Gain Properties of the Single Cell of a One-Dimensional Photonic Crystal with PT Symmetry

Abstract

In this paper, an analysis of gain properties of a single primitive cell of a one-dimensional photonic crystal with parity–time symmetry is demonstrated for the first time. The proposed simple model makes it possible to study the transmission and amplification properties of the investigated cell made of a wide range of optical materials, taking into account the refractive index of the surrounding medium. This analysis is carried out with the use of a transfer matrix method. The obtained characteristics allow indicating the optimal size of the studied structure providing wave amplification, i.e., a transmittance greater than unity. In this case, the increase in the wave intensity in the gain layer exceeds its decrease in the loss layer. This effect is illustrated with the distributions of the electromagnetic field of waves propagating inside the cell.

1. Introduction

An investigation of parity–time (PT) symmetric structures began in 1998 with work [1], which showed that even non-Hermitian Hamiltonians can exhibit entirely real spectra as long as they fulfill the conditions of PT symmetry.

In 2007, work [2] initiated the study of optical one-dimensional structures, which are made from the same amount of two artificial materials, amplifying and absorbing, with different refractive indexes. In general, these coefficients are complex n = n_Re + in_Im and satisfy the condition n*(−z) = n(z) (the asterisk denotes a complex conjugate). Thus, the PT symmetry is achieved when n_Re(z) = n_Re(−z) and n_Im(z) = −n_Im(−z). This implies that the real parts of refractive indices of such materials are equal, and the imaginary parts are equal with respect to their absolute values (i.e., for a gain layer n_Im < 0 and for a loss layer n_Im > 0).

PT symmetric structures are investigated in various arrangements: photonic crystals [3], parallel coupling such as PT symmetry lattice [4], optical waveguide networks [5,6], or in lasers as an amplifying medium [7,8]. These structures are being studied for their very intriguing properties: beam refraction [9], nonreciprocity of light propagation [9], unidirectional invisibility [10], and coherent perfect absorption [11]. Such structures are made in the semiconductor III-V technology [8], an example of which is an electron beam lithography process that contains multiple electron beam evaporation steps and inductively-coupled plasma etching [12]. In these devices, the semiconductor junction is the gain layer, and the loss layer is obtained by creating an additional metallic grating [8] or selective doping of Cr/Ge [12].

So far, the literature has not shown a detailed analysis of the distribution of the electromagnetic field in a single cell of a one-dimensional photonic crystal (1-D PC) with PT symmetry, enabling the study of its gain properties. Therefore, this work aims to present an analysis of these properties of such a single primitive cell. This goal is reached by proposing an original simple model, which enables the study of the transmission and amplification properties of the investigated cell made of any optical material, taking into account the refractive index of the surrounding medium. The property analysis presented in the paper is carried out using a modified transition matrix method [13,14]. An analysis for a PT primitive cell is provided to numerically obtain characteristics showing the reflectances and transmittances, and the electromagnetic field distribution. The study illustrates that an argument of the reflections at all boundaries between the cell layers (gain and loss) and the material surrounding the cell and the cell layers causes the gain layer’s amplification of the electromagnetic wave to be higher than the loss layer’s attenuation. The next section presents the theory describing the 1-D PC with PT symmetry. In Section 3, the characteristics showing the reflectances and transmittance as a function of the ratio of the grating period to the operating wavelength, and the electromagnetic field longitudinal distributions are demonstrated. Section 4 presents the conclusions.

2. Methods

The structure of an analyzed single cell of a one-dimensional photonic crystal with PT symmetry is presented in Figure 1. Each layer of the cell is characterized by the refractive index n₁ or n₂ and a width w_a or w_b in a gain (the light red background) and a loss (the light blue background) layer, respectively. The length of the investigated cell is denoted by Λ. In the case of the PT cell, it equals Λ_PT = w_a + w_b, where both layers have the same width, i.e., w_a = w_b. For the homogeneous gain or loss cell, the length is denoted by Λ_g = w_a, or Λ_α = w_b, respectively.

