1. Introduction

Controlling the transport behavior of electromagnetic (EM) waves in media to achieve amusing propagation phenomena has always been an important topic in wave physics. In the last decade, optical zero-index metamaterials (ZIMs) [1] have provided a feasible platform for controlling the transmission and scattering of EM waves as well as radiation from sources, resulting in many novel effects [2–20], such as tunneling effects [2,3], perfect transmission [4,5], beam splitting [6], bound states in the continuum (BIC) [7–9], and cloaking [10–12]. Recently, the quasi-static properties of ZIMs have made it possible to extend the concept of doping in semiconductor physics from electronic systems to photonic systems, resulting in the concept of ZIM doping [21], which opens a new pathway for engineering EM metamaterials to manipulate EM propagation [22–26]. In most of the studies reported so far, the doped defects inside ZIMs are either purely dielectric particles without loss or dielectric particles with only loss or gain. The EM transport behavior of ZIM doped simultaneously with gain and loss dielectric defects has rarely been reported, especially when the doped ZIM meets optical parity-time (PT) symmetry. Non-Hermitian photonic systems with balanced gain and loss have attracted increasing attention due to their many potential applications in communications and lasing [27,28]. In particular, the exploration of PT symmetry in classical photonic platforms not only deepens the understanding of quantum physics but also leads to many novel optical phenomena, such as coherent perfect absorption [29–31], nonreciprocal light propagation [32,33], and unidirectional invisibility [34–36]. Inspired by these advances, in this work, we introduce the concept of optical PT symmetry into ZIM doping to show the influence of PT symmetry in ZIMs on EM wave transport. Although optical non-Hermitian doping was proposed in Ref. [37], the doped ZIMs do not satisfy the PT symmetry.

Unlike 3D ZIM [38,39], it is widely believed that the anomalous phenomena induced by two-dimensional (2D) ZIM are independent of the location of the defects [1,21–24]. This is a result of the properties of quasi-static uniform electric or magnetic fields in 2D ZIM. This common physics is also the cornerstone of photonic doping with 2D ZIM, in which the doped dielectric defects are spatially disordered distributions. Therefore, the PT-ZIM issue raised above is usually considered trivial because the quasi-static uniform features ensure little interaction between all doped defects, regardless of their doping location, number, and geometry. The lost energy when an EM wave is transmitted through the PT-ZIM junction is always completely offset by the gain, leading to a barren outcome.

Here, we present an exception to this common sense by designing and studying a PT-ZIM junction. The designed junction supports an ideal geometry symmetry-free BIC mode when their gain and loss are zero [9], i.e., corresponding to the Hermitian case. We will show that as the PT symmetry is introduced (i.e., the gain/loss varies from zero), the spatial symmetry of the two cylinders is broken, resulting in the ideal BIC modes in the photonic ZIM junction being reduced to a quasi-BIC mode. The excitation of quasi-BIC modes can produce a perfect transmission resonance in the total reflection background, and its resonance width or Q factor is determined by the loss/gain level. Significantly, it is found that this wave transport feature is highly dependent on the locations of the two cylinders in the ZIM junction, leading to an unbalanced total energy. The reason is that although the refractive index of ZIM is very small, the small phase difference between the two cylinders cannot be ignored when the quasi-BIC mode is excited. These results provide a route for the study of non-Hermitian physics in ZIMs, opening up new possibilities for many potential devices and applications.

2. Models and theory

Figure 1 schematically shows the considered two-dimensional waveguide system with a PT-ZIM junction. The junction consists of a ZIM host embedded with two nonmagnetic dielectric cylinders with the same radius of R, and the PT symmetry is introduced by adding a gain and a loss to the two cylinders. Generally, in optical systems, the PT symmetry requires a complex refractive index potential of $n(x)$ satisfying $n(x )= {n^ \ast }({ - x} )$ [40]. This means that the gain and loss are balanced, and the real part of the refractive index is spatially symmetric. In the microwave frequency band, the ZIM environment can be achieved by combing the waveguide dispersion and photonic doping [21]. The non-Hermitian materials can be obtained by using circuit-based metamaterials [41–43]. In higher frequencies such as visible and near infrared, the photonic crystal with Dirac-like dispersion also can be used as ZIMs. Loss can be easily obtained by natural materials and gain can be achieved by using dielectric crystals, glasses, ceramics, semiconductors, and dyes [44–49]. These may provide a reliable experimental platform for this work. Here, for the sake of analysis, we assume that the permittivity of the two cylinders is ${\varepsilon _{\textrm{d,1}}} = {\varepsilon _\textrm{d}} - i\gamma $ (the red cylinder) and ${\varepsilon _{\textrm{d,2}}} = {\varepsilon _\textrm{d}} + i\gamma $ (the blue cylinder), respectively, where $\gamma $ indicates the loss or gain coefficient. The center coordinates of the two cylinders are $({{x_1},{y_1}} )$ and $({{x_2},{y_2}} )$, respectively. For convenience, we set ${x_1} ={-} {x_2}$ and ${y_1} ={-} {y_2}$ to ensure that the two cylinders are centrosymmetric with respect to the origin of the whole system.

 

Fig. 1. Schematic of an air-ZIM-air waveguide structure. The upper and lower boundaries are PECs. Two nonmagnetic cylinders are embedded in the ZIM host. A TM wave is incident from the left. The width and height of the ZIM junction are d and h, respectively. The radii of the two cylinders are R, and their permittivities are ${\varepsilon _{\textrm{d,}1}} = {\varepsilon _\textrm{d}} - i\gamma $ and ${\varepsilon _{\textrm{d,2}}} = {\varepsilon _\textrm{d}} + i\gamma $, respectively.

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We consider a transverse magnetic (TM) wave with ${\mathbf{H}_{\textrm{inc}}} = \hat{{\bf z}}{H_0}\exp [{i{k_0}({x + {d / 2}} )} ]$ incident from the left inside the waveguide, where ${k_0} = {\omega / c}$ is the wave vector and $\omega = 2\pi f$ is the angular frequency. The EM wave in each region is subject to the Ampere-Maxwell equation,

According to Eq. (1), the electric field in region 1 is

3. Perfect resonances induced by PT symmetry

Based on Eq. (10), the transmission coefficient is a function of ${{{J_1}({{k_{\textrm{d},i}}R} )} / {{J_0}({{k_{\textrm{d},i}}R} )}}$. Therefore, the changes in transmission results from the variations in ${k_{\textrm{d},i}}$, which can be controlled by adjusting both ${\varepsilon _\textrm{d}}$ and $\omega $. Due to the narrow bandwidth of ZIM, we fixed the working frequency (i.e., $f = 15\textrm{ GHz}$) and the radius of the two cylinders (i.e., $R = 8\textrm{ mm}$), and varied only the permittivity (both real and imaginary parts) of the cylinders to study the transmission response of the system. Similar to the Q factor in frequency domain, the Q factor of resonance linewidth can also be defined as $Q = {{{n_{\max }}} / {\Delta n}}$, where ${n_{\max }}$ is the refractive index corresponding to transmission peak, and $\Delta n$ the full width half maximum bandwidth. Therefore, a vanishing linewidth can indicate the presence of a BIC.

To investigate the EM properties of the system, we calculate the transmission change as ${\varepsilon _\textrm{d}}$ and $\gamma $, shown in Fig. 2(a) and (b). The analytical results are obtained by using Eq. (10). In addition, we performed numerical simulations by using the finite element method software COMSOL Multiphysics. For a fixed $\gamma $, the appearance of a transmission resonance with perfect efficiency (i.e., $\Im = 1$) around ${\varepsilon _\textrm{d}} = 4.818$ is clearly seen. In particular, as $\gamma $ tends to zero, the resonance peak narrows and eventually vanishes at $\gamma = 0$, leading to an infinite Q factor in theory. This means that such a ZIM junction without gain and loss supports an ideal BIC mode decoupled from the external incident wave, which agrees with the results in Ref. [9]. This BIC is a Friedrich-Wintgen type [50], resulting from the interference cancellation of resonance radiation in the two dielectric cavities. Once $\gamma \ne 0$, the PT symmetry makes the ideal BIC turn into a quasi-BIC, and its excitation produces a perfect transmission resonance with a controlled Q factor. As a demonstration, Fig. 2(b) shows the analytical and simulated transmission for different losses/gains, i.e., $\gamma = 0$ (the blue curve and circles) and $\gamma = 0.05$ (the red curve and circles). The simulation results are in good agreement with the analysis results. In the case of $\gamma = 0.05$, the symmetry of the two cylinders is broken, i.e., one cylinder is loss and the other is gain; then, the balanced gain and loss induce a perfect transmission resonance around ${\varepsilon _\textrm{d}} = 4.818$, which is analogous to electromagnetically induced transparency in optics [51]. Figure 2(c)-(e) displays the corresponding simulated field pattern at the three special points. At transmission dips (see Fig. 2(c) and (e)), the fields at both the transmission and incident ends are weak. Note that the reflection field and the incident field interfere with each other, resulting in a very weak total magnetic field at the incident side. At the transmission peak (see Fig. 2(d)), the field passes through ZIM without any reflection. Due to the zero curl of magnetic field inside 2D ZIMs, the magnetic field in the ZIM remains constant in the ZIM. In addition, the white color region indicates that the field amplitude is out of the color bar, which means that the quasi-BIC resonance leads to a great enhancement of the EM field inside the cylinder. The right panel of Fig. 2(c)-(e) shows the enlarged field pattern in the ZIM junction, illustrating that two identical but out-of-phase monopole mode resonances are excited. For the dip-one and dip-two situations, the magnetic field inside the cylinder is approximately 5 times the incident magnetic field. However, at the peak, it increases to 20 times the incident magnetic field. Such strong resonance has great potential for lasing and nonlