Planar Bilayer PT-Symmetric Systems and Resonance Energy Transfer

Abstract

Parity-time (PT) symmetry provides an outstanding improvement of photonic devices’ performance due to the remarkable physics behind it. Resonance energy transfer (RET) as an important characteristic mediating the molecules that can be tailored in the PT-symmetric environment, too. We study how planar bilayer PT-symmetric systems affect the process of resonance energy transfer occurring in the vicinity thereof. First, we investigate the reflectance and transmittance spectra of such systems by calculating reflection and transmission coefficients as well as total radiation amplification as functions of medium parameters. We obtain that reflectance and total amplification are greatest near the exceptional points of the PT-symmetric system. Then, we perform numerical calculations of the RET rate and investigate its dependence on the complex permittivity of the PT-symmetric medium, dipole orientation, frequency of radiation and layer thickness. Optically thick PT-symmetric systems may operate at lower gain at the expense of the appearance of chaotic-like behaviors. These appear owing to the dense oscillations in the reflectance and transmittance spectra and vividly manifest themselves as stochastic-like positions of the exceptional points for PT-symmetric bilayers. The RET rate, being a result of the field interference, can be significantly amplified and suppressed near exceptional points exhibiting a Fano-like lineshape.

1. Introduction

In their 1998 paper, Bender and Boettcher conjectured that quantum–mechanical systems described by non-Hermitian Hamiltonians may, under certain conditions, possess entirely real-valued spectra [1]. For this, it is necessary and sufficient that its eigenfunctions are invariant under the action of parity-time (PT) operator [2]. Later, through the formal analogy between the stationary Schrödinger equation and the Helmholtz equation in electrodynamics, the concept of the PT symmetry migrated to optics [3,4,5]. By definition, an optical system is called PT-symmetric if its permittivity $\epsilon$ satisfies the following condition

$${\epsilon }^{\mathrm{*}}\left(\mathbf{r},\omega \right)=\epsilon \left(-\mathbf{r},\omega \right),$$

(1)

where $\mathbf{r}$ is the radius-vector, $\omega$ is the angular frequency of the incoming electromagnetic field and * denotes complex conjugation. Despite difficulties with experimental implementation [4], PT-symmetric systems have attained a great amount of interest, because they possess a number of peculiar properties, like double and asymmetric refraction in optical lattices and unidirectional invisibility [3,6]. Exceptional points play a tremendous part in non-Hermitian photonics, offering a new physics due to the interplay of the loss and gain components [7]. Recently, exceptional points were observed along a synthetic orbital angular momentum dimension [8] and were experimentally found above the lasing threshold due to the cavity detuning [9]. Perfectly chiral exceptional points were realized using the mirror-symmetry-broken metasurface through the chiral quasi-bound state in the continuum [10]. Higher-order exceptional points are able to enhance the sensibility of non-Hermitian sensors [11,12] due to the intriguing increase in the spectral response strength [13].

In the present work, we explore the influence of spectral properties of optically thick bilayer PT-symmetric systems on the process of resonance energy transfer (RET). Resonance energy transfer is a quantum–electrodynamic process of non-radiative energy transfer between two quantum systems, one of which, initially in an excited state (donor), releases a virtual photon, which is subsequently absorbed by another system, initially in the ground state (acceptor). As was originally shown by Förster using semi-classical approximation [14], the RET rate depends on the distance R between the donor and acceptor as $R^{-6}$, being very short ranged. Nevertheless, it plays an important role in the energy exchange between organic molecules [15]. The RET rate increases near the exceptional points; the greater the order of the exceptional point, the larger the RET rate [16].

This paper is organized as follows. Section 1 contains an introduction. In Section 2, we calculate and analyze the power reflection $r$ and transmission $t$ coefficients of planar PT-symmetric bilayers as functions of layer thickness for different values of loss and gain and investigate their behavior for the case of optically thick layers. We relate the maxima of reflectance and total intensity of the scattered radiation with the exceptional points of PT-symmetric systems. In Section 3, we carry out numerical calculations of the spectral density function $T\left(\omega \right)$ that defines the effect of the environment on the RET rate and study its correlation to the exceptional points. In Section 4, we discuss the obtained results. Section 5 concludes the paper.

2. Reflective and Transmissive Properties of Planar PT-Symmetric Media

2.1. General Remarks

Figure 1a illustrates the simplest example of a planar medium satisfying the PT-symmetry condition (1). It consists of two bulks of the same thickness $L$ and permittivities ${\epsilon }^{’}+i{\epsilon }^{’’}$ and ${\epsilon }^{’}-i\epsilon ’’$ of lossy and gain media, respectively. An obvious generalization, illustrated in Figure 1b, is a periodic planar system comprising $N$ layers, the permittivities of which are equal to ${\epsilon }_j={\epsilon }^{’}+{\left(-1\right)}^{j+1}i{\epsilon }^{’’},j=1,\dots ,N$.

Before proceeding further, we would like to briefly mention three points. The first one is the fact that the definition of PT-symmetric media depends on the choice of a coordinate frame: its origin has to be placed at the interface between two layers in Figure 1a to satisfy Equation (1). Obviously, it makes little sense for a system to be PT-symmetric in one set of coordinates and not to be such in another. Therefore, any system composed of alternating loss and gain layers of equal thickness, such that the absolute value of $\epsilon ’’$ is the same for each layer, will be referred to as being PT-symmetric. The second point concerns the negative sign of $\epsilon ’’$, that is, the gain. If plane waves are described by $\exp \left[i\left(\mathbf{k}\mathbf{r}-\omega t\right)\right]$, then the electromagnetic radiation propagating through a layer with $\epsilon ’’<0$ is being exponentially amplified. Any amplification requires a mechanism that provides a source of energy for the propagating wave, and a negative imaginary part of $\epsilon$ is an effective macroscopic way of describing this mechanism [17]. The examples of such a mechanism are the stimulated emission of radiation and coupling to an external gain source. The amplified radiation may greatly affect the properties of the medium, including permittivity $\epsilon$, to the point when amplification may disappear altogether. The precise intensity of radiation at which this happens has to be determined either experimentally or from the microscopic properties of the system under study. In what follows, we assume that the value of the permittivity is unaffected by the propagating radiation. Finally, the last point concerns the frequency dependence of $\epsilon$. Complex permittivity is fundamentally related to the dispersive properties of media [17] and, strictly speaking, we need to remember that permittivity depends on the frequency $\omega$ of the incoming radiation. It was shown in [4], using the Kramers–Kronig relations, that the PT-symmetry condition, defined by Equation (1), holds only for a discrete set of frequencies. Thus, it makes sense to perform all the calculations for some fixed frequency $\omega$ (or wavelength $\lambda$), assuming that the system is PT-symmetric at that frequency. In Section 2.2, when we plot the dependencies versus $L/\lambda$, it is implied that the thickness of layers $L$ rather than the wavelength $\lambda$ is being varied.

In the next subsection, we investigate power reflection $r$ and transmission $t$ coefficients of PT-symmetric layered systems. Along with them, we calculate $r+t$, which shows how the total intensity of the incoming radiation changes after interacting with the layered system. Since a medium with $\epsilon ’’\ne 0$ can both amplify and absorb the incoming radiation, there is no reason for $r+t$ to be equal to 1, as in the case of pure dielectrics, or to decay exponentially, as in the case of pure metals. Instead, a complex interplay between loss, gain and interference effects takes place, which leads to non-trivial results.