The refractive indices are complex for both layers and are given by:

$$\begin{array}{l}{n}_1=n_{\mathit{Re}}-i{n}_{\mathit{Im}},\\ n_2=n_{\mathit{Re}}+i{n}_{\mathit{Im}},\end{array}$$

(1)

where n_Re and n_Im are the real and imaginary parts of the refractive indices, respectively. The real part n_Re characterizes the material from which the investigated structure is made. The imaginary part n_Im of a linear single cell is related to a small-signal gain coefficient g₀ for the gain layer and a small-signal loss coefficient α₀ for the loss layer:

$$n_{\mathit{Im}}=\left|g_0/k_0\right|=\left|{\alpha }_0/k_0\right|,$$

(2)

where k₀ is the wave number in free space. The refractive index of the surrounding medium is denoted by n₀.

The reflection r and transmission t of the light incident on an interface between two optical media layers are complex and described by the following Fresnel equations:

• The interface between the gain layer and the surrounding medium:

$$r_{10}=\frac{n_1-n_0}{n_1+n_0},\hspace{1em}{r}_{01}=\frac{n_0-n_1}{n_0+n_1},\hspace{1em}{t}_{10}=\frac{2n_1}{n_1+n_0},\hspace{1em}{t}_{01}=\frac{2n_0}{n_0+n_1},$$

(3)

• The interface between the loss and gain layers:

$$r_{21}=\frac{n_2-n_1}{n_2+n_1},\hspace{1em}{r}_{12}=\frac{n_1-n_2}{n_1+n_2},\hspace{1em}{t}_{21}=\frac{2n_2}{n_2+n_1},\hspace{1em}{t}_{12}=\frac{2n_1}{n_1+n_2},$$

(4)

• The interface between the surrounding medium and the loss layer:

$$r_{02}=\frac{n_0-n_2}{n_0+n_2},\hspace{1em}{r}_{20}=\frac{n_2-n_0}{n_2+n_0},\hspace{1em}{t}_{02}=\frac{2n_0}{n_0+n_2},\hspace{1em}{t}_{20}=\frac{2n_2}{n_2+n_0}.$$

(5)

The electric field distribution within the gain layer E₁(x) and the loss layer E₂(x) can be expressed as a sum of a right-going plane wave (positive direction of the X axis) and a left-going plane wave (negative direction of the X axis), where each wave has a complex amplitude. According to [14], they can be written down as:

$$\begin{array}{l}{E}_1(x)=a\cdot e^{(i{k}_0{n}_{1}x)}+b\cdot e^{(-i{k}_0{n}_{1}x)},\\ E_2(x)=c\cdot e^{(i{k}_0{n}_{2}x)}+d\cdot e^{(-i{k}_0{n}_{2}x)},\end{array}$$

(6)

where a and b are the electric field complex amplitudes of waves travelling in opposite directions in the gain layer, and c and d are the electric field complex amplitudes of waves traveling in opposite directions in the loss layer. The amplitudes of waves close to the interface between the different layers are as follows:

• The gain layer and the surrounding medium—a_R and b_R in the gain layer, and c₀ and d₀ in the surrounding medium;
• The loss and gain layers—c_R and d_R in the loss layer, and a_L and b_L in the gain layer;
• The surrounding medium and the loss layer—a₀ and b₀ in the surrounding medium, and c_L and d_L in the loss layer.

