1. Introduction

Generally, by adding complex potentials, Hermitian systems are converted into the non-Hermitian ones which do not support real eigenvalues and orthogonal eigenstates [1–4]. Note that a non-Hermitian system becomes parity-time (PT) symmetry on condition that the complex potentials results in the invariance of the intrinsic Hamiltonian after the parity reflection and time reversal operations, in both [5]. Moreover, PT-symmetric systems sustain real eigenvalues, and experience a phase transition from the PT-unbroken to the PT-broken one across the so-called exceptional point (EP) [6,7]. Due to such intriguing properties, the notion of PT-symmetry has been extended into various realms, for example, in optical systems wherein the dielectric constant plays the role of the complex potentials [8]. The existence of an EP and the resulting phase transition have been extensively studied and many interesting phenomena have been discovered, such as unidirectional transmission [9,10], lasing states [11,12], coherent absorption [13], and others [14–17].

Since the balanced gain and loss is not easy to achieve experimentally, several alternative techniques have been presented to avoid the employment of gain in optical systems according to the gauge transformation [18,19]. However, non-uniform gain and loss or asymmetric losses have significant effects on the original decoupled resonances or the propagation modes [20]. Besides, providing a system with many modes coupling, multiple EPs tend to emerge and their interactions occur under parameter variation [21–23]. Except for conventional variation of gain/loss, various photonic crystals (PCs) with PT-symmetry are typical paradigms: for each Bloch k, the coalescence of two eigenstates in a discrete spectrum can also be treated as an EP [22]. Thus, PT phase diagram can be found through the evolution of complex band structures in PT-symmetric PCs [24,25].

Photonic cavities possess high quality factors and small mode volume, which is used to confine the optical power in a small volume with low loss rate [26,27]. As a consequence, it provides a wonderful platform to enhance the interaction between light and matter. Coupled waveguide-cavity structures have been utilized to build various optical devices, such as optical switcher, optical filter, and optical sensor [28,29]. Moreover, directly coupled microresonators have been numerically and experimentally explored in the PT-symmetric context [10,30]. These structures show exceptional properties such as asymmetric transmission and unidirectional invisibility.

In this work, we investigate the evolution of complex band structures of PT-symmetric polariton crystal as the non-Hermiticity increases. The prototype system is a period chain of resonators coupled to a waveguide. Different gain and loss are supposed in the nearest-neighboring resonators for setting up the PT-symmetric complex potentials. In such 1D periodic system, the resonator acts as a discrete “polariton-type” resonance which modulates the spectrum of the waveguide. Moreover, interactions between the polariton state and the Bloch states give rise to unusual band structure evolution, e.g., a defect-like miniband may arises [31]. The miniband fundamentally depends on the polariton-like resonance and the system can be referred to as “polariton crystal” due to the analogue to the band structure of polariton resonance induced anticrossing. We show that for relative small quantity of non-Hermiticity, EPs emerge at the center and boundary of the irreducible Brillouin zone, coalesce at a point far from the boundary, and finally become absent when the non-Hermiticity is beyond a threshold value. As such, the EPs have significant effects on the transmission properties.

2. Formulation of the model transfer matrix

Our system is shown schematically in Fig. 1, which is the unit cell of the prototype polariton crystal. It consists of two cavities coupled to a waveguide. An individual unit cell structure also expresses intriguing properties (Appendix A). We assume that the side cavities support only the fundamental mode, and the adjacent cavities have no direct interaction, i.e., the cavities are optically connected only by the indirect path through the waveguide channel. Based on the coupled mode theory (CMT), the incident and scattering fields at the reference line of the cavity (see dashed lines in Fig. 1) can be described as [31,32]:

 

Fig. 1 Schematic picture of a unit cell of the polariton crystal. The solid lines mark the dual ports of the unit cell, and the dashed lines mark the reference line of coupling resonator.

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In our unit cell, the $L^{\prime }$ corresponding to${M^{\prime }}_1$, ${M^{\prime }}_2$, and ${M^{\prime }}_3$ is $L/2$, $L$ and $L/2$, respectively. The band structure in terms of continuous Bloch wave number $k$ can be obtained by applying Bloch theorem [33,34]: $\left|M_{tot}-e^{jkp}I\right|=0$, where $I$ is the identity matrix,$p=2L$ is the period length for dual cavities (see Fig. 1). The corresponding band structure is shown in Fig. 2, for different Bragg resonance frequency. The details are discussed in the following section.

 

Fig. 2 Complex band structure of the polariton crystal for $\kappa =0.03{\omega }_0$, ${\gamma }_1={\gamma }_2=0$ and${\omega }_1={\omega }_2={\omega }_0$. The lowest order Bragg resonance is ${\omega }_1={\omega }_2={\omega }_0$ (a), ${\omega }_0$ (b), and $1.9{\omega }_0$ (c). The orange vertical dash-dotted line represents the Brillouin boundary $k=\pi /2L$ corresponding to the unit cell with dual cavities (cf. Figure 1). The grey-shaded zone marks the polariton gap by the localized ‘polariton’ resonance. In panels (a) and (c), the brown-shaded zone denotes the Bragg gap. The dashed lines are the results of the band fold of the solid lines due to lattice constant extended to double, while the different color denotes different band index.

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3. Complex band structures and the evolution with non-Hermiticity

Before examining the evolution of the complex band structure for${\gamma }_n\ne 0$, we firstly construct the band structure using the Bloch state with ${\gamma }_n=0$ as a basis (see the solid lines in Fig. 2). In our previous work [31], we have examined the band structure without additional gain and loss modulation. Here as we turn on the gain and loss in adjacent cavities, the non-Hermitian periodicity becomes an integer multiple. As such, the Bragg band structures are folded back at the Brillouin boundary$k=\pi /2L$, transformed to the dashed line in Fig. 2. Figures 2(a)-2(c) are for three selective situations where the polariton gap overlaps with the second Bragg band, falls in the Bragg gap, and overlaps with the first Bragg gap. The operation of folding creates a natural set of degeneracies whose eigenstates are identical at the boundary.

When PT-symmetry potential is involved in, the unit cell now contains dual cavities and the band structure becomes much more complicated. Figures 3 and 4 exactly show the evolution of both the real and imaginary parts of the eigenfrequency for different ${\gamma }_1=-{\gamma }_2=-\gamma$ ($\gamma >\text{0}$). Here we purposely selected Bragg resonance frequency $\omega =0.77{\omega }_0$ (Fig. 3) and $\omega =1.9{\omega }_0$ (Fig. 4) that represent typical cases. We would mainly focus on the band structure in the vicinity of the polariton gap (the gray region in Fig. 2). Figure 3 shows the complex band structure for Bragg resonance $\omega =0.77{\omega }_0$ with four different setting:$\gamma =0.01{\omega }_0$, $\gamma =0.04{\omega }_0$, $\gamma =0.09{\omega }_0$, and $\gamma =0.1{\omega }_0$. Since the folded band has zero Bragg gaps at the zone boundary, a tiny nonzero $\gamma$ results in the emergence of EP [see Figs. 3(a) and 3(b)], in sharp contrast to the original band structure [see Fig. 2(a)]. However, the polariton gap at the zone center keeps opened until $\gamma$ is beyond a critical value${\gamma }_c=0.035{\omega }_0$. As a consequence, the EP emerges [see Figs. 3(c) and 3(d)]. These EPs are marked by the letters ‘M’, ‘N’, and ‘S’ in Fig. 3. Further, for an particular$\gamma$, the M and N EPs collide first. Figure 3(e) shows that this critical point is near ${\gamma }_c=0.09{\omega }_0$ where these two EPs coalesce near $k=0.6\pi \text{/2}L$ [see Figs. 3(e) and 3(f)]. As $\gamma$ further increases to $\gamma =0.1{\omega }_0$, the M and N EPs disappear, while the S EP still exists [see Figs. 3(g) and 3(h)]. In addition, the bands, including the red solid band and the blue dashed one, are fully overlapped for their real part of the eigenfrequency.

 

Fig. 3 Complex band structures at $\gamma =0.01{\omega }_0$ (a), (b), $\gamma =0.04{\omega }_0$(c), (d), $\gamma =0.09{\omega }_0$(e), (f), and $\gamma =0.1{\omega }_0$ (g), (h). The Bragg resonance is identical to that of Fig. 1(a). (a), (c), (e), (g) are the real parts, (b), (d), (f), (h) are the imaginary part. (M), (N), (S) denote the EPs.

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Fig. 4 Complex band structures at $\gamma =0.01{\omega }_0$ (a), (b), $\gamma =0.04{\omega }_0$ (c), (d), $\gamma =0.155{\omega }_0$ (e), (f), and $\gamma =0.18{\omega }_0$ (g), (h). The Bragg resonance is identical to that of Fig. 1(c). (a), (c), (e), (g) are the real parts, (b), (d), (f), (h) are the imaginary part. (M), (N), (S) denote the EPs.

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Figure 4 shows the complex band structure evolution for Bragg resonance at $\omega =1.9{\omega }_0$ corresponding to Fig. 2(c). It is to some extend similar to the above situation in Fig. 3. As $\gamma$ increases, the polariton gap narrows and finally closes for $\gamma >{\gamma }_c=0.03{\omega }_0$, whereas the Bragg gap at the zone boundary is always closed [see Figs. 4(a)-4(d)]. Furthermore, the N and S EPs approach to each other and finally coalesce at $\gamma \approx 0.155{\omega }_0$ [see Figs. 4(e) and 4(f)]. For further increased $\gamma$, the N and S EPs disappear, while the M EP persists. There exists two different EP phenomena: (1) the EP appears in the polariton gap when $\gamma >{\gamma }_c$; (2) In contrast, the folded Bragg gap at the boundary of the Brillouin zone undergoes a thresholdless PT transition, as long as the frequency is in the band of the polariton state.

The above observations clearly suggest that the complex band structures are fully controlled by the independent parameters$\gamma$, $k$ and $L$. We note that $L$ plays an important role to modulate the interaction of the Bragg resonance and polariton resonance, which determines the fundamental band structure. In addition, the coalescence of states in different original bands can happen by varying either $\gamma$ or $k$. Figures 5(a) and 5(b) show the trajectories of the EPs (e.g., M, N, and S) in ($k$, $\gamma$) space for the Bragg resonance at $\omega =0.77{\omega }_0$ and $\omega =1.9{\omega }_0$, respectively. It is clearly seen that the N EP initially appears at $k=0$, whereas the M and S EPs show up firstly at $k=\pi \text{/2}L$. The difference is that the emergence of N requires that $\gamma$ reaches a critical value, however, M and S are present at any nonzero$\gamma$. This is due to the intrinsic properties of the polariton gap and naturally zero width of the Bragg gap at the zone boundary due to band folding. Moreover, as $\gamma$ increases, a pair of EPs at the zone center and boundary approach to each other, coalesce at a specific $k$, and finally disappear. The phase transition between the PT-broken phase and PT-symmetry phase in the ($k$, $\gamma$) space is therefore evidently sketched by Fig. 5.

 

Fig. 5 The respective trajectories of the EPs in ($k,\gamma$) space for the parameters presented in Fig. 3(a) and Fig. 4 (b). The gray region stands for the PT-unbroken phases.

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It is well known that the eigenstates and eigenvalues at an EP are both identical, and can be identified by the phase rigidity [4,21,22] or Petermann factor [35]. As a matter of fact, these two are consistent in nature. Therefore, to measure the degree of state identity, we study the phase rigidity which is defined as $r_m\left(k\right)={〈{\tilde{u}}_{mk}^R|{\tilde{u}}_{mk}^R〉}^{-1}$, where $|{\tilde{u}}_{mk}^R〉$ are the normalized right eigenstates. Phase rigidity measures the degree of mixing of two states near an EP. Here, the eigenstates for the proposed polariton crystal is characterized by the amplitudes of the two resonances ${\left[\begin{array}{cc}{a}_1& a_2\end{array}\right]}^T$ (Appendix B).

Figure 6 shows the phase rigidity for the complex bands in Figs. 3(c) and 3(d). The phase rigidity is supposed to vanish at the EPs (M, N, and S) [22,36]. However, for the polariton crystal discussed here, this is not the case (see Fig. 6). The reason is due to the fact that the normalization [21,22] of right eigenstates depending on the corresponding left eigenstates for the Hamiltonian H [see Eq. (13) in Appendix B]. More specifically, the right $u_R$ and left $u_l$ eigenstates are respectively Eq. (16) and Eq. (17).

 

Fig. 6 Phase rigidity of the eigenstates on the bands shown in Figs. 3(c) and 3(d). The dotted lines correspond to the position of the EPs (M), (N), and (S).

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In typical non-Hermitian systems, the left and right eigenstates are orthogonal at the EP, i.e., $〈u_l|u_r〉=0$, as a result the phase rigidity extremely approaches zero [22,36]. Whereas in the polariton crystal, they are no longer orthogonal at the EP, i.e., $〈u_l|u_r〉\ne 0$, leading to nonvanishing phase rigidity across the EPs (see Fig. 6).

4. Transmission in 1D PT symmetric polariton crystal

To this end, we have investigated the complex band structure and their evolution as a function of $\gamma$ for the proposed polariton crystal. In the following, we mainly focus on the transmission properties of such polariton crystal. The transfer matrix for extended structure with finite period $n$ can be obtained by means of Eq. (4): $M=M_{tot}^n$. Then the relation of the fields at left/right port can be written as

Figure 7 shows the transmission spectra for the system with ten periods, corresponding to the Bragg resonance of $\omega =0.77{\omega }_0$ and $\omega =1.9{\omega }_0$, respectively. It is seen that the transmission for the frequency in the gap of the band structure is extremely small, while in the vicinity of the EPs, the transmission is beyond unity. The transmission properties can be considered in two different regions: (1) before the emergence of N EP [Figs. 7(a) and 7(c)] and (2) after the N EP emergence [Figs. 7(b) and 7(d)]. The transmission nearby the M and S EPs keep growing for small gain/loss factor $\gamma$, as marked by the orange dashed arrow in Figs. 7(a) and 7(c). When the N EP emerges in the polariton gap, the total transmission grows up rapidly. The transmission would be reduced for $\gamma$ exceeding a particular value, when N EP moves towards another EP (e.g., M for $\omega =\text{0}\text{.77}{\omega }_0$, S for $\omega =\text{1}\text{.9}{\omega }_0$) [see Figs. 7(b) and 7(d)]. Besides, when two EPs become spectrally close, the interaction between them cannot be ignored, and the transmission spectrum oscillates and finally becomes smooth after the EPs coalesce. Moreover, when the S EP in the case of $\omega =\text{0}\text{.77}{\omega }_0$is apart from the other EPs, the transmission is high and the bandwidth is narrow [marked by the upward arrow in Fig. 7(b)].

 

Fig. 7 Transmission spectra for two different Bragg resonances: $0.77{\omega }_0$ (a), (b), $1.9{\omega }_0$ (c), (d), wherein (a) and (c) express the transmission before the emergence of EP (N), whereas (b) and (d) express the transmission at following cases, including the existence and coalescence of EP (N). The orange dashed arrows specify the tendency of the peaks with the increasing $\gamma$.

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Demonstrated by both the complex band structure and the transmission spectra, it is clear that the properties of the PT-symmetric polariton crystal are significantly influenced by the value of gain/loss and the propagation phase delay between the adjacent cavities. In addition to the Bragg resonance ${\omega }_{\text{B}}=0.77{\omega }_0$ and ${\omega }_{\text{B}}=1.9{\omega }_0$, results for several other situations are shown in Appendix C.

We then design a metal-dielectric-metal (MDM) waveguide system [31] to verify the characteristic features of the proposed theory and numerically verify the behaviors by the finite element method (FEM, COMSOL Multiphysics). Figure 8 shows the geometry of the system. The imaginary parts of the relative permittivity of materials filled in the side-way cavities play the role of the gain/loss coefficient $\gamma$. The cavity serves as the polariton resonance and the length between the adjacent cavities determines the fundamental Bloch states and the Bragg resonance. The relative permittivity of silver is described by the Drude model${\epsilon }_m\left(\omega \right)={\epsilon }_{\infty }-{\omega }_p^2/\left({\omega }^2+j\omega \Gamma \right)$, where${\epsilon }_{\infty }=3.7$, ${\omega }_p=9.1$eV, and $\Gamma =0.018$eV. For simplicity, the intrinsic loss of the silver is ignored, i.e., $\Gamma =0$. The relative permittivity of left (right) cavity with additional gain (loss) is set as ${\epsilon }_1=2.25+j{\epsilon }_I$ (${\epsilon }_1=2.25-j{\epsilon }_I$). To calculate the transmission spectra, only single port is excited.

 

Fig. 8 The proposed MDM waveguide structure. The parameters are defined: $d=25$_.nm, $h=50$nm, $l=175$nm. The cladding metal is silver (Ag), the embedded dielectric is silica.

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Note that the lattice constant of the original structure, $L$, is set up for different Bragg resonance frequency [c.f. Equation (3)]. Therefore, Bloch resonances of the cases in which $L=161$nm and $L=\text{483}\text{.8}$nm are far away from the polariton resonance, a reminiscent to the band structure shown in Fig. 2(c). While for $L=\text{355}$nm, these dual resonances are close to each other, whose band structure is accord to the case in Fig. 2(a). Figure 9 shows the results for MDM waveguiding systems with length $L=161$nm, $L=\text{355}$nm, and $L=\text{483}\text{.8}$nm, respectively. The transmission spectra are basically consistent with the theoretical analysis. Notice that the transmission is enhanced at both sides of the Bragg gap all the way, and raises beyond unity at the central band region, before it tends to decline for a larger ${\epsilon }_I$ [see Figs. 9(a) and 9(c)]. Particularly, Fig. 9(b) shows the case when the Bragg resonance and the polariton resonance are close to each other. The transmission spectra show only one extremely narrow hump which is absent for the case with larger ${\epsilon }_I$.

 

Fig. 9 Transmission spectra obtained by full-wave numerical simulation corresponding to three different $L$ selections: $L=161$nm (a), $L=\text{355}$nm (b), $L=\text{483}\text{.8}$nm (c). The orange dashed arrow is used to label the tendency of the peaks.

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In order to compare the results obtained by theoretical calculation [c.f. Equation (6)] and numeric simulation, we take the transmission for $L=483.8$nm as an example, as shown in Fig. 10. The parameters are obtained by the curve fitting technique through the unit cell transmission. It is clear that the results agree well with the numeric simulation results.

 

Fig. 10 The transmission spectrum (the colored circle) obtained by CMT analysis [Eq. (6)] for the configuration of Fig. 9(c) corresponding to different non-Hermiticity ${\epsilon }_I$. The parameters are obtained by curve fitting technique through unit cell numeric simulation. The solid lines are the duplicate of Fig. 9.

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In contrast to the previous add-drop resonators [10,30], the polariton crystal does not exhibit asymmetric transmission, i.e., the complete identity of transmission for either left port excitation or the right one. The reason can be illustrated by the consistent format Eq. (6) within alternative port excitation. In Appendix D, the concrete derivation is demonstrated. The transmission for right port excitation can be regarded as that with gain/loss exchange. As a result, for this polariton crystal, it cannot be distinguished for different port excitation.

5. Conclusions

In conclusion, we have studied the complex band structure and the associated phase diagram in 1D PT-symmetric polariton crystal. As the gain/loss parameter varies, the original polariton gap tends to close, and form an EP at the Brillouin zone center. Meantime, the thresholdless EP formed at the zone boundary Bragg gap (zero-width due to the band folding) approaches to the EP formed at the zone center. They coalesce and eventually disappear beyond a critical point in the ($k,\gamma$) space. The transmission properties of the polariton crystal are therefore quite complicated. We construct a MDM waveguide and demonstrate the EPs emergence, coalescence and elimination. The structures are ready to fabricate and the results are helpful to achieve usual transmission control and possible lasing state.

Appendix

A. Property of the waveguide-cavity coupled system with only one unit cell

Based on Eq. (1), the coupled mode equations for this unit cell composed of two cavities (as shown in Fig. 11) can be written as

(7)$$\begin{array}{c}\frac{d{a}_1}{dt}=-j{\omega }_1{a}_1-\left({\gamma }_1+\kappa \right)a_1+\sqrt{\kappa }{S}_{1+}+\sqrt{\kappa }\left(S_{2+}-\sqrt{\kappa }{a}_2\right)e^{j\theta }\\ \frac{d{a}_2}{dt}=-j{\omega }_2{a}_2-\left({\gamma }_2+\kappa \right)a_2+\sqrt{\kappa }{S}_{2+}+\sqrt{\kappa }\left(S_{1+}-\sqrt{\kappa }{a}_1\right)e^{j\theta }\\ S_{2-}=\left(S_{1+}-\sqrt{\kappa }{a}_1\right)e^{j\theta }-\sqrt{\kappa }{a}_2\\ S_{1-}=\left(S_{2+}-\sqrt{\kappa }{a}_2\right)e^{j\theta }-\sqrt{\kappa }{a}_1\end{array}$$

Here, $\theta =\beta L$ is the propagation phase induced by the waveguide mode.

 

Fig. 11 Schematic figure of the proposed structure consisting of two resonators coupled to a waveguide. $a_1$ and $a_2$ are the amplitudes of dual resonators. $S_{1(2)+}$ ($S_{1(2)-}$) are the incident (scattering) waveguides mode amplitude.

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Firstly, we focus on the dynamics without external excitation. Therefore, the Hamiltonian of the system derived from Eq. (7) can be written as

(8)$$j\frac{d}{dt}\left(\begin{array}{c}{a}_1\\ a_2\end{array}\right)=\left(\begin{array}{cc}{\omega }_1-j\left({\gamma }_1+\kappa \right)& -j\kappa e^{j\theta }\\ -j\kappa e^{j\theta }& {\omega }_2-j\left({\gamma }_2+\kappa \right)\end{array}\right)\left(\begin{array}{c}{a}_1\\ a_2\end{array}\right)=H\left(\begin{array}{c}{a}_1\\ a_2\end{array}\right)$$

The eigenvalues of H describe the evolution of the eigenstates:

(9)$$\omega =\frac{\left({\omega }_1+{\omega }_2\right)-j\left({\gamma }_1+{\gamma }_2+2\kappa \right)\pm j\sqrt{4e^{j2\theta }{\kappa }^2+{\left[{\gamma }_1-{\gamma }_2+j\left({\omega }_1-{\omega }_2\right)\right]}^2}}{2}$$

The real part of these eigenvalues characterizes the eigen-frequency of the eigenmodes, while the imaginary part characterizes the time evolution of the energy stored in the dual resonator system. The realization of exceptional point requires the zero of the square root of Eq. (9). So two distinct conditions can be obtained: (1) ${\gamma }_1={\gamma }_2$, $\theta =n\pi$ (n being an integer), and ${\omega }_1-{\omega }_2=\pm 2\kappa$. (2) ${\omega }_1={\omega }_2$, $\theta =\left(2n+1\right)\pi /2$ (n being an integer), and ${\gamma }_1-{\gamma }_2=\pm 2\kappa$. Here, we must emphasize that these distinct standpoints are based on the contrasting selection of propagation phase, in other words, as a function of resonant modes frequency or the intrinsic medium gain/loss.

In terms of the frequencies of the two modes (assuming${\gamma }_1={\gamma }_2=\gamma$), then the eigenvalues are $\omega =\left[\left({\omega }_1+{\omega }_2\right)-j\left(2\gamma +2\kappa \right)\pm j\sqrt{4e^{j2\theta }{\kappa }^2-{\left({\omega }_1-{\omega }_2\right)}^2}\right]/2$. We plot the eigenvalues $\omega$ as a function of $\left|{\omega }_1-{\omega }_2\right|/\kappa$ for three different phase $\theta$ (see Fig. 12). The degenerate point (generally called exceptional point) only occurs for specific choice $\theta =n\pi$ [see Figs. 12(e) and 12(f)]. This is exactly meaning the pure PT-symmetry. In addition, with the choice of $\theta =\pi /2$ [see Figs. 12(a) and 12(b)], the imaginary part of $\omega$ keeps identical, no experiencing any phase transition. As $\theta$ is apart from $n\pi$ [see Figs. 12(a)-12(d)], this evolution behaves less and less like perfect PT symmetry, which is analogous to the previous coupled waveguides with different modes coupling.

 

Fig. 12 The real (upper panel) and imaginary (lower panel) part of the eigenvalues with the increase of absolute difference between the dual resonant modes’ frequency corresponding to distinguished propagation phase, respectively, $\theta =\pi /2$ (left column), $\theta =3\pi /4$ (central column), and $\theta =\pi$ (right column).

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In terms of the intrinsic gain/loss of the cavities (assuming${\omega }_1={\omega }_2={\omega }_0$), then the eigenvalues of the system becomes$\omega =\left[2{\omega }_0-j\left({\gamma }_1+{\gamma }_2+2\kappa \right)\pm j\sqrt{4e^{j2\theta }{\kappa }^2+{\left({\gamma }_1-{\gamma }_2\right)}^2}\right]/2$. We also plot the eigenvalues as a function of $\left|{\gamma }_1-{\gamma }_2\right|/\kappa$ with three phase values $\theta$ (see Fig. 13). Under this condition, the system experiences the phase transition from unbroken PT phase to broken one for $\theta =\pi /2$[see Figs. 13(a) and 13(b)]. It is evident that the transition behavior differs from the above frequencies case [Figs. 12(e) and 12(f)], wherein experiences the inverse transition from the broken phase to the unbroken phase. Besides, for $\theta =\pi$, the real parts of $\omega$ are identical to ${\omega }_0$, while the imaginary parts are splitting [see Figs. 13(e) and 13(f)], which is also contrary to the above frequencies case [Figs. 12(a) and 12(b)]. Except for $\theta =n\pi /2$, the properties of the system behaves far away from the perfect PT symmetry [Figs. 13(c)-13(f)].

 

Fig. 13 The real (upper panel) and imaginary (lower panel) part of the eigenvalues with the increase of absolute difference between the gain/loss coefficients embedded in the dual cavities corresponding to distinguished propagation phase, respectively, $\theta =\pi /2$ (left column), $\theta =3\pi /4$ (central column), and $\theta =\pi$ (right column).

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B. Left and right eigenstates for the polariton crystal

The transfer matrix for the unit cell is written as Eq. (4). By the combination of Eq. (4) and the Bloch theorem, we have obtained the band structure. While for the eigenstates of the system, we should describe the Hamiltonian in terms of the amplitudes of resonance supported in the resonance, i.e., $a_1$ and $a_2$. The detailed derivation is as follows. Based on the transfer matrix, we obtain

(10)$$\left(\begin{array}{c}{S}_{6+}\\ S_{6-}\end{array}\right)=\left(\begin{array}{cc}{e}^{j2\theta }& 0\\ 0& e^{-j2\theta }\end{array}\right)\left(\begin{array}{c}{S}_{1+}\\ S_{1-}\end{array}\right)+\left(\begin{array}{c}-e^{j3\theta /2}\\ e^{-j3\theta /2}\end{array}\right)\sqrt{\kappa }{a}_1+\left(\begin{array}{c}-e^{j\theta /2}\\ e^{-j\theta /2}\end{array}\right)\sqrt{\kappa }{a}_2$$

Applying the Bloch theorem, Eq. (10) is transformed into

(11)$$\left(\begin{array}{cc}{e}^{jkL}-e^{j2\theta }& 0\\ 0& e^{jkL}-e^{-j2\theta }\end{array}\right)\left(\begin{array}{c}{S}_{1+}\\ S_{1-}\end{array}\right)=\sqrt{\kappa }\left(\begin{array}{cc}-e^{j3\theta /2}& -e^{j\theta /2}\\ e^{-j3\theta /2}& e^{-j\theta /2}\end{array}\right)\left(\begin{array}{c}{a}_1\\ a_2\end{array}\right)$$

Based on Eq. (1), the coupled mode equations can be written as

(12)$$\frac{d}{dt}\left(\begin{array}{c}{a}_1\\ a_2\end{array}\right)=\left(\begin{array}{cc}-j{\omega }_1-\left({\gamma }_1+\kappa \right)& -\kappa e^{j\theta }\\ -\kappa e^{j\theta }& -j{\omega }_2-\left({\gamma }_2+\kappa \right)\end{array}\right)\left(\begin{array}{c}{a}_1\\ a_2\end{array}\right)+\left(\begin{array}{cc}\sqrt{\kappa }{e}^{j\theta /2}& \sqrt{\kappa }{e}^{j3\theta /2}\\ \sqrt{\kappa }{e}^{j3\theta /2}& \sqrt{\kappa }{e}^{j\theta /2}\end{array}\right)\left(\begin{array}{c}{S}_{1+}\\ S_{6-}\end{array}\right)$$

Substitute Eq. (11) into Eq. (12), we can get

(13)$$\frac{d}{dt}\left[\begin{array}{c}{a}_1\\ a_2\end{array}\right]=\left[\begin{array}{cc}\frac{\kappa \sin \left(2\theta \right)}{\cos \left(kL\right)-\cos \left(2\theta \right)}-j{\omega }_1-{\gamma }_1& \frac{-2j\kappa \left(1+e^{jkL}\right)\sin \theta }{1+e^{j2kL}-2e^{jkL}\cos \left(2\theta \right)}\\ \frac{-j\kappa \left(1+e^{jkL}\right)\sin \theta }{\cos \left(kL\right)-\cos \left(2\theta \right)}& \frac{\kappa \sin \left(2\theta \right)}{\cos \left(kL\right)-\cos \left(2\theta \right)}-j{\omega }_2-{\gamma }_2\end{array}\right]\left[\begin{array}{c}{a}_1\\ a_2\end{array}\right]=H\left[\begin{array}{c}{a}_1\\ a_2\end{array}\right]$$

The achievement of the eigenstates including left and right eigenstates should be by means of the known complex band structure, for example, the transformation Eq. (14) from Eq. (13) can be used to obtain the right eigenstates, and the other transformation Eq. (15) to obtain the left eigenstates.

(14)$$\left[\begin{array}{cc}\frac{\kappa \sin \left(2\theta \right)}{\cos \left(kL\right)-\cos \left(2\theta \right)}+j\omega -j{\omega }_1-{\gamma }_1& \frac{-2j\kappa \left(1+e^{jkL}\right)\sin \theta }{1+e^{j2kL}-2e^{jkL}\cos \left(2\theta \right)}\\ \frac{-j\kappa \left(1+e^{jkL}\right)\sin \theta }{\cos \left(kL\right)-\cos \left(2\theta \right)}& \frac{\kappa \sin \left(2\theta \right)}{\cos \left(kL\right)-\cos \left(2\theta \right)}+j\omega -j{\omega }_2-{\gamma }_2\end{array}\right]\left[\begin{array}{c}{a}_1\\ a_2\end{array}\right]=0$$

