1. Introduction

In quantum mechanics, it is well known that Hamiltonians with Hermiticity can guarantee a real energy value in a physical system. However in 1998, Bender [1] et.al found that the real eigenvalue spectra could also be exhibited in a non-Hermitian Hamiltonian with parity-time (PT) symmetry, where parity and time mean spatial reflection and time reversal, respectively. In optic framework, by spatially modulating loss and gain in materials, when a complex refractive index distribution is satisfied, i.e.,$n(x)=n{(-x)}^{\ast }$, the concept of PT symmetry can be perfectly translated from quantum mechanics [2] to optics domain [3,4]. By means of applying PT symmetry into materials, many interesting features have been demonstrated, such as unidirectional invisibility phenomena [5–7], coherent perfect absorption (CPA) [8–10], nonreciprocity of light propagation [11–13], PT-scattering configurations [14–16], extraordinary nonlinear effects [17,18]. Over the past years, considerable research efforts have also been invested in zero index metamaterials (ZIMs) [19–24] due to their unusual properties which is that electromagnetic field in such media is constant field. More recently, some interesting work connecting ZIMs with PT symmetry [25–27] have been proposed. For example, S. Savoia et.al. found that in epsilon-near-zero (ENZ) bilayers with PT symmetry, tunneling phenomena [25] are mediated by excitation of a surface wave at the interface between the gain and loss regions. They also have shown that ENZ bilayer structures can support bound and radiative modes [26] at the gain-loss interface, which depends on a critical threshold. When the loss/gain level exceeds the critical threshold, bound modes can be excited, otherwise there will be radiative modes. Moreover, a new type of nonreciprocal modes [27] has been observed at the interface between two PT symmetric magnetic domains (MDs) for mu-near-zero metamaterials. For two semi-infinite MDs, this mode is a nonpropagating one, while for finite MDs, it becomes a propagating one.

In fact, in these structures (e.g., bilayers) with PT symmetry, more or less, there are air gaps in the middle of them. Therefore, we would like to take advantage of the air gap to explore unknown physics in a PT symmetry structure. We proceed from this point, by combining ZIMs with PT symmetry, and propose a new waveguide system, which is comprised of two straight waveguides connected by a waveguide junction. For the waveguide junction, it is composed of three parts: the middle one is made of an air gap, while for the left and right areas, they are ZIMs with PT symmetry (i.e., one is ZIMs with gain, the other is ZIMs with loss). By analytical calculations and numerical simulations, we discover that there are two exceptional points in such a waveguide system, which can induce unidirectional transparency effect. Thanks to the existence of the air gap, we can control some interesting effects of the PT symmetry system. For example, when the phase difference in the air gap is a half-integer multiple of $\pi$, it is possible to achieve the singularity point in PT broken phase for a specific value of loss/gain, which can excite CPA-laser modes [8,9]. More interestingly, when such a phase difference in the air gap is an integer multiple of $\pi$, i.e., it satisfies the Fabry-Pérot (FP) resonance condition, the PT symmetry system will be suppressed. As a result, no matter how the value of loss/gain applied in ZIMs changes, the transmission and reflection do not change. To be exact, when FP resonances take place in the air gap, total transmission without any reflection can be realized in the waveguide system regardless of incident directions and the value of loss/gain. In addition, the resonance conditions of the above phenomena (e.g., unidirectional transparency and CPA-laser mode) are only determined by the size of waveguide system, which eases our design.

2. Theoretical model and analytical approach

Let us consider a two dimensional waveguide structure with ZIMs as schematically plotted in Fig. 1, where region 1 and region 5 are two straight empty waveguides with the width of a₁, connected by a rectangle waveguide junction whose width and length are a₂ and L, respectively. In such a waveguide junction we have three components: the middle one (region 3) is made of an air gap with a length of d, while for the left (region 2) and right (region 4) areas with an equal length, they are filled with ZIMs ($\epsilon =\mu =\alpha (\alpha \to 0)$). As we know, for realizing a PT-symmetry system in optics, the complex refractive index distribution n(x) needs to obey $n(x)=n{(-x)}^{\ast }$. In the above waveguide structure, we suppose the permeability of ZIMs has the following spatial distribution:

 

Fig. 1 The schematic description of the two dimensional waveguide structure, where region 1 and region 5 are empty waveguides connected by a waveguide junction (with its width and length a2 and L, respectively). For the waveguide junction, it consists of three components: the middle one (region 3) is an air gap; the left one (region 2) is ZIMs with gain; the right one (region 4) is ZIMs with loss. The black lines are outer perfect electric conductor (PEC) boundaries of the waveguide.

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For region 1 and region 5, i.e.,$\left|x\right|\ge L/2$, the general solutions of magnetic field could be expressed as, $H_z^1=f_L\exp (i{k}_{0}x)+b_L\exp (-i{k}_{0}x)$ and $H_z^5=f_R\exp (i{k}_{0}x)+b_R\exp (-i{k}_{0}x)$. Likewise, for region 3 ($-d/2\le x\le d/2$), the magnetic field is written as $H_z^3=A\exp (i{k}_{0}x)+B\exp (-i{k}_{0}x).$ While for region 2 and region 4, which comprise ZIMs with $\epsilon =\alpha (\alpha \to 0),$ the magnetic field should be constants to obtain a finite electric field [19,20], i.e.,$H_z^2=H_2$ and $H_z^4=H_4$, respectively. All the corresponding electrical fields can be expressed by $E=\nabla \times H/(-i\omega \epsilon )$. As we know, in a general case (e.g., the periodic PT symmetric system), by employing the transfer matrix M, the amplitudes of the forward and backward propagating waves are related to each other, i.e.,

In order to reveal the underlying properties of such a waveguide system with PT symmetry, we analytically show in Fig. 2 the corresponding results of transmission ($T={\left|t\right|}^2$) and reflection ($R_L={\left|r_L\right|}^2$,$R_R={\left|r_R\right|}^2$) related to two variables $\chi$ and $\phi$, where $\chi$ is the related value of loss/gain, $\phi =k_{0}d$ is the total phase difference of the wave propagating through region 3 (the air gap). In this work, we assume that the working wavelength of the waveguide and ZIMs is $\lambda$ (ZIMs always function for a narrow band of frequencies). The corresponding parameters of the waveguide structure are $a_1=\lambda {,}_{}{a}_2=4\lambda$. We set the length of each ZIM region invariable so that their area $A_s$ also is a constant value. The variables $\chi$ and $\phi$ are dependent of $\delta$ (the value of loss/gain) and d (the length of region 3), respectively.

 

Fig. 2 The analytical results of transmission (T) and reflection (R_L, R_R) related to the variables $\chi$ and $\phi$. (a) is analytical result of transmission (T). (b) and (c) are the corresponding analytical results of the left and right reflections (R_L, R_R), respectively. The white regions mean that the amplitudes are beyond the maximum value of the color bar.

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From Fig. 2, we find that there are quite intense transmission and reflection near the position $\chi =4.12\lambda$, $\phi =\pi /2$(see the black circles), which can be deduced from the variable $\Gamma$ in Eq. (6). The variable $\Gamma$ is a complex number, whose real and imaginary parts are $2a_1{a}_2\gamma$ and $-\sin (k_{0}d)(a_1^2+a_2^2-{\chi }^2)$, respectively. When the variable $\gamma$ is equal to zero, i.e., $k_{0}d=(2m-1)\pi /2$ (m is an integer), $\Gamma$ is an imaginary number. In addition, such a number could be zero for the case of $\chi =\delta k_0{A}_s=\sqrt{a_1{}^2+a_2{}^2}$. As a consequence, there is a singularity in such a waveguide system with PT symmetry due to $\Gamma =0$, which brings about quite intense transmission and reflection independent of incident directions. Moreover, the reflectionless phenomenon for $R_L$ happens at the condition of $\chi =3\lambda$, which is marked by the black dashed line in Fig. 2(b). In such a case, from the information of transmission in Fig. 2(a), we find that the transmission is unity. However, the right reflection $R_R$ in Fig. 2(c) is nonzero. Hence, the unidirectional transparency [5] could be obtained for the case of $\chi =3\lambda$. Likewise, the similar unidirectional transparency effect could be found in the case of $\chi =5\lambda$. By observing the right reflection $R_R$ in Fig. 2(c), the reflection is zero at $\chi =5\lambda$, which is depicted by the green dashed line. While for transmission and left reflection, from the corresponding results in Figs. 2(a) and 2(b), we detect that the transmission is unity and the reflection is nonzero. Therefore, the unidirectional transparency also could be realized at another condition with $\chi =5\lambda$. In fact, the unidirectional transparency can be easily deduced from Eq. (6). For example, we perceive that $\left|t_L\right|=1,$$\left|r_L\right|=0$,$\left|r_R\right|\ne 0$ for the case of $\chi =\delta k_0{A}_s=a_2-a_1$, while for $\chi =\delta k_0{A}_s=a_2+a_1$, the corresponding transmission and reflection are $\left|t_R\right|=1$, $\left|r_R\right|=0$, $\left|r_L\right|\ne 0$, respectively. Therefore, the positions of unidirectional transparency and singularity effect are only determined by the waveguide structure, i.e., a₁ and a₂. Furthermore, we find another interesting phenomenon from Eq. (6). When $\kappa =i\sin (k_{0}d)=0$, i.e.,$\phi =k_{0}d=m\pi$(m is an integer) are corresponding to Fabry-Pérot (FP) resonances, the elements of the S-matrix will be constants: $t_L=t_R=1$ and $r_L=r_R=0$, which means that total transmission without any reflection will take place in this waveguide system, independent of incident directions and the related value of loss/gain $\chi$. For example, when $\phi =\pi$, the transmission is unity and the left and right reflections are zero, which can be found in Fig. 2.

To verify the validity of the analytical results in Eq. (6), we test the transmission and reflection of the waveguide system from numerical simulations with$\phi =\pi /2$, by using COMSOL Multiphysics. In such a case, we can observe more information of the waveguide system with PT symmetry, including the unidirectional transparency and the singularity effect. We show the analytical and numerical results of reflections in Fig. 3(a), where the corresponding results of transmission are inserted. By comparing the analytical results (the solid curves) and the numerical results (the hollow shapes), we find that both of them are consistent, hence showing the validity of Eq. (6). In a PT symmetry system, a more generalized energy-conservation relation is [29],

 

Fig. 3 The properties of S-matrix. (a) are the reflection amplitudes (in a logarithm scale) of the S-matrix, where the blue and red solid curves are corresponding to the analytical results of the left (R_L) and right (R_R) reflections, respectively. The blue up triangle and red down triangle data points are the corresponding numerical results of the left (R_L) and right (R_R) reflections. The insert in (a) is the result of transmission (T), where the black solid curve is the analytical result, while the green circles are the numerical one. (b) are the phases of left (r_L), right (r_R) reflection coefficients and transmission coefficient (t), represented by the blue dashed, red solid and green solid curves, respectively. (c) are the absolute value of the eigenvalues of the S-matrix in a logarithm scale. (d) is the second component of the eigenstates ${\nu }_2$, where the real and imaginary parts of ${\nu }_2$ are represented by the red solid and blue dashed curves, respectively. All the results are related to the value of loss/gain $\chi$.

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In fact, in a PT symmetry system, the transition from the PT symmetry phase to the PT broken phase is related to the variation of transmission [29]. When T<1, the phases of the left reflection and right reflections are equivalent, corresponding to the PT symmetry phase. While for T>1, there is a $\pi$phase difference between the left and right reflections, related to the PT broken phase. Moreover, when the transmission is unity, i.e., T = 1, the corresponding points $\chi =3\lambda {,}_{}5\lambda$ are related to the exceptional (EP) points [28], i.e., the transition between the PT symmetry and broken phases. Such transitions can be demonstrated by exploring the properties of the S-matrix, including the eigenvalues $S_{\pm }=t\pm \sqrt{r_L{r}_R}$and the eigenstates ${\nu }_{\pm }={(1\pm \sqrt{r_L/r_R})}^T$. In general, for a PT symmetry phase, each eigenstate ${\nu }_{\pm }$ is itself symmetric, i.e.,$PT{\nu }_{\pm }\equiv {\nu }_{\pm }$, thus the eigenstates do not display dissipation nor amplification. Therefore, the eigenvalues of S-matrix are unimodular and nondegenerate, i.e.,$\left|S_{\pm }\right|=1$. While for a PT broken phase, the eigenstates will be transformed to each other under the PT operation, i.e.,$PT{\nu }_{\pm }\equiv {\nu }_{\mp }$, the eigenvalues are degenerate and non-unimodular, i.e.,$S_{\pm }=1/S_{\mp }{}^{\ast }$. We display the absolute value of the eigenvalues of the S-matrix in Fig. 3(c), where the EP points can be observed at $\chi =3\lambda$ and $5\lambda$, marked by the black circle points as the eigenvalues meet and bifurcate at such points. At the EP points, the unidirectional transparency could be obtained in the waveguide system. When the system is below the first EP point $\chi =3\lambda$ or above the second one $\chi =5\lambda$, the eigenvalues in a logarithm scale are zero, i.e., unimodular, which means that the waveguide system is under the PT symmetry phase. However, when the system is between these two EP points, i.e., $3\lambda <\chi <5\lambda$, the eigenvalues will be degenerate and non-unimodular, implying that the system is in the PT broken phase. We also find that there is a singularity point at $\chi =4.12\lambda$ in Fig. 3(c). The related eigenvalues either go to zero or to infinity, which respectively correspond to a PT-symmetric coherent perfect absorber (CPA) mode or laser mode [8,9]. At this point, the waveguide system can behave simultaneously as a CPA (i.e., absorbing incoming coherent waves) and a laser oscillator (that is, emitting outgoing coherent waves). In addition, the transition between the PT symmetry and broken phases also could be found from the eigenstates. For example, we plot the second component of the eigenstates, i.e.,${\nu }_2=\sqrt{r_L/r_R}$, as shown in Fig. 3(d). If ${\nu }_2$ is a real number, each eigenstate ${\nu }_{\pm }$ is itself under the PT operation, which means that the system is in the PT symmetry phase. However, if ${\nu }_2$ is an imaginary number, the eigenstates are conjugated to each other. Hence, the eigenstates will be mapped to each other under the PT operation, which implies that the system is in the PT broken phase. From Fig. 3(d), we find that ${\nu }_2$ is an imaginary number between the EP points, i.e.,$3\lambda <\chi <5\lambda$, hence it is related to the PT broken phase. Otherwise, ${\nu }_2$is a real number corresponding to the PT symmetry phase. As a consequence, the EP points can be acquired by observing the ${\nu }_2$ transition from a real number to an imaginary one, see the black points in Fig. 3(d).

3. Numerical verification

Now we will display the field pattern for the above interesting phenomena by carrying out numerical simulations. For example, the length of region 3 is set as $d=2.25\lambda$ in the simulations, so that the phase difference in region 3 is $\phi =4.5\pi$. For the parameters of the waveguide structure, they are given as $L=4\lambda$, $a_1=\lambda {,}_{}{a}_2=4\lambda$. In addition, the area of each ZIM region is $A_s=a_2(L-d)/2=3.5{\lambda }^2$. By choosing the required values of loss/gain$\delta$, we can obtain the unidirectional transparency in the waveguide system with PT symmetry. For example, when $\delta$ is about 0.1364, which is corresponding to $\chi =3\lambda$, total transmission without reflection for the left incident wave happens in the waveguide structure, which can be found from the field pattern in Fig. 4(a). In the simulations, the amplitudes for the incident wave are set as unity. To make it more straightforward, we plot the field amplitude distribution from the position $x=-4\lambda$ to $x=4\lambda$ at $y=0$, as displayed in the lower part of Fig. 4(a), where the field amplitude is unity inside the whole waveguide, implying that total transmission occurs without any reflection. While for the right incidence, there is reflective wave in the right side of the waveguide, which could be observed from the field pattern and field amplitude distribution in Fig. 4(b). The transmitted field in the left side of the waveguide is unity. Likewise, when $\delta$ is about 0.2274, which is corresponding to $\chi =5\lambda$, total transmission for the right incident wave will take place in the waveguide without any reflection, which could be observed from the field pattern and field amplitude distribution in Fig. 4(c). While for the left incidence, the reflected wave can be seen in the left side of the waveguide from the field pattern and field amplitude distribution in Fig. 4(d). Hence, the unidirectional transparency in the waveguide system with PT symmetry could be achieved if the required $\delta$ is satisfied.

 

Fig. 4 The field pattern and field amplitude distribution for the unidirectional transparency. (a) and (b) are corresponding to the left and right incidences for the case of $\chi =3\lambda$, respectively. (c) and (d) are corresponding to the right and the left incidences for the case of $\chi =5\lambda$, respectively.

