1. Introduction

Designing and fabricating man-made optical materials or structures that can control the propagation of light waves is an unremitting goal all the time. After decades of efforts, people have made many remarkable achievements, such as photonic crystals [1], photonic crystal fibers [2], negative-index metamaterials [3], and so on. However, it is noteworthy that these remarkable works mainly focus on the modulation of the real part of optical parameters to obtain optical properties not found in nature. In recent years, a new family of artificial optical structures, which aims to explore novel optical properties in the complex domain, has attracted considerable attention. This class of optical structures takes advantage of the concept of PT-symmetry in quantum mechanics [4, 5] to attain new exotic optical functionalities and properties. These PT-synthetic structures rely on judiciously balanced gain and loss regions such that the complex refractive index is required to satisfy the condition: $n\left(\overrightarrow{r}\right)=n^{*}\left(-\overrightarrow{r}\right)$. Under this condition, the optical modes can be created and absorbed in a controlled manner. Researchers have always been interested in these PT-symmetric optical systems because it can exhibit some intriguing optical phenomena that are not available in the Hermitian system. These phenomena include double refraction [6], power oscillations [6–8], absorption enhanced transmission [9, 10], unidirectional invisibility [11–14], and coherent perfect absorbers and lasers (CPA-Lasers) [15, 16], etc. Moreover, within the framework of PT-symmetry, other properties also intrigue people’s interests, such as Bloch oscillations [6, 12, 17–20], the Talbot effect [21], andthe realization of optical isolators and circulator [22–25].

In many practical applications, such as metamaterials and plasmonics [3, 26, 27], the loss is an inevitable problem and impedes the progress of applied physics. The concept of PT-symmetry may provide an effective way to overcome losses. However, in practice, it is a very challenging task to fabricate an artificial optical structure with delicately balanced gain and loss elements due to technical difficulties. As a result, many exotic phenomena, like the PT-phase transition or CPA-Lasers, can be experimentally observed only in elemental two-component systems [7, 9, 28, 29]. On the other hand, although there exist many studies on the PT-symmetric periodic systems, most of them mainly focused on layered periodic structures. Therefore, it will be very important to implement PT-symmetry concepts on a variety of new platforms or optical structures. It will help to design new optical devices and materials, such as unidirectional on-chip devices and laser oscillators.

In this paper, we construct a 1D PT-symmetric network which is connected by 1D gain or loss waveguide segments of different length in a periodic manner. This periodic PT-symmetric structure can be described by a simple network equation. Utilizing the network equation and the transfer matrix method, we obtain three expressions for total transmission, left reflection and right reflection coefficients, which depends on the single cell transmission and reflection coefficients, the dimensionless Bloch wavevector, and the total number of unit cells. For this periodic PT-symmetric system, new criteria for the spontaneous PT-symmetry breaking transition are analytically derived. These criteria indicate that the exceptional point is only related to the unit cell structure, and the cell number has no impact on the frequency position and number of the exceptional points. According to these criteria and expressions, the relationships between the transmittances (reflectances) and the cell number are systematically studied. Moreover, the conditions for producing ultrastrong transmission are found for any finite periodic PT-symmetric network. We also demonstrate that our designed PT-symmetric network can display singular scattering characteristics such as unidirectionalor bidirectional transparency. The conditions and related properties of unidirectional and bidirectional transparencies are introduced in detail. Finally, we find that the way of adjusting cell number is an effective route to design a one-way invisible network near the exceptional point. Our work may contribute to the design and manufacture of large-scale PT-symmetric optical lattices or networks, such as two-dimensional or three-dimensional optical networks. This kind of PT-symmetric structure can be arranged more flexible. Therefore, it can be used to implement various PT-synthetic devices or networks with new characteristics and functionalities.

This paper is organized as follows: The model and main equations of the finite periodic PT-symmetric network is introduced in Section 2. In Section 3, we discuss the extraordinary properties of the finite periodic PT-symmetric network and show the results of numerical simulations. Finally, the conclusion is drawn in Section 4.

2. Model and main equations

The designed 1D PT-symmetric periodic optical waveguide network is schematically shown in Fig. 1. This periodic structure contains N cells, and is composed of three kinds of waveguide segments: passive (involving no gain or loss) segments (black lines), loss segments (red lines), and gain segments (green lines). The refractive indices of them are, respectively, $n_0=2$, $n_1=2+0.005\iota$ and $n_2=2-0.005\iota$, and the refractive index distribution of the whole structure satisfies the condition:$n\left(x\right)=n^{*}\left(-x\right)$. The lengths of the two passive segments connected to nodes 0 and 2N are d. The lengths of the two loss or gain segments between adjacent nodes (blue dots) are d₁

($d_1=d$) and d₂, and their length ratio is $d_1:d_2=1:2$.

 

Fig. 1 Schematic of the 1D PT-symmetric periodic optical waveguide network containing N cells. Red lines labeled n₁ are the loss segments $\left(n_1=2+0.005\iota \right)$, green lines labeled n₂ are the gain segments $\left(n_2=2-0.005\iota \right)$, and black lines labeled n₀ are the passive segments $\left(n_0=2\right)$. The lengths of the two loss (gain) segments between adjacent nodes (blue dots) are d₁ and d₂, and their length ratio is $d_1:d_2=1:2$. The lengths of the two passive segments are d.

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In this PT-symmetric optical network, the wave function ${\phi }_{ij}\left(x\right)$ in any segment between nodes i and j obeys the Helmholtz equation:

When the optical waves are, separately, incident from the left or right side, the transmission and reflection coefficients for left (L) and right (R) incidence waves are defined as $t_{N\left(L\right)}={\alpha }_2/{\alpha }_1$, $r_{N\left(L\right)}={\beta }_1/{\alpha }_1$, $t_{N\left(R\right)}={\beta }_1/{\beta }_2$ and $r_{N\left(R\right)}={\alpha }_2/{\beta }_2$, and we have the boundary conditions: ${\beta }_2=0$ for left incidence and ${\alpha }_1=0$ for right incidence. In terms of the transfer matrix $\left(M_N\right)$ elements, these coefficients can be expressed as follows [30]:

In order to derive the transfer matrix M_N, we will analyze the PT-symmetric waveguide network. The wave function ${\phi }_{ij}\left(x\right)$ in any passive, loss or gain segments can be decomposed into two opposite traveling waves:

Then, from Eq. (2), we have$\left(\begin{array}{c}{\phi }_0\\ {\phi }_{-1}\end{array}\right)=\left(\begin{array}{cc}1& 1\\ e^{-\iota kd}& e^{\iota kd}\end{array}\right)\left(\begin{array}{c}{\alpha }_1\\ {\beta }_1\end{array}\right)=Q_L\left(\begin{array}{c}{\alpha }_1\\ {\beta }_1\end{array}\right),$ and

Combining Eqs. (3), (10) and (11), we obtain the transfer matrix M_N:

Thus, the transfer matrix for one cell is $M_1=Q_R^{-1}{\mathrm{\Gamma }}_2{\mathrm{\Gamma }}_1{\mathrm{\Gamma }}_0{Q}_L$. Because the PT-symmetric optical network is periodic, these recursion matrices satisfy the following relation:

By use of the generalized Floquet-Bloch theorem [31], one can derive the dispersion relation: $\cos K=u$, where K is the dimensionless Bloch wavevector. At present, combining Eqs. (12), (14), and (15), we can rewrite the transfer matrix M_N as:

It is found that the transmission coefficient t_N for N cells is obtained as a function of t₁, K, and N, while the left (right) reflection coefficient $r_{N\left(L\right)}$ ($r_{N\left(R\right)}$) for N cells is derived as a function of $r_{1\left(L\right)}$ ($r_{1\left(R\right)}$), t₁, K, and N. These expressions will help us accurately analyze the extraordinary optical properties and numerically calculate the transmittance $T=|t_N{|}^2$ and reflectances $R_L=|r_{N\left(L\right)}{|}^2$ and $R_R=|r_{N\left(R\right)}{|}^2$.

3. Discussions and numerical results

3.1. PT-symmetry breaking transition

For the PT-symmetric optical network, the amplitudes of the forward and backward traveling optical waves in passive segments are also related through the scattering matrix $S\left(\omega \right)$:

From this rigorous result, we can conclude that for a finite periodic PT-symmetric network, (i) when $T_1<1$, the eigenvalues are unimodular, and the system is in the PT-symmetric phase; (ii) when $T_1>1$, the eigenvalues are nonunimodular (except the point $\sin NK=0$), and the system is in the broken phase. (iii) when $T_1=1$ or $\sin NK=0$, the eigenvalues are degenerate. But the symmetry-breaking transition can only happen at $T_1=1$; (iv) At the exceptional point, T₁ and T_N are equal to 1. $T_1=T_N=1$ is a necessary condition for the spontaneous PT-symmetry breaking transition. These criteria indicate that for a finite periodic PT-symmetric optical waveguide network, the exceptional point is determined by its unit cell structure, and the cell number N has no effect on the exceptional point.

 

Fig. 2 Logarithm of the absolute values of the eigenvalues of the corresponding scattering matrix for the finite periodic PT-symmetric network with different cell numbers. Red solid lines correspond to N = 1, blue dashed lines correspond to N = 4, and black dashed lines correspond to N = 8.

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In order to understand these criteria more intuitively and elucidate the influence of the cell number on the exceptional points and the eigenvalues of the scattering matrix, we plot the logarithm of the absolute values of the eigenvalues of the scattering matrix for the finite periodic PT-symmetric network with three different cell numbers (N= 1, 4, 8) in Fig. 2. From Fig. 2, we observe that no matter what the cell number N is, there only exists two exceptional points between frequency $62.12\pi c/d$ and $62.20\pi c/d$, and these two exceptional points are, respectively, located at frequency $62.1287\pi c/d$ and $62.1895\pi c/d$. Between these two exceptional points, the eigenvalues are generally nonunimodular, and the system is in the broken phase. However, when $N>1$, the eigenvalues become degenerate and are unimodular at the point $\sin NK=0$. On both sides of these two exceptional points, the eigenvalues are unimodular, and the system is in the PT-symmetric phase. Similarly, in the PT-symmetric phase, the eigenvalues can be also degenerate when $\sin NK=0$. Although the eigenvalues become degenerate at the point $\sin NK=0$, the symmetry-breaking transition does not happen. Thus, we can conclude that the cell number has no effect on the number and frequency location of the exceptional points. From Fig. 2(a), one can also observe that the cell number N has great influence on the eigenvalues of the scattering matrix. When N = 4, the absolute value of one eigenvalue of the scattering matrix is very large, while the other approaches zero at frequency $62.177\pi c/d$. However, when N is equal to 1 or 8, the absolute value of one eigenvalue is relatively small and the other is relatively big at this frequency.

Next, we will show the influence of the cell structure on the PT-symmetry breaking transition. In Fig. 3(b), we plot the logarithm of the absolute values of the eigenvalues of the scattering matrix for the PT-symmetric network with different waveguide length ratios. Here, the PT-symmetric network contains only one cell. From Fig. 3(a), we observe that when the length ratio is $d_1:d_2=1:1$, the eigenvalues are unimodular, and the system is in the PT-symmetric phase. When we keep the length d₁ unchanged and extend the length d₂, the frequency region where the system is in the broken phase becomes larger and larger. From Fig. 3(b), we observe that when the length ratio is $d_1:d_2=1:5$, the system is completely in the broken phase. when the length ratio is $d_1:d_2=1:8$, the system is still in the broken phase, and the eigenvalues of these two cases are degenerate at the same frequency. If we further extend the length d₂, the absolute values of the eigenvalues are approximately equal to the case: $d_1:d_2=1:8$. (In the following investigation, we still focus on the PT-symmetric network with $d_1:d_2=1:2$)

 

Fig. 3 Logarithm of the absolute values of the eigenvalues of the corresponding scattering matrix for the PT-symmetric network with different length ratios. Here the PT-symmetric network contains only one cell. (a) Green solid lines correspond to $d_1:d_2=1:1$, magenta solid lines correspond to $d_1:d_2=1:3$, and blue solid lines correspond to $d_1:d_2=1:4$. (b) Cyan solid lines correspond to $d_1:d_2=1:5$, black solid lines correspond to $d_1:d_2=1:8$, and red dashed lines correspond to $d_1:d_2=1:17$.

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3.2. The influence of cell number on transmission and reflection

After introducing the PT-symmetry breaking transition of the finite periodic PT-symmetric optical network, it is necessary to investigate the transmission and reflection properties related to the number of unit cells. From Eq. (17), the total transmission and reflection for a finite periodic PT-symmetric network are obtained as a function of the cell number N.