2.2. Calculations and Discussion

The derivation of $r$ and $t$ coefficients is presented in Appendix A. The power coefficients are defined as the moduli squared of the diagonal elements of the matrices (A9) and (A10); the latter generalize the Fresnel coefficients. Since we consider the case of normal incidence and all layers are isotropic, the power reflection and transmission coefficients are the same for TE- and TM-polarized modes. We distinguish two cases for the bilayer system In Figure 1a: (i) the incoming wave hits the gain layer first and (ii) it hits the loss layer first. Once the real part of the permittivity $\mathrm{R}\mathrm{e}\left(\epsilon \right)$ and geometry of the system are fixed, there are only three essential parameters to vary: the thickness of layer $L$, wavenumber of the incoming wave $k$ and imaginary part of the dielectric permittivity $\epsilon \mathrm{’}\mathrm{’}$. Noticing that the wavenumber and thickness appear only as a product $kL=2\pi L/\lambda$ in the definition of the reflection and transmission coefficients (see Appendix A), it makes sense to investigate the dependence of $r$ and $t$ coefficients on the ratio $L/\lambda$, i.e., on the thickness, measured in the units of wavelength of the incoming wave. We take $\epsilon \mathrm{’}=3$ and consider four values of ${\epsilon }^{\mathrm{’}\mathrm{’}}$, namely, $0.03,0.05,0.1$ and $0.3$. The results for $r$ and $t$ coefficients are presented in Figure 2 and Figure 3. The plots look solid, but in fact they are highly oscillatory with a period $\propto \mathrm{R}\mathrm{e}(1/\sqrt{\epsilon })$. In Figure 4, we investigate asymptotic behavior $L/\lambda \to \infty$ (that is, the case of very thick layers) of $r+t$ with respect to $\epsilon \mathrm{’}\mathrm{’}$ for different values of $\epsilon \mathrm{’}$.

The first thing one observes when comparing Figure 2 and Figure 3 is the obvious fact that the order of loss and gain layers matters. Being reciprocal, the system has the same transmittance independent of the order of layers; however, the reflectance is drastically different.

In both cases of layer ordering, there exists a layer thickness, at which the reflectance, transmittance and intensity amplification have sharp spikes. When the gain layer meets the electromagnetic wave first, the maximum amplification is greater than in the case of the opposite sequence of layers by about an order of magnitude (cf. Figure 2a and Figure 3a). Although, as the calculation shows, the maximum amplification reaches the values ${10}^4-{10}^5$ in one case and ${10}^3-{10}^4$ in the other, one should keep in mind that these values are likely unattainable in practice due to the arguments noted in Section 2.1. One may still expect that a spike of amplification (most probably, less pronounced) will be observed in the actual experiment near the layer thicknesses predicted by the theory. In any case, the practical realization of the two-layer system under study requires fine tuning due to the highly oscillatory nature of $r$ and $t$.

Transmittance (Figure 2c and Figure 3c) is the same in both cases of layer ordering: at small values of $L/\lambda$ (precisely which values are ‘small’ is different for different permittivities), the larger part of the incoming radiation passes through the medium; then, $t$ has a sharp spike, after which it quickly decreases to zero. However, the behavior of reflectance is qualitatively different for the two cases. If the loss layer is placed first, $r$ is almost zero for small $L/\lambda$; then, it has a sharp spike, after which it drops to small but non-vanishing values. If, instead, the gain layer is placed first, then at large $L/\lambda$, it does not fall off but rather remains near 10. The asymptotic behavior of $r+t$ (effectively $r$ alone, since for large $L/\lambda$, transmittance quickly vanishes in both cases) illustrated in Figure 4 has the following properties:

• It depends significantly on the permittivity of the first layer: if the gain layer is first, then there is a net intensity gain, while if the loss layer is first, then the intensity is absorbed rather than amplified.
• The smaller the value of $\mathrm{R}\mathrm{e}\left(\epsilon \right)$ at $\epsilon ’’<0$ ($\epsilon ’’>0$), the larger (smaller) the asymptotic intensity gain $r+t$.
• The total intensity gain $r+t$ tends to $1$ for very large values of $|\epsilon ’’|$.

We also notice that for large values of $L/\lambda$, the amplitude of oscillations of both $r$ and $t$ significantly decreases. Thus, if one aims at constructing a layered PT-symmetric system with a given level of amplification, it is more expedient to use thick layers with the value of $\epsilon ’’$ which provides the desired level of amplification. Although the asymptotic amplification is much smaller than the maximum amplification, the former is substantially easier to achieve, since no fine tuning is required.