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In addition, when $\delta$ is about 0.1874, which is corresponding to $\chi =4.12\lambda$, the CPA-laser mode will be excited as the phase difference in region 3 is $\phi =4.5\pi$. To realize a CPA mode, we need two-port coherent incident waves with specific amplitude and phase [8,9], while to obtain a laser mode, we just need a single-port incident wave or two-port incoherent incident waves. For simplicity, we would like to verify the CPA-laser mode in our proposed waveguide structure by exciting a laser mode from a single-port incident wave. From the field pattern in Figs. 5(a) and 5(b), it is obvious that the transmission and reflection are immensely enhanced as the laser mode is excited, which is independent of the incident directions. To be exact, for the case of the left incidence in Fig. 5(a), due to the intense reflection, the power flow marked by the black arrows in the left side of the waveguide is backward. Moreover, the transmitted and reflected waves are separated by the gain region of ZIMs, which is well depicted from the information of power flows. When the incoming wave is incident, the waveguide system will be at resonance, i.e., the laser mode is excited. Quite dramatic resonance will take place in the gain region of ZIMs so that very forceful transmitted and reflected waves can emit from such a region. The case of the right incidence in Fig. 5(b) is similar to that of the left incidence. Because of the resonance in the gain region of ZIMs, intense transmission and reflection could be observed from the field pattern and the power flows in Fig. 5(b). It is noted that the CPA-laser mode could be modulated by changing the phase difference $\phi$ in region 3, e.g., changing the length of region 3, or filling dielectric media in region 3 and so on. For example, when the phase difference $\phi$ is not a half-integer multiple of $\pi$, the CPA-laser mode will never be excited, as observed in Fig. 2.

 

Fig. 5 The field pattern for the laser mode. (a) and (b) are the corresponding field patterns of the left and right incidences for the case of $\chi =4.12\lambda$, respectively.

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Finally, we discuss another phenomenon, where the FP resonances take place in region 3, i.e., $\phi =k_{0}d=m\pi$(m is an integer). Without loss of generality, we choose m = 4 as an example to illustrate this interesting effect. All the parameters of the waveguide structure are the same as those in Figs. 4 and 5, except the length of region 3. Here the length is set as $d=2\lambda$ to guarantee $\phi =k_{0}d=4\pi$, i.e., satisfying the FP resonance conditions. As we have described it in Fig. 2, when FP resonances are satisfied, total transmission without any reflection will take place in the waveguide, independent of incident directions and the related value of loss/gain $\chi$. The conclusion is well confirmed by the numerical results in Figs. 6(a) and 6(b), where we find that the transmission is unity, while the reflection is zero, and such result is indeed independent of incident directions and the related value of loss/gain $\chi$. For example, when the value of loss/gain is set as $\delta =0.3$, which is corresponding to $\chi =7.536$, and the corresponding field patterns for the left and right incidence are shown in Figs. 6(c) and 6(d), where the incident wave can pass through the waveguide without any reflection regardless of incident directions. From the power flows in Figs. 6(c) and 6(d), when FP resonances happen inside region 3, the power flow comes from the loss region to gain region, which is independent of incident directions. Such a unidirectional energy flow between the loss region and gain region will make a balance for the loss and gain media, i.e., the dissipative energy in the loss region is exactly equal to the amplifying energy in the gain region. The waveguide system will thereby suppress the influence of PT symmetry under the condition of FP resonances, so that it will not produce energy dissipation nor amplification. Hence, when FP resonances happen in the middle region of the waveguide junction, total transmission without any reflection could be realized in the waveguide system regardless of incident directions and the value of loss/gain. It is noted when the phase difference $\phi$ is not an integer multiple of $\pi$, i.e., FP resonances do not occur, the waveguide system is still under the effect of PT symmetry.

 

Fig. 6 The transmission (reflection) and field pattern for the case of FP resonances. (a) and (b) are the corresponding transmission and reflection related to the value of loss/gain $\chi$ for cases of the left and right incidence, respectively. (c) and (d) are the corresponding field patterns of the left and right incidences for FP resonances, respectively.

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4. Conclusion

In conclusion, we have demonstrated that there are two exceptional points in a ZIM waveguide system with PT symmetry, which can induce unidirectional transparency effect. In addition, by introducing an air gap inside ZIMs with PT symmetry, we can control CPA-laser mode, or even the PT symmetry system. When the phase difference in the air gap is a half-integer multiple of $\pi$, the CPA-laser mode could be excited for a required value of loss/gain. What's more, when FP resonances happen in the air gap, the properties of such a waveguide system with PT symmetry will be suppressed. Under such condition, the value of loss/gain from PT symmetry cannot affect the transmission and reflection of the waveguide system and perfect bidirectional transmission occurs. For a fixed waveguide system with PT symmetry, by filling required dielectric materials inside the air gap or changing the length of air gap, we could turn on or off FP resonances and CPA-laser modes, which is a method for designing optical switches. Our work thereby open up a new way in investigating the physics for a PT symmetry system.