Thus, we may argue that the transmission and reflection curves should oscillate as the cell number N increases. However, in fact, the results may be different from our intuition. According to the discussions in Subsection 3.1, we have known that T_N and T₁ are equal to 1 at the exceptional point. Combining this condition and Eq. (17), we can deduce that when $T_1=1$, T_N will be equal to 1 whether the point $T_1=1$ is the exceptional point or not. Therefore, the transmission curve does not oscillate with the cell number N when $T_1=1$. When $T_1\ne 1$, the transmission curve will oscillate as the cell number N is increasing. Moreover, according to Fig. 2, we can also deduce that the transmission T_N for N cells satisfies the following relations: (i) when $T_1<1$, $T_N\le 1$ ; (ii) when $T_1>1$, $T_N\ge 1$. Both cases are equal to 1 at the point $\sin NK=0$. In Fig. 4, we have, respectively, calculated the transmittance, left reflectance and right reflectance of the finite periodic PT-symmetric network with the cell number N increasing from 1 to 30 around the point $T_1=1$. These results in Fig. 4(a) are in excellent agreement with our prediction. From Fig. 4(b) and 4(c), we observe that if the point $T_1=1$ is the exceptional point, the left reflection is always zero regardless of the increment of the cell number N whereas the right reflection curve oscillates with the cell number N; if the point $T_1=1$ is not the exceptional point, both left and right reflections are always zero no matter what the cell number N is. In the PT-symmetric phase ($T_1<1$) or the broken phase ($T_1>1$), both left and right reflection curves will oscillate with the cell number N, and they will vanish at $\sin NK=0$.

 

Fig. 4 (a) Transmittance (T_N), (b) left reflectance ($R_{N\left(L\right)}$), and (c) right reflectance ($R_{N\left(R\right)}$) for the finite periodic PT-symmetric network as a function of cell number N. The black dot-solid lines correspond to the PT-symmetric phase $(T_1<1)$, the red circle-solid lines correspond to the exceptional point $\left(T_1=1\right)$, the blue dot-solid lines correspond to the broken phase $(T_1>1)$, and the cyan dot-dashed lines correspond to the degeneracy point ($T_1=1$, not the exceptional point).

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3.3. Conditions for generating ultrastrong transmission

In Fig. 2, when N = 4, the absolute value of one eigenvalue of the scattering matrix is very large at frequency $62.177\pi c/d$. It implies that the transmission for N = 4 is very strong at this frequency. However, when $N=$1 or 8, the transmission is relatively weak at frequency $62.177\pi c/d$. Here, we will analyse this phenomenon and derive the conditions for generating ultrastrong transmission. For 1D PT-symmetric system, the elements of the transfer matrix M_N have the property: $m_{N\left(22\right)}\left(\omega \right)=m_{N\left(11\right)}^{*}\left({\omega }^{*}\right)$ [14]. Using this property and Eq. (4), we can rewrite the transfer matrix M₁, which has the form: $M_1=\left(1/t_1^{*},r_R/t_1;-r_L/t_1,1/t_1\right)$. Meanwhile, the eigenvalues of M₁ are written as ${\lambda }_1=1/{\lambda }_2=\exp (\iota K)$. According to the theory of matrices, we can deduce $1/t_1=\cos K+\iota \text{Im}\left(1/t_1\right)$. Substituting this equation into Eq. (17), the transmission coefficient t_N can be rewritten as

Since $\cos NK$ and $\sin NK$ cannot be equal to zero at the same time, from this equation, the transmission coefficient t_N goes to infinity only when $\cos NK=0$ ($\sin NK=\pm 1$) and $\text{Im}\left(1/t_1\right)$=0. In Fig. 5(a) and 5(b), we show the imaginary part of $1/t_1$ and the values of $\sin NK$ with $N=34$ in the frequency range $62.10\pi c/d\sim 62.40\pi c/d$. The transmittances of the periodic PT-symmetric network for N = 1 and $N=34$ are displayed in Fig. 5(c). From Fig. 5(a) and 5(b), we observe that at the frequency $62.351\pi c/d$, the imaginary part of $1/t_1$ is zero and the value of $\sin 34K$ is approaching zero. The ultrastrong transmission (red line) can be clearly observed near the frequency $62.351\pi c/d$ in Fig. 5(c). However, the transmission is relatively weak at frequency $62.177\pi c/d$ due to $\sin 34K\approx 0.4$. For a finite periodic PT-symmetric network, the condition $\text{Im}\left(1/t_1\right)=0$ designates the frequency region where the ultrastrong transmission can be created. Then, we can obtain ultrastrong transmission by adjusting the cell number to satisfy the condition: $\sin NK=\pm 1$ ($\cos NK=0$). Under these two conditions ($\text{Im}\left(1/t_1\right)=0$, $\sin NK=\pm 1$/$\cos NK=0$), the ultrastrong transmission can be created in a controlled manner. It is noteworthy that these two stong transmissions in Fig. 5(c) are realized in the broken phase ($T_1>1$).