Having investigated the reflective and transmissive properties of the bilayer PT-symmetric system, the next step is to investigate how these properties affect the RET rate. In particular, it is interesting to check whether there is an amplification of RET rate at the points of maximum reflection and whether the asymptotic behavior of $r$ manifests itself in the asymptotic behavior of the RET rate. But before proceeding further, let us relate the obtained spectral properties to the exceptional points of the PT-symmetric system in question.

2.3. Connection to Exceptional Points

In the context of photonics, the exceptional points of a multilayer PT–symmetric system are defined through the analysis of the eigenvalues of the scattering matrix (S-matrix) [18]. The eigenvalues of the scattering matrix are normally complex numbers of absolute value equal to one. However, by tailoring the system’s parameters (for example, $\epsilon ’’$ and layer thickness $L$), one is able to approach an exceptional point, where the system undergoes a PT symmetry breaking [19], and the absolute value of each eigenvalue is not the unity anymore. For example, consider the bilayer system in Figure 1a characterized by parameters $\mathrm{R}\mathrm{e}\left(\epsilon \right)=3$ and $L=7.5\mathrm{\mu }\mathrm{m}$, which scatters the incoming radiation of wavelength $500$ nm. For the ratio $L/\lambda =15$, an exceptional point appears at $\mathrm{I}\mathrm{m}\left(\epsilon \right)\approx 0.09$ (see Figure 5). Here, we use the S-matrix eigenvalues of the form ${\lambda }_{\mathrm{1,2}}=(R_L+R_R)/2\pm \sqrt{1-{\left(R_L+R_R\right)}^2/4}$, where $R_L$ and $R_R$ are, respectively, the field reflection-to-the left and reflection-to-the-right coefficients [18].

From Figure 2 and Figure 3, we notice that the $r$ and $t$ power coefficients have a sharp spike at approximately $L/\lambda =15$ when $\epsilon ’’=0.1$, i.e., the spikes in Figure 2 and Figure 3 are related to the exceptional points of the bilayer system defined by ${\left(R_L+R_R\right)}^2/4=1$. Exactly in this case, the eigenvalues of the S-matrix coalesce, ${\lambda }_1={\lambda }_2$. As discussed in Ref. [18], the exceptional points of the multilayer system can be introduced twofold. The first-type exceptional point corresponds to the transfer from the regime of balanced loss and gain to the regime of imbalanced loss and gain. An exceptional point of the second type (considered in our manuscript) is a lasing predictor, in the vicinity of which the instability is developed, while the reflectance and transmittance take large but finite values. The reflectance enhancement can be also expected from the exceptional point condition $R_L+R_R=\pm 2$, which says that the reflectance becomes great and has a tendency to increase. The locations of the exceptional points in the $(\epsilon ’’,L/\lambda )$ coordinates are presented in Figure 6 as blue dots. For all the considered values of $\mathrm{I}\mathrm{m}\left(\epsilon \right)$ from Figure 2 and Figure 3, the spikes correspond to the exceptional points in Figure 6. The positions of the blue points in Figure 6 are not sufficiently regular, though they are concentrated in the vicinity of the fitting curve of the form $y\left(x\right)=a+b{x}^{-1}+c{x}^{-2}$, where $a,b$ and $c$ are constants. Irregularity can be associated with dense oscillations in the spectral reflection and transmission dependences, when a tiny variation of the thickness $L/\lambda$ results in the great change in the reflectance and transmittance. The stochastic-like positions of points in Figure 6 is a manifestation of the dynamic chaos known in the theory of dynamic systems. The picture shown in Figure 6 is a result of a particular discretization of the normalized length $L/\lambda$ (similar discretization can be performed for $\epsilon ’’$). Such a chaotic-like nature imposes restrictions on the experimental realization of the PT-symmetric systems with predicted behaviors.

