function of the normalized frequency,

$\frac{\omega L}{c}$, at incidence angle

${\theta}_{in}={45}^{\xb0}$ and three different cases regarding chirality: (a) Without chirality

${\kappa}_{A}={\kappa}_{{\rm B}}=0$, (b) with chirality that respects PT-symmetry (i.e.,

${\kappa}_{A}=-0.04-0.04i,{\kappa}_{{\rm B}}=0.04-0.04i),$ and (c) with

${\kappa}_{A}=-0.06-0.02i,{\kappa}_{{\rm B}}=-0.04-0.01i$ (i.e., beyond PT-symmetry). It can be observed that a consequence of obliquely incident waves is the appearance of mixed phases, where one pair of eigenvalues is unimodular while the other is not. Moreover, in contrast to what happens for normal incidence, for oblique incidence the positions of the EPs (two in this case) are strongly affected by chirality.

This implies that one important impact of chirality on the PT-related features of our structure is the tuning of the EPs.To analyze further the impact of chirality on the different PT–related phases of a chiral bilayer, we investigate the different PT-related phases for a system with permittivities and permeabilities, as in

Figure 3, by scanning the PT-obeying chirality parameter (both real and imaginary part) at a fixed frequency,

$\omega L/c=15.5,$ and incidence angle

${\theta}_{in}={45}^{\xb0}$. The results for the eigenvalues of the scattering matrix as a function of chirality are illustrated in

Figure 4a. In

Figure 4a we can see that, as we increase the chirality, the system passes from PT-symmetric phase to mixed PT-symmetric phase (light grey) and, with further increase of the chirality, to PT-broken phase, indicating a possibility of a full control of exceptional points (EPs) and the associated PΤ phases by changing the chirality. Surprisingly, it is possible to achieve analogous control by changing only the real or only the imaginary part of the chirality, as illustrated in

Figure 4b,c, respectively. All of the above reveal a very rich behavior and possibility for different phases and phase re-entries as one changes the system chirality.

Although for oblique incidence the PT-related features of a chiral bilayer can be highly controlled by chirality, chirality cannot offer an external dynamic control. Such a control can be offered by the angle of incidence, as we show in

Figure 5. In

Figure 5, we plot the phase diagrams of the same systems as in

Figure 3a (see

Figure 5a,b) and

Figure 3b (see

Figure 5c,d) as we change the incidence angle. As can be observed there, the frequency positions of the exceptional points and, consequently, the frequency extent of the different PT-phases are highly dependent on the incidence angle. Moreover, incident angle controllable mixed phases and phase re-entries (see panel (d)) can be achieved. The possibility of PT-feature tuning by the incidence angle offers a great and practical way for dynamic control of chiral PT-symmetric systems.

As was mentioned already, among the interesting and particularly useful characteristics associated with chiral media are the optical activity and the circular dichroism (CD), which for our double-layer chiral slab and for normal incidence depend exclusively on the chirality parameter of the two layers (see Equations (25) and (26)). In the case of oblique incidence, where in the chiral bilayer there are cross-polarized transmission terms even for circularly polarized incident waves, the optical activity and the transmitted wave ellipticity (a measure of the CD response) cannot be obtained anymore by the simple relations (25) and (26). They can be calculated, though, through the Stokes parameters [

27,

29], which describe completely the polarization state of a wave. The four Stokes parameters for the transmitted wave in our case are defined by

${S}_{0}={E}_{\perp}{E}_{\perp}^{*}+{E}_{\parallel}{E}_{\parallel}^{*}$,

${S}_{1}={E}_{\perp}{E}_{\perp}^{*}-{E}_{\parallel}{E}_{\parallel}^{*}$,

${S}_{2}=2\mathrm{Re}\left[{E}_{\perp}{E}_{\parallel}^{*}\right],$ and

${S}_{3}=2\mathrm{Im}\left[{E}_{\perp}{E}_{\parallel}^{*}\right]$, where the subscript

$\parallel $ indicates the transmitted electric field component that lies on the plane of incidence, while the subscript

$\perp $ indicates the perpendicular component. Through Stokes parameters the optical activity is given by

and the ellipticity by

To be able to calculate

θ and

η employing the above formulas and to have a full picture for our system potential for polarization manipulation, one needs to calculate the scattering coefficients (transmission and reflection) for linearly polarized incident waves. These coefficients can be directly obtained from the circularly polarized reflection and transmission data according to the equations [

30]

and

In Equations (29) and (30), as in the Stokes parameters, the subscripts $\parallel $ and $\perp $ indicate the components parallel and perpendicular to the plane of incidence, respectively, while, as in the circular polarization case, the first subscript refers to the transmitted (or reflected) component and the second to the incident one.

In

Figure 6, we show the transmitted and reflected power coefficients (

${T}_{ij}={\left|{t}_{ij}\right|}^{2}$,

${R}_{ij}={\left|{r}_{ij}\right|}^{2}$,

$i,j=\left\{\parallel ,\perp \right\}$, respectively) for linearly polarized waves as well as the corresponding optical activity and ellipticity for the chiral PT-symmetric system examined in

Figure 2 under oblique incidence. In particular, we present those quantities at

$\omega L/c=15.5$ as a function of chirality (left two columns) and as a function of the incidence angle (right two columns), keeping all the other parameters constant. Our results reveal a very rich behavior of propagation characteristics, including asymmetric (i.e., side dependent) transmission (note the asymmetry in the

${T}_{\parallel \perp}$ between panels (a) and (e), and between (i) and (m); also the asymmetry in

${T}_{\perp \parallel}$ between panels (c) and (g) and between (k) and (o)), asymmetric reflection (compare

${R}_{\perp \perp}$ in panels (a) and (e), and in panels (i) and (m), as well as

${R}_{\parallel \parallel}$ in (c) and (g) and between (k) and (o)), as well as asymmetric optical activity, and ellipticity (compare panels (b), (d), (j), (l) with (f), (h), (n), and (p), respectively). Moreover, as can be seen in

Figure 6, all the above mentioned asymmetric effects are not only chirality dependent but also angle dependent, offering additional degrees of freedom for controlling the scattering and polarization properties of electromagnetic waves.