 

Fig. 5 (a) The imaginary part of $1/t_1$ between frequency $62.10\pi c/d$ and $62.40\pi c/d$, (b) the values of $\sin NK$ with $N=34$ in the same frequency range, and (c) the transmittance of the periodic PT-symmetric network with two different cell numbers, where the green line corresponds to N = 1, and the red line corresponds to $N=34$

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3.4. Unidirectional and bidirectional transparencies

In PT-symmetric system, we have $S^{*}\left(\omega \right)=PTS\left(\omega \right)PT=S^{-1}\left(\omega \right)$. From this relation, one can deduce that $r_{N\left(L\right)}{r}_{N\left(R\right)}^{*}=1-|t_N{|}^2$, and this equation implies that $\sqrt{R_{N\left(L\right)}{R}_{N\left(R\right)}}=\left|T_N-1\right|$ [35]. When we have perfect transmission T_N = 1, the geometric mean of the two reflectances $\sqrt{R_{N\left(L\right)}{R}_{N\left(R\right)}}$ is required to vanish. This constraint can be satisfied by the following two cases: the first case is that only one reflection becomes zero, and the second case is that both reflections are all zero. From Eqs. (4) and (17), the left reflectance $R_{N\left(L\right)}$ is generally different from the right reflectance $R_{N\left(R\right)}$ due to $r_{1\left(L\right)}\ne r_{1\left(R\right)}$, and the left and right reflectances can be equivalent at some frequencies (i.e. $\sin NK=0$). Therefore, these two cases can be realized in the PT-symmetric system. When only one reflection vanishes, we have reflectionless perfect transmission in one direction but not the other, and this phenomenon is the unidirectional transparency. When both reflections vanish, we have reflectionless perfect transmission in each direction, and this phenomenon is the bidirectional transparency.

We first examine the unidirectional transparency properties of the PT-symmetric network. This phenomenon is closely related to the PT-symmetric structures, and it can only happen at the exceptional point. Based on Eq. (19), we have calculated the logarithm of the absolute values of the eigenvalues of the scattering matrix for the finite periodic PT-symmetric network with four cells in Fig. 6(a). The transmittance, left reflectance and right reflectance of optical waves propagating through the PT-symmetric network are shown in Fig. 6(b). The results in Fig. 6(a) and 6(b) show that the exceptional point is at $62.1895\pi c/d$, and the transmittance is unitary at the frequency $\omega =62.1895\pi c/d$. Below the exceptional point, the eigenvalues are nonunimodular and the system is in the PT-broken phase. The transmittance is superunitary. While above the exceptional point, the eigenvalues are unimodular and the system is in the PT-symmetric phase. The transmittance is subunitary. At the exceptional point, the reflectance for left incident waves vanishes whereas the reflectance for right incident waves is $R_{N\left(R\right)}\approx 15$. This phenomenon is the so-called unidirectional transparency mentioned above. From Eq. (17), the left and right reflection coefficients for N cells have the same term: $1-\left[t_1\sin (N-1\right)K]/\sin NK$. Therefore, the conditions for unidirectional transparency can be expressed as $\left|t_1\right|=1$, $r_{1\left(L\right)}=0$ and $r_{1\left(R\right)}\ne 0$, or $\left|t_1\right|=1$, $r_{1\left(R\right)}=0$ and $r_{1\left(L\right)}\ne 0$.

 

Fig. 6 Unidirectional transparency characteristics of the finite periodic PT-symmetric network with four cells. (a) Logarithm of the absolute values of the eigenvalues of the corresponding scattering matrix. (b) Transmittance, left reflectance and right reflectance. Phases and delay times of transmitted, left-reflected, and right-reflected waves of the finite periodic PT-symmetric are shown in (c) and (d).