3. Resonance Energy Transfer in the Presence of a Bilayer PT-Symmetric Medium

The rate of the resonance energy transfer is given by the relationship [20]

$${\gamma }_{D\to A}={\int }_0^{\mathrm{\infty }}{\sigma }_{em}\left(\omega \right){\sigma }_{abs}\left(\omega \right)T\left(\omega \right)d\omega ,$$

(2)

where ${\sigma }_{em}\left(\omega \right)$ and ${\sigma }_{abs}\left(\omega \right)$ are the donor luminescence and acceptor absorption spectra, respectively, and the spectral function

$$T\left(\omega \right)=\frac{2\pi }{{\mathrm{\hslash }}^2}{\left(\frac{{\omega }^2}{{\epsilon }_{0}c}\right)}^2{\left|{\mathbf{d}}_{\mathrm{D}}G\left(\omega ,{\mathbf{r}}_{\mathrm{D}},{\mathbf{r}}_{\mathrm{A}}\right){\mathbf{d}}_{\mathrm{A}}\right|}^2.$$

(3)

Here, $G$ is the dyadic Green’s function of the system (see Appendix B), ${\mathbf{d}}_{\mathrm{A}}$ and ${\mathbf{d}}_{\mathrm{D}}$ are the electric dipole moments of, respectively, the acceptor and donor, ${\mathbf{r}}_{\mathrm{A}}$ and ${\mathbf{r}}_{\mathrm{D}}$ are their radius-vectors, and $\omega$ is the angular frequency associated with the released photon. From Equation (3), one can immediately deduce that Green’s function specifies the dependence of the RET rate on the surrounding environment. Since PT-symmetric media may both amplify and absorb radiation depending on the wavelength and layer thickness, one could expect a similar kind of behavior when calculating the RET rate. However, it is hard to tell a priori which values of the parameters of the system correspond to amplification or attenuation because of the aforementioned interplay of loss, gain and interference effects.

The configuration of the system we consider is as follows (see Figure 7): two quantum systems with electric dipole moments ${\mathbf{d}}_A=d_A{\mathbf{n}}_A$ and ${\mathbf{d}}_D=d_D{\mathbf{n}}_D$ (${\mathbf{n}}_{A,D}$ are unit vectors) are placed inside a semi-infinite medium with $\epsilon =1$ (air or vacuum) near the interface of a bilayer PT-symmetric medium. The parameters that determine the intensity of the RET rate are shown in Figure 7:

• Magnitudes of the dipole moments $d_{A,D}$ and orientations of the unit vectors ${\mathbf{n}}_{A,D}$.
• Distances $z_A$ and $z_D$ from, respectively, acceptor and donor to the interface.
• Lateral separation $r_{\perp }$ between the donor and acceptor systems.
• Wavelength $\lambda$ of the mediated photon.
• Permittivity $\epsilon$ and thickness $L$ of layers.

We want to compare the RET rate with and without the layered medium (in the latter case, both dipoles are surrounded by vacuum). To that end, the exact values of $d_{A,D}$ are irrelevant, since they cancel out in the ratio of spectral functions T/T₀. The parameters take the following values: $\lambda =500$ nm, $z_A=z_D=\lambda /10=50$ nm and $r_{\perp }=\lambda /50=10$ nm. As in Section 2, we consider four values of $\epsilon ’’:0.03, 0.05 ,0.1$ and $0.3$.

The results are presented in Figure 8 and Figure 9 for the two cases when both dipoles are oriented along either $z$ or $x$-axis, respectively.

We see that the dependence of the spectral function T on $L/\lambda$ roughly resembles that of the intensity gain (cf. Figure 2a and Figure 3a). The positions of the spikes in Figure 8 and Figure 9 correspond to those in Figure 2 and Figure 3. This means that the RET rate is directly correlated with the scattering properties of the surrounding environment and in the case of the PT-symmetric layered system, it is greatest at the exceptional points.