The investigated PT symmetry cell is examined with the help of the modified transfer matrix method [13,14]:

$$\left[\begin{array}{c}{a}_0\\ b_0\end{array}\right]=M\cdot \left[\begin{array}{c}{c}_0\\ d_0\end{array}\right]$$

(7)

and the total transfer matrix M of this cell has the following form:

$$M=J_{02}{P}_2{J}_{21}{P}_1{J}_{10},$$

(8)

where J are the matrices describing the behavior of the electromagnetic wave at the interface between:

• The gain layer and the surrounding medium:

$$J_{10}=\frac{1}{t_{10}}\cdot \left[\begin{array}{cc}1& -r_{01}\\ r_{10}& t_{01}\cdot t_{10}-r_{01}\cdot r_{10}\end{array}\right],$$

(9)

• The loss and gain layers:

$$J_{21}=\frac{1}{t_{21}}\cdot \left[\begin{array}{cc}1& -r_{12}\\ r_{21}& t_{12}\cdot t_{21}-r_{12}\cdot r_{21}\end{array}\right],$$

(10)

• The surrounding medium and the loss layer:

$$J_{02}=\frac{1}{t_{02}}\cdot \left[\begin{array}{cc}1& -r_{20}\\ r_{02}& t_{20}\cdot t_{02}-r_{20}\cdot r_{02}\end{array}\right].$$

(11)

The matrices P₁ and P₂ describe the wave propagation through the gain and loss layer, respectively:

$$P_1=\left[\begin{array}{cc}{\mathrm{e}}^{-i{k}_0{n}_1{w}_a}& 0\\ 0& {\mathrm{e}}^{i{k}_0{n}_1{w}_a}\end{array}\right],\hspace{1em}{P}_2=\left[\begin{array}{cc}{\mathrm{e}}^{-i{k}_0{n}_2{w}_b}& 0\\ 0& {\mathrm{e}}^{i{k}_0{n}_2{w}_b}\end{array}\right].$$

(12)

The field amplitudes of waves at the interface between two different media in the PT primitive cell are related in the following way:

$$\left[\begin{array}{c}{a}_0\\ b_0\end{array}\right]=J_{02}\cdot \left[\begin{array}{c}{c}_L\\ d_L\end{array}\right],\hspace{0.17em}\hspace{0.17em}\left[\begin{array}{c}{c}_L\\ d_L\end{array}\right]=P_2\cdot \left[\begin{array}{c}{c}_R\\ d_R\end{array}\right],\hspace{0.17em}\hspace{0.17em}\left[\begin{array}{c}{c}_R\\ d_R\end{array}\right]=J_{21}\cdot \left[\begin{array}{c}{a}_L\\ b_L\end{array}\right],\hspace{0.17em}\hspace{0.17em}\left[\begin{array}{c}{a}_L\\ b_L\end{array}\right]=P_1\cdot \left[\begin{array}{c}{a}_R\\ b_R\end{array}\right],\hspace{0.17em}\hspace{0.17em}\left[\begin{array}{c}{a}_R\\ b_R\end{array}\right]=J_{10}\cdot \left[\begin{array}{c}{c}_0\\ d_0\end{array}\right].$$

(13)

Using components of the total transfer matrix M, a related scattering matrix S is defined as follows:

$$S=\frac{1}{M_{11}}\cdot \left[\begin{array}{cc}{M}_{21}& M_{21}\cdot M_{11}-M_{12}\cdot M_{21}\\ 1& -M_{12}\end{array}\right]=\left[\begin{array}{cc}{r}_{PT\alpha }& t_{PT}\\ t_{PT}& r_{PTg}\end{array}\right]$$

(14)

where the reflections r_PTα and r_PTg, and the transmission t_PT coefficients of the whole cell are complex. In general, the two reflectances and transmittance are related to the reflections r_PTα and r_PTg, and transmission t_PT coefficients in the following way:

$$R_{PT\alpha }={\left|r_{PT\alpha }\right|}^2,\hspace{1em}{R}_{PTg}={\left|r_{PTg}\right|}^2,\hspace{1em}{T}_{PT}={\left|t_{PT}\right|}^2.$$

(15)

The eigenvalues λ₁ and λ₂ of the S matrix are the solutions of the characteristic equation det(S − λ_eI) = 0, where I is an identity matrix and λ_e is an eigenvalue vector [15], and are described as follows:

$${\lambda }_{1,2}=\frac{1}{2}\left(r_{PT\alpha }+r_{PTg}\pm \sqrt{r_{PT\alpha }^2+r_{PTg}^2+4t_{PT}^2-2r_{PT\alpha }{r}_{PTg}}\right).$$

(16)

The obtained eigenvalues allow to determine whether a given PT cell satisfies the PT symmetry conditions [16,17]. In a PT symmetric phase, the eigenvalues λ₁ and λ₂ are unimodular, i.e., |λ₁| = |λ₂| = 1. For such an operating mode, the PT operation maps each scattering eigenstate back to itself. However, in a broken symmetry phase with reciprocal moduli of eigenvalues, i.e., |λ₁| = 1/|λ₂|, |λ₁| < 1, |λ₂| > 1, each scattering eigenstate is mapped to the other.

In order to describe the lasing threshold in the examined PT symmetric cell, the condition for reproducing the wave (its amplitude and phase) after its full circulation through the structure should be written as:

$$c_L=d_L\cdot r_{20}.$$

(17)

After taking into account the appropriate dependencies for the reflection coefficients and the field amplitudes (Equations (5) and (12) respectively), the following transcendental dependence is obtained:

$$e^{-i{k}_0\left(n_1+n_2\right)\frac{{\Lambda }_{PT}}{2}}-r_{12}{r}_{10}{e}^{i{k}_0\left(n_1-n_2\right)\frac{{\Lambda }_{PT}}{2}}=r_{21}{r}_{20}{e}^{-i{k}_0\left(n_1-n_2\right)\frac{{\Lambda }_{PT}}{2}}+r_{10}{r}_{20}\left(t_{12}\cdot t_{21}-r_{12}\cdot r_{21}\right)e^{i{k}_0\left(n_1+n_2\right)\frac{{\Lambda }_{PT}}{2}}$$

(18)

In Equation (18), all reflection and transmission coefficients and refractive indices are complex.

In the case of the homogeneous gain cell, Equation (17) is simplified to the following form:

$$a_L=b_L\cdot r_{10}\hspace{1em}\Rightarrow \hspace{1em}{e}^{-i{k}_0{n}_1{\Lambda }_g}=r_{10}^2{e}^{i{k}_0{n}_1{\Lambda }_g}.$$

(19)

The integral mean of the PT cell’s intensity was introduced to compare its electric field longitudinal distribution. Such mean is obtained for both layers of the PT cell separately, i.e., for the gain layer ‖E₁(x)‖ and the loss layer ‖E₂(x)‖, and has the following form:

$$\Vert E_1(x)\Vert =\frac{1}{w_a}{\displaystyle \underset{w_a}{\int }\left({\left|a(x)\right|}^2+{\left|b(x)\right|}^2\right)dx},\hspace{1em}\Vert E_2(x)\Vert =\frac{1}{w_b}{\displaystyle \underset{w_b}{\int }\left({\left|c(x)\right|}^2+{\left|d(x)\right|}^2\right)dx}.$$

(20)

The characteristics showing the eigenvalues, reflectances and transmittances as a function of the ratio of the grating period to the operating wavelength, and the electromagnetic field longitudinal distribution are presented in the next section.

3. Results and Discussion

Numerical analysis of the single cell of a one-dimensional photonic crystal with PT symmetry is performed with the assumption that the refractive index’s real part is equal to n_Re = 3.165 for a semiconductor material InP [18], and the imaginary part is n_Im = 0.1 [7,15,17]. The analyzed cell is surrounded by air, i.e., the refractive index is n₀ = 1. The wavelength is λ = 1.55 μm (the third telecommunication window). Moreover, the PT structure is only excited by the wave from one side. Performing the calculations requires the assumption of a steady state, i.e., the fulfillment of the modified transfer matrix method—Equation (7).

The reflection and transmission complex coefficients for the investigated PT cell were determined at all boundaries between the cell layers (gain and loss) and the material surrounding the cell.

Table 1. Moduli and arguments of reflection and transmission coefficients.

Coefficient	Modulus	Argument [πrad]
r02	0.5202	−0.9929
t02	0.4801	−0.0076
t20	1.5201	0.0024
r20	0.5202	0.0071
r21	0.0316	0.5000
t21	1.0005	0.0101
t12	1.0005	−0.0101
r12	0.0316	−0.5000
r10	0.5202	−0.0071
t10	1.5201	−0.0024
t01	0.4801	0.0076
r01	0.5202	0.9929

presents the moduli and arguments of these coefficients obtained from Equations (3)–(5); the light blue background indicates the coefficients in the loss layer, and the light red one in the gain layer.

It is worth noting that, at the boundary between the layers forming the PT cell, the reflection coefficients are negligibly small and the transmission coefficients are dominant and equal unity. Moreover, the arguments of the reflection and transmission complex coefficients in the gain layer are negative, whereas in the loss layer they are positive. This behavior of the coefficient arguments is caused by the differences between the imaginary parts of the refractive index of both layers forming the PT cell.

3.1. Eigenvalues of the S Matrix

To indicate whether the investigated cell meets the conditions of the PT symmetric phase, it is necessary to analyze the eigenvalues λ₁ and λ₂ (see Equation (16)) as a function of a ratio of the length of the PT cell to the operating wavelength Λ_PT/λ, see Figure 2.

The eigenvalues λ₁ and λ₂ are unimodular for small values of Λ_PT/λ. After reaching the point Λ_PT/λ = 6.650, bifurcation occurs and the eigenvalues have reciprocal moduli. The value of the mentioned point is consistent with the work [19], where the condition for the unimodular eigenvalues of the S matrix is shown as a function of the reflection and transmission coefficients (see Equation (10) in [19]). For larger values of Λ_PT/λ, the investigated eigenvalues indicate that symmetry has been broken.

3.2. Reflectance and Transmittance

To examine the reflectance and transmittance of the PT cell, Equation (15) is used. The figures showing the reflectances and the transmittances as functions of the ratio of the length of the cell to the operating wavelength Λ/λ, calculated for the independent gain and loss cells, and the entire PT structure, are presented below (see Figure 3 and Figure 4).

Figure 3 shows the reflectance of the homogeneous (gain or loss) media and the PT cell. In the calculation, it was assumed that the entire PT cell is two times bigger than the homogeneous cell.

In general, oscillations are observed in the characteristics, and they result from the constructive and destructive interferences of the counter running waves inside the cell. For the larger values of the ratio Λ/λ, the oscillations become smaller due to the increasing difference between the intensities of these waves.

In the case of the homogeneous loss cell (see Figure 3a), as the ratio Λ_α/λ increases, the reflectance R_α (blue line) decreases and the oscillations disappear. For the larger Λ_α/λ (i.e., for the larger cell size), the waves inside the loss cell are more strongly damped. In this situation, the reflectance will depend only on the Fresnel reflection from the boundary of the media r₀₂ (determined by the refractive indices’ contrast between n₀ and n₂).

For the homogeneous gain cell (see Figure 3a), as the ratio Λ_g/λ increases, the reflectance R_g (red line) also increases up to the maximal value at a point of a lasing threshold Λ_g/λ = 1.107 obtained from Equation (19). In such circumstances, the waves inside the gain cell are more strongly amplified. At the point of a lasing threshold, the total gain of the waves in the cell is equal to the total loss (the outcoupled waves). For values of the ratio Λ_g/λ above the lasing point, the wave entering the structure must be growing larger to maintain the steady state. In this situation, the waves propagating inside the gain cell are suppressed by interference with the entering wave. This effect is observed in Figure 3a as decreasing of the reflectance R_g, which tends to the value of reflectance being equal to the inverse of the Fresnel formula for r₁₀ [16].