(15)$$\left[\begin{array}{cc}{a}_1& a_2\end{array}\right]\left[\begin{array}{cc}\frac{\kappa \sin \left(2\theta \right)}{\cos \left(kL\right)-\cos \left(2\theta \right)}+j\omega -j{\omega }_1-{\gamma }_1& \frac{-2j\kappa \left(1+e^{jkL}\right)\sin \theta }{1+e^{j2kL}-2e^{jkL}\cos \left(2\theta \right)}\\ \frac{-j\kappa \left(1+e^{jkL}\right)\sin \theta }{\cos \left(kL\right)-\cos \left(2\theta \right)}& \frac{\kappa \sin \left(2\theta \right)}{\cos \left(kL\right)-\cos \left(2\theta \right)}+j\omega -j{\omega }_2-{\gamma }_2\end{array}\right]=0$$

Then, the right eigenstate reads

(16)$$\left[\begin{array}{c}{a}_1\\ a_2\end{array}\right]=\left[\begin{array}{c}\frac{2j\kappa \left(1+e^{jkL}\right)\sin \theta }{1+e^{j2kL}-2e^{jkL}\cos \left(2\theta \right)}\\ \frac{\kappa \sin \left(2\theta \right)}{\cos \left(kL\right)-\cos \left(2\theta \right)}+j\omega -j{\omega }_1-\gamma \end{array}\right]$$

And, the left eigenstate reads

(17)$${\left[\begin{array}{cc}{a}_1& a_2\end{array}\right]}^T=\left[\begin{array}{c}\frac{j\kappa \left(1+e^{jkL}\right)\sin \theta }{\cos \left(kL\right)-\cos \left(2\theta \right)}\\ \frac{\kappa \sin \left(2\theta \right)}{\cos \left(kL\right)-\cos \left(2\theta \right)}+j\omega -j{\omega }_1-{\gamma }_1\end{array}\right]$$

C. Complex band structure for different Bragg resonance positions

Figure 14 shows that the complex band structure for the Bragg resonance ${\omega }_0$ equivalent to the polariton resonance. In this case, the zero-width of the two-fold Bragg gap (at the boundary of the Brillouin zone) approximately is not working. Importantly, the bands emerging in the center of the polariton gap play significant effects on the evolution of the band structure when the gain/loss $\gamma$ is introduced. The complete overlap of the bands in the gap gives rise to the fully broken phase.

 

Fig. 14 Complex band structure for polariton crystal (Bragg resonance${\omega }_0$) with different gain/loss coefficients: (a, b)$\gamma =0.01{\omega }_0$, (c, d) $\gamma =0.04{\omega }_0$.

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Figure 15 shows the transmission spectra for the polariton crystal (just as Fig. 14 expresses) with only eight periods. Interestingly, the original band stop spectra arise an extremely narrow transmission, which is analogous to the phenomena of electromagnetic-induced-transmission.

 

Fig. 15 Transmission spectrum with the parameters consistent with Fig. 14.

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D. Transfer matrix and transmission for alternative port excitation

Based on Eq. (4), the corresponding transfer matrix for the unit cell as shown in Fig. 1, is finally written as

(18)$$M_{unit}=\left[\begin{array}{cc}a& b\\ c& d\end{array}\right].$$

Note that the system is with PT complex potentials, i.e., ${\gamma }_1=-{\gamma }_2$. Providing ${\omega }_1={\omega }_2$, exchanging the position of gain and loss (${\gamma }_1\leftrightarrow {\gamma }_2$) results into the new transfer matrix form:

(19)$${M^{\prime }}_{unit}=\left[\begin{array}{cc}a& -c\\ -b& d\end{array}\right].$$

From Eq. (6), we realize that the transmission property is tightly associated with the element $d$ of the transfer matrix. If the structure is extended to $n$ periods, then the whole transfer matrix is transformed to $M_{unit}^n$ and ${M^{\prime }}_{unit}^n$, respectively. Both of them have the same fourth element of the matrix.

So the transmissions for the finite periods with different gain/loss sequence are exactly same.

In theory, we have concluded that the polariton crystal have the same transmission for either left or right port excitation. Next, it can be verified by the numerical simulation, here, we take $L=483.8$ nm as an example as shown in Fig. 16. It is obvious that they completely agree well with each other.

 

Fig. 16 The transmission spectrum for the case $L=483.8$nm, obtained by the numeric simulation including left port and right port excitation.

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Funding

Shenzhen Municipal Science and Technology Plan (Grant No. JCYJ20150513151706573); Natural Science Foundation of Guangdong Province (Grant No. 2015A030313748).

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