When the gain layer is placed first, one observes a significant amplification of the spectral function T owing to the electromagnetic energy concentration in the gain slab, while when the loss layer is placed first, the amplification is scarce and barely noticeable for all thicknesses except for the chosen ones (see Figure 8 and Figure 9). The asymptotic behavior is similar to that in Figure 2 and Figure 3: the order of loss and gain layers matters. The asymptotic amplification is substantial only if the gain layer is placed first. It is worth noting that even when the loss layer is placed first, the spectral function in the PT-symmetric environment is still slightly amplified (cf. Figure 4, where radiation is attenuated when $\mathrm{I}\mathrm{m}\left(\epsilon \right)>0$). The increase in T is of the same order of magnitude for both orientations of dipoles.

According to Ref. [16], the spectral density function $T\left(\omega \right)$ maximizes in the vicinity of the exceptional point, though it may depart from the exact position of the exceptional point due to the perturbation caused by dipoles. In Figure 8 and Figure 9, we also observe suppression of the spectral density $T$ to the values below $T_0$. The availability of the enhancement and suppression is explained by the interference of two electromagnetic fields, the first of which propagates directly from the donor to the acceptor, while the second field originates from the reflection at the layer interface. Near the exceptional point in the loss–gain configuration: both strong enhancement and strong suppression are realized, resembling a Fano lineshape.

4. Discussion

All the results presented above have been obtained using numerical calculations. While performing these calculations, we used the following assumptions: (1) no matter how high the radiation amplification in a PT-symmetric medium, the material parameters remain the same; (2) the value of the RET rate is given by expressions (4) and (5); (3) the presence of dipoles does not modify the environment. The validity of the first assumption has to be determined either experimentally or using the microscopic theory of the PT-symmetric medium under study, as discussed in Section 2.1. The amplitudes of waves in the experiments aimed at determining the reflective properties of media are by far not sufficient to alter their characteristics, so this assumption is justified. (Such an experiment may be realized, for example, by placing a sample of medium inside a waveguide connected to a vector network analyzer). For the second assumption to be valid, the dipole approximation has to be true, since it played a key role in the derivation of Equation (2) (see [20]). This means that the quantity $kd$ is sufficiently small, where $k$ is the wavenumber of the emitted photon and $d$ is the characteristic size of the donor and acceptor. Moreover, all the characteristic length scales in the problem should be sufficiently greater than interatomic distances, so that the effective macroscopic treatment of medium using dyadic Green’s function is applicable. Finally, the third assumption is not strictly true, since the presence of dipoles should at least slightly modify the environment, resulting in the departure from the perfect PT symmetry of the considered system. The perturbation theory can be exploited in order to account for the RET more accurately, as seen in [16]. The perturbation changes the positions of the exceptional points as well as Green’s function and the RET rate in the vicinity of exceptional points. Thus, the maximum of the spectral function $T\left(\omega \right)$ shifts. However, the RET rate as an integral (4) will include all frequencies, and the effect of the maximum displacement will likely not be dramatic, being exhibited as a perturbative correction.

The shape of the spectral function amplification curves resembles a Fano lineshape, which is a result similar to that obtained in Ref. [20], where the RET rate between two molecules located near a planar dielectric half-space was calculated as a function of the transition frequency. There, the maximum amplification rate is of the order ${10}^2$, which is a result similar to ours. However, a PT-symmetric environment causes residual amplification in the asymptotic case of thick layers (or, alternatively, in the case of short wavelengths of mediated photons), which is a result that is absent when a dielectric half-space is considered.

The numerically obtained maximum amplification of the spectral function is of the order 10². It is reasonable to ask whether it is experimentally possible to obtain such levels of amplification. In Ref. [21], the authors report an amplification of the RET rate by a factor of 10², but it was achieved by placing the donor and acceptor inside a dielectric particle of diameter $\approx \mathrm{}$10 μm rather than by placing them near a layered medium. The recent study on the RET rate near a planar metasurface reports an amplification by a factor of $1-10$ depending on the geometry of the experiment [22]. This level of amplification is lower than that obtained in the present paper, but the experiment was conducted in the microwave range, whereas we took the wavelength of the mediated photon to be in the optical ($500$ nm) range. While various experimental difficulties may prevent the achievement of the RET rate amplification by a factor of ${10}^2$, this possibility should not be discarded on a priori grounds.