Figure 3b presents the reflectances of the PT cell for the loss layer R_PTα (blue line) and the gain layer R_PTg (red line). In the case of the loss layer, the reflectance R_PTα is less than unity for the ratio Λ_PT/λ, smaller than around Λ_PT/λ = 5.4. However, the reflectance R_PTα demonstrates six maxima exceeding unity for bigger values of Λ_PT/λ. The highest reflectance occurs around the point of the lasing threshold Λ_PT/λ = 7.032, obtained from Equation (18). As the ratio Λ_PT/λ increases, the reflectance above the mentioned point is less than unity and decreases towards the Fresnel reflection value r₀₂. In the case of the gain layer, with increasing of the ratio Λ_PT/λ, the reflectance R_PTg oscillates up to the lasing point, where it reaches a maximum value. For larger values of Λ_PT/λ, the reflectance R_PTg decreases to a value of the inverse of the Fresnel formula for r₁₀.

The transmittances for the independent gain and loss cells, and the PT structure, are demonstrated in Figure 4. All presented characteristics of the transmittances exhibit oscillations. In the case of the homogeneous loss cell (see Figure 4a), as the ratio Λ_α/λ increases, the transmittance T_α (blue line) tends to zero and the oscillations disappear. The reason for such behavior is the same as for the diminishing of the reflectance R_α. In the case of the homogeneous gain cell (see Figure 4a), the transmittance T_g (red line) oscillates and increases until reaching a maximum value at the point of the lasing threshold. For larger values of the ratio Λ_g/λ, it decreases towards zero and the oscillations vanish. The waves propagating inside the gain cell, above the lasing point, are suppressed by interference with the incident wave, thus the transmittance tends to zero.

The transmittance T_PT for the PT cell is shown in Figure 4b. For the ratio Λ_PT/λ smaller than Λ_PT/λ = 4, the transmittance maxima values slightly exceed unity. For the ratio range between Λ_PT/λ = 4 and Λ_PT/λ = 6.650, the magnitude of successive maxima increases and the PT cell demonstrates gain properties. Simultaneously, the investigated structure satisfies the PT symmetry conditions below Λ_PT/λ = 6.650. For further increasing of the ratio Λ_PT/λ, up to the point of the lasing threshold Λ_PT/λ = 7.032, the PT symmetry is broken and the magnitude of the observed maxima increases. In this situation, the PT cell continues demonstrating the gain properties. At the point of the lasing threshold, the transmittance T_PT reaches a maximal value and the PT structure has the highest gain properties. For the larger PT cell size (i.e., for the larger ratio Λ_PT/λ), the PT symmetry is broken, the transmittance T_PT decreases towards zero and its oscillations reduce. Thus, the increase in the wave intensity caused by passing through the gain layer is lost during its travel through the loss layer.

Equation (14) is used to obtain the complex transmission coefficient t_PT. Figure 4c shows an argument of this transmission coefficient arg(t_PT) as a function of the ratio Λ_PT/λ. The periodic variations of the argument of the transmission are observed for the ratio Λ_PT/λ smaller than its value at the point of the lasing threshold Λ_PT/λ = 7.032. The argument’s oscillations disappear and the argument tends to a constant value for the greater ratio Λ_PT/λ (above the lasing point). This value is the constant difference between the input and output waves’ phases of the PT cell.

3.3. Electromagnetic Field Distribution

The longitudinal field distribution in the investigated PT cell is calculated using Equations (6)–(13) with the following assumptions: the PT cell is only illuminated from one side with a wave of intensity I_inc = |a₀|², the intensity of the reflected wave from the same side is I_ref = |b₀|², output intensity equals I_out = |c₀|² = 1 W/cm², and simultaneously d₀ = 0. This distribution is calculated in two steps. In the first one, the intensity values of the incident wave I_inc and reflected wave I_ref from the same side are obtained using the transfer matrix method with the assumed value of the output wave intensity I_out. In the second step, the field distributions in each layer are calculated as a function of the X-coordinate according to Figure 1.

Figure 5 and Figure 6 present the obtained characteristics of the longitudinal field distribution: the amplitudes (|a₀|, |b₀|, |c|, |d|, |a|, |b|, |c₀|) and phases (φ(a₀), φ(b₀), φ(c), φ(d), φ(a), φ(b), φ(c₀)) of the counter running waves versus the position in the PT cell.