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Unidirectional transparency is a singularity with highly directional responses. The physical mechanism of this singularity is the resonance trapping [36]. The occurrence of resonance trapping for transmission and reflection can be judged from the divergence of the corresponding delay time. The delay time is associated with the time experienced by a wave packet inside the potential before it exits from the scattering region, and it is defined as ${\tau }_{t,r}=d{\varphi }_{t,r}/d\omega$, where ${\varphi }_{t,r}$ is the phase of the transmission or reflection coefficient. In Fig. 6(c), we plot the phases of transmitted, left-reflected, and right-reflected waves. At the exceptional point ($\omega =62.1895\pi c/d$), there is an abrupt change of π in the phase of left-reflected waves. It indicates that the delay time for the left reflection diverges. In fact, in Fig. 6(d), the delay time for left reflection also behaves like a $\delta \left(\omega \right)$ function due to the step-function behavior of the phase. As a result, the light reflecting from the left side will be trapped in the lossy waveguide segment for a long time and get absorbed completely by the loss. While the light reflecting from the right side will delay in the gain waveguide segment for a relatively short time and is enhanced. It is also necessary to note that the phase relation between transmitted and reflected waves. From Fig. 6(c), in the broken phase, there is a π phase difference between the left-reflected and right-reflected waves, while in the PT-symmetric phase, the phase of the left-reflected waves is the same as that of the right-reflected waves. Moreover, there is a $\pm \pi /2$ phase difference between transmitted waves and reflected waves either in the broken phase or in the PT-symmetric phase. These results can be also derived from the above equation, that is, $r_{N\left(L\right)}{r}_{N\left(R\right)}^{*}=1-|t_N{|}^2$. Since transmitted and reflected waves have such phase relations, the delay times for transmission and reflections are exactly equivalent in the broken phase and the PT-symmetric phase [Fig. 6(d)].

 

Fig. 7 Bidirectional transparency characteristics of the finite periodic PT-symmetric network with four cells in the PT-symmetric phase. (a) Logarithm of the absolute values of the eigenvalues of the corresponding scattering matrix. (b) Transmittance, left reflectance and right reflectance. Phases and delay times of transmitted, left-reflected, and right-reflected waves of the PT-symmetric are shown in (c) and (d).

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Fig. 8 Bidirectional transparency characteristics of the finite periodic PT-symmetric network with four cells in the broken phase. (a) Logarithm of the absolute values of the eigenvalues of the corresponding scattering matrix. (b) Transmittance, left reflectance and right reflectance. Phases and delay times of the transmitted, left-reflected, and right-reflected waves of the PT-symmetric are shown in (c) and (d).

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Next, we analyze the bidirectional transparency properties. From Eq. (17), the conditions for this phenomenon are written as $r_{1\left(L\right)}=r_{1\left(R\right)}=0$ or $\sin NK=0$. The first case is related to the unit cell structure of the finite periodic PT-symmetric network, and the second case is related to the periodicity of the PT-symmetric network. The number of points corresponding to bidirectional reflectionless increases with the number of cells. According to Subsection 3.1, the eigenvalues of the scattering matrix can be degenerate in the PT-symmetric phase and the broken phase, thus bidirectional transparency can happen in the PT-symmetric phase and the broken phase. In this paper, we show the bidirectional transparency properties of the finite periodicPT-symmetric network with four cells in the PT-symmetric phase (Fig. 7) and the broken phase (Fig. 8). From Fig. 7, one can observe that the eigenvalues of the corresponding scattering matrix are unimodular and degenerate at $\omega =62.902\pi c/d$, as marked by a darkened dot. At the degeneracy point $\omega =62.902\pi c/d$, the transmittance T_N is unitary. Away from this point, the transmittance T_N are subunitary. Although the left and right reflectances are quite different inthe frequency domain $\left[62.88\pi c/d,62.902\pi c/d\right)$ and $\left(62.902\pi c/d,62.92\pi c/d\right]$, they both vanish at the degeneracy point $62.902\pi c/d$. Since the system is in the PT-symmetric phase, the phases of left-reflected and right-reflected waves must be equivalent. the phase difference between transmitted waves and reflected waves is still $\pm \pi /2$. At the degeneracy point, the phases of two reflected waves experience an abrupt change of π simultaneously. Therefore, the delay time for both reflections all behaves as a delta function. Both left-reflected and right-reflected waves are trapped in the PT-symmetric network for a long time and get absorbed. Another case for the bidirectional reflectionless is in the broken phase. From Fig. 8, one can observe that the absolute values of a scattering matrix is unimodular and degenerate at the frequency $62.1474\pi c/d$. On both sides of this frequency $62.1474\pi c/d$, the absolute values of the eigenvalues are nonunimodular, and the transmission is superunitary. The phases of the two reflected waves all have an abrupt change of π, but the directions of two abrupt changes are opposite. Thus the delay time of two reflected waves satisfies the relation: ${\delta }_L\left(\omega \right)=-{\delta }_R\left(\omega \right)$ [Fig. 8(d)]. It is necessary to note that all bidirectional transparencies are caused by Bragg resonances, which are mainly attributed to the increase of the cell number.