It is possible to predict the location of amplification spikes in bilayer PT-symmetric systems using the obtained results. To that end, one needs to construct the scattering matrix of such a system, calculate its eigenvalues for some set of parameters and, by varying parameters separately, determine the points of bifurcation; the obtained points are approximately the desired points of maximum spectral amplification and RET rate. But, unfortunately, due to the highly oscillatory behavior of reflection and transmission coefficients, even in the case of an ideal bilayer medium, the procedure described above may require serious fine tuning. In general, optically thick PT-symmetric systems demonstrate a chaotic-like behavior clearly exhibiting itself in the positions of exceptional points and in ‘noisy’ curves for the spectral function of the RET rate. Interestingly, both strong enhancement and strong suppression near the exceptional points are available. Oscillations also decay as the thickness of layers increases, which means no fine tuning is required if thick layers are chosen in the experiment setup. Thus, instead of trying to achieve maximum amplification at the location of exceptional points, it might be more expedient to obtain a lesser but stably predictable amplification in the asymptotic (large $L$) case. To that end, one only has to numerically determine which values of $\epsilon \mathrm{’}\mathrm{’}$ produce the desired asymptotic behavior of $T\left(\omega \right)$ and use the materials with the obtained parameters in an experiment.

Although we did not study multilayered systems with the number of layers $N>2$ herein, it is natural to ask what would differ in that case. Moreover, one could ask how the obtained results would change in the case of non-planar (e.g., spherical or cylindrical) and/or anisotropic PT-symmetric media. To encompass such systems, the appropriate Green functions should be determined. Since the set of systems allowing the closed-form Green functions is limited, the numerical methods of finding the Green functions might be used. Perhaps it is natural to conjecture that a behavior similar to that observed in the present paper is also true for such systems, i.e., the location of amplification spikes corresponds to the location of the exceptional points of PT-symmetric systems. In any case, these non-trivial questions deserve a separate study.

5. Conclusions

We have numerically investigated the reflective and transmissive properties of a bilayer PT-symmetric system paying attention to its influence on the resonance energy transfer rate between the donor and acceptor quantum systems. The main results are as follows:

• The order of loss and gain layers matters greatly: it affects both the total intensity gain and RET rate amplification for a given thickness of layers and the asymptotic amplification at $L\to \infty$.
• In the ‘gain slab’–‘loss slab’ sequence of layers (Figure 2), the maximum reflection and amplification is of the order ${10}^5$, while in the opposite sequence of layers (Figure 3), the maximum reflection and amplification is of the order ${10}^4$, which is an order of magnitude lower. The asymptotic amplification (Figure 4) is of the order $10$ in the ‘gain slab’–‘loss slab’ case (radiation is amplified) and of the order $0.1$ in the ‘loss slab’–‘gain slab’ case (radiation is attenuated).
• The pattern of the spectral function amplification T/T₀ of the RET rate resembles that of the intensity gain and reflection: whenever the radiation is highly amplified and reflected, the RET rate also rises.
• In the ‘gain slab’–‘loss slab’ sequence of layers, the maximum RET spectral function amplification is of the order $100$ and the asymptotic amplification is around $20$, while in the opposite sequence of layers, the amplification is barely noticeable: maximum amplification is of the order of unity, and asymptotic amplification is practically absent (Figure 8 and Figure 9). These results are true for both orientations of the dipole moments (whether along the $z$ or $x$-axis).
• The reflection and amplification of radiation by PT-symmetric systems are greatest at the values of media parameters corresponding to the exceptional points of such systems. Hence, if there is a PT-symmetric medium located near interacting dipoles, then the RET rate is also greatest at the exceptional points.