Figure 5 presents the field distribution for the ratio Λ_PT/λ = 0.158. It is the smallest value of the ratio Λ_PT/λ for which the first peak of the transmittance greater than unity occurs (Figure 4b), and the investigated structure satisfies the PT symmetry conditions. In particular, Figure 5a,b shows the amplitudes and Figure 5c,d shows the phases of the longitudinal field distribution. The characteristics placed in the left column were obtained for the PT cell illuminated from the loss layer, while the characteristics in the right column were obtained for the PT cell illuminated from the gain layer.

As shown in Figure 5a,b, in the gain layer, the amplitudes of the propagating waves are amplified (amplitude |a| in the positive direction of the X axis—red line, and amplitude |b| in the negative direction of the X axis—pink line) as expected. The opposite situation is observed in the loss layer, where the amplitudes of the propagating waves are suppressed (amplitude |c| in the positive direction of the X axis—blue line, and amplitude |d| in the negative direction of the X axis—cyan line). Comparing the behavior of the field distribution in oppositely oriented single PT cells, for the assumed intensity I_out (green line), shows that the incident wave amplitudes are the same (orange line). Simultaneously, the amplitudes of the reflected waves (dark yellow line) are marginally different and the wave reflected from the loss layer is greater, which is consistent with the reflectance’s distribution (see Figure 3b).

Moreover, comparing the behavior of the wave phases in both single PT cells (illuminated from different layers; see Figure 5c,d), for the assumed value of the phase of the output wave φ(c₀) = 0 (green dotted line), shows that the phases of the input waves (orange dotted line) are equal. On the other hand, the phases of the reflected waves (dark yellow dotted line) differ exactly by π, which results from the interaction of the wave with the entire PT cell. At the same time, the phases of the waves propagating through the loss and the gain layers change monotonically. The step change of the mentioned phases, caused by a change in refractive index, is observed on the border between the different media.

In the case of the wave reflection from the boundary between two different media with the real indices of refraction, the reflected wave does not change its phase (when the wave is incident from a high to a low refractive index medium) or changes it by π (when the wave is incident from a low to a high refractive index medium). The behavior of the phases between the wave incident on and reflected off the same layer of the PT cell (Figure 5c,d, respectively) is different than in the case of media with the real indices of refraction. When the PT cell is illuminated from the loss layer, the phases differ by 3π/2 (see Figure 5c). In the case of the PT cell illuminated from the gain layer, the phases differ by π/2 (see Figure 5d). These phase values are related to the interaction of the incident wave with the entire PT cell.

In Figure 6, when the PT cell is only illuminated from the loss layer, the field distribution is presented for three different values of the ratio of the length of the PT cell to the operating wavelength Λ_PT/λ:

• The smallest value, for which the first peak of the transmittance greater than unity occurs Λ_PT/λ = 0.158 (Figure 6a,b), and the PT cell satisfies the PT symmetry conditions;
• The value at the point of the lasing threshold, when the highest reflectance and transmittance occurs, Λ_PT/λ = 7.032 (Figure 6c,d), and the PT symmetry is broken;
• The value, for which the transmittance is lower than 10⁻³, Λ_PT/λ = 12.0 (Figure 6e,f), and the PT symmetry is broken.

Characteristics placed in the left column present the amplitudes of the longitudinal field distribution, while the phases are presented in the right column. The individual amplitudes and phases of the waves are marked with the same colors as in the previous figure.

In general, the distributions of field amplitudes inside the layers of the PT cell are of the same nature as shown in Figure 5. The increase in the ratio Λ_PT/λ causes much greater changes in the waves’ amplitudes. Moreover, for the assumed I_out value, increasing the size of the PT cell (i.e., the ratio Λ_PT/λ) causes non-linear changes in the incident and reflected waves. In particular, for the smallest value of Λ_PT/λ = 0.158 (see Figure 6a), the incident wave’s amplitude is smaller than the reflected one’s. In this situation, the transmittance is slightly greater than unity and the whole PT cell amplifies an electromagnetic wave. In the case of the point of the lasing threshold (Λ_PT/λ = 7.032) (see Figure 6c), the reflected wave’s amplitude is greater than the incident one’s. The PT cell amplifies the electromagnetic wave the most, because the transmittance and reflectance are the highest in this case (see Figure 3b and Figure 4b).

Additionally, the whole PT cell acts as a single gain layer, and the transmittance and reflectance are greater than unity simultaneously (see Figure 3a and Figure 4a). In the third case of the ratio Λ_PT/λ’s value (Λ_PT/λ = 12.0) (see Figure 6e), the reflected wave’s amplitude is smaller than the incident one’s, and the transmitted wave’s amplitude is smaller than the incident wave’s and reflected wave’s amplitudes. Thus, the whole PT cell acts as a single loss layer, and the transmittance and reflectance are much lower than unity simultaneously (see Figure 3a and Figure 4a).

Taking the dependence of the wave’s phase on the ratio Λ_PT/λ into consideration, the results show that progressively more halves of the propagating wavelength are included in the analyzed PT cell when increasing the ratio. In particular, for the smallest investigated ratio (see Figure 6b), there is only one half of the wavelength in the PT cell. For the higher values of the ratio Λ_PT/λ, there are noticeably more halves of the wavelength (see Figure 6d,f).

Further, for the assumed value of the output wave’s phase, the difference between the phases of the incident and reflected waves has the following values for the indicated ratios: Λ_PT/λ = 0.158—the phases differ by 3π/2 (see Figure 6b), Λ_PT/λ = 7.032—the phases are almost the same (see Figure 6d), Λ_PT/λ = 12.0—the phases differ by π (see Figure 6f). In the last two cases, the phase differences mentioned above have values close to the differences that occur when analyzing the incident and reflected waves in materials with real refractive indices.

The expression for the integral mean (Equation (20)) was used to investigate the gain properties of the analyzed PT cell. The integral mean of the intensity of the electric field’s longitudinal distribution of each PT layer was calculated using this formula. The obtained results are presented in

Table 2. The integral mean of intensity of field distribution in each layer [W/cm²].

ΛPT/λ	Loss Layer	Gain Layer
0.158	0.535057	0.535126
7.032	2.186726	2.269591
12.0	84.567893	29.266907

for different values of the ratio Λ_PT/λ, as shown in Figure 6.

In general, for the ratio value smaller or equal to Λ_PT/λ at the point of the lasing threshold, the integral mean of the intensity of the electric field is greater in the gain layer than in the loss one (see Table 2, where larger mean values are indicated in light orange). As a result, the entire PT cell demonstrates gain properties, which are related to the values of the reflection and transmission coefficients (see Table 1). The differences in the investigated integral means between the loss and the gain layers are small, but the typical PT structures are made of many elementary cells, which leads to the enhancement of this effect. Further increasing of the ratio Λ_PT/λ beyond the point of the lasing threshold causes the structure to lose its gain properties. It begins to strongly absorb the wave, which is confirmed by a very large difference in the integral means for Λ_PT/λ = 12.0 between the loss and the gain layers.

4. Conclusions

This work shows the analysis of the gain properties of a single primitive cell of a one-dimensional photonic crystal with parity–time symmetry. In particular, the reflectance and transmittance were obtained for a wide range of the ratio of the PT cell’s length to the operating wavelength. The gain properties of such a PT cell occur when the transmittance is greater than unity. This effect is strongly dependent on the length of the PT cell and is the strongest for the length related to the point of the lasing threshold. A longer PT cell loses its gain properties. Moreover, it is shown that the gain properties are caused by the phases of the reflections at the boundary between two different media. This effect is illustrated with the distributions of the electromagnetic field of waves propagating inside the cell and calculated values of the integral mean of the electric field’s intensity of each PT layer. The presented model of the PT cell can help in the design of the telecommunication system’s elements.