3.5. Unidirectional invisibility

Over the past decades, optical cloaking has attracted considerable attention, specifically in connection to transformation optics [37, 38]. In recent years, a new notion of unidirectional invisibility, which results from a completely different physical process, was proposed [11]. This kind of invisibility is based on the concept of PT-symmetry and arises because of the spontaneous PT-symmetry breaking. It is well known that at the exceptional point, the PT-symmetric network can display unidirectional transparency when the light is incident from different sides of the structure. However, unidirectional transparency is not in general equivalent to unidirectional invisibility. This is due to the fact that the phase of transmitted waves might be different from the phase of the waves after passing through a homogeneous background medium with the same length. For example, in Fig. 6(c), the transmission is exactly unitary at the exceptional point and the phase of the transmitted wave is $0.26\pi$. However, for a homogeneous background medium, the phase of transmitted waves is $0.5\pi$. In this case, simple time-of-flight measurements can sense the existence of the PT-symmetric optical network, and the PT-symmetric optical network cannot be viewed as an invisible structure. Therefore, if we want to design an invisible structure, it is necessary to examine the phase of the transmitted wave and compare it with the phase of a wave propagating in a homogeneous waveguide network. Using Eq. (17), we deduce that the phase ${\varphi }_{t_N}$ for N cells is

One can observe that ${\varphi }_{t_N}$ is closely related to the number of cells, and the cell number has no impact on the frequency positon and the number of the exceptional points. It signifies that we can design a unidirectionally invisible PT-symmetric network by adjusting the cell number of the structure at the exceptional points. In order to compare the phases of the transmitted waves traveling through the PT-symmetric network (${\varphi }_{\text{PT}}$) with those of the transmitted waves traveling through a homogeneous background medium (ϕ), we plot the absolute value of the phase difference of these two phases ($\left|{\varphi }_{\text{PT}}-\varphi \right|$) as a function of the cell number at the exceptional point ($\omega =62.1895\pi c/d$) in Fig. 9. From this figure, one can observe that the phase difference has an approximate periodicity and approaches zero when the cell number N is 1, 25, 53 and 79. Therefore, the finite periodic PT-symmetric network with such cell number can be seen as a unidirectionally invisible structure. In summary, adjusting cell number is a valid way to realize optical cloaking in PT-symmetric networks at given frequencies.

 

Fig. 9 The absolute value of the phase differences for transmitted waves traveling through the PT-symmetric network (${\varphi }_{\text{PT}}$) and those traveling through a homogeneous background medium (ϕ) vary with cell number N at $\omega =62.1895\pi c/d$.

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4. Conclusions

We have investigated the singular properties of the finite periodic PT-symmetric network. Based on theoretical derivation, we obtain three expressions for transmission and left and right reflection coefficients. New criteria for the PT-symmetry breaking transition of the periodic PT-symmetric network are analytically derived. These criteria indicate that the exceptional points of the finite periodic PT-symmetric network are only related to the single cell structure, and the cell number has no effect on the frequency location and the number of the exceptional points. In terms of these criteria, the relationship between the transmittance (reflectances) and the cell number is systematically examined. More importantly, we find the conditions for ultrastrong transmission, which are applicable to an arbitrary 1D periodic PT-symmetric network. Besides, we also study the phenomenon of unidirectional and bidirectional transparencies. Using the transmission and reflection coefficient expressions, the conditions forunidirectional and bidirectional transparencies are analytically obtained, and the related optical properties are well introduced. Finally, we find that the finite periodic PT-symmetric network with 1, 27, 53, or 79 unit cells is unidirectionally invisible at the exceptional point. Our work will deepen people’s understanding of the optical properties of PT-symmetric waveguide network. We envision that PT-symmetric networks will open a new route for designing various optical structures or networks with new characteristics and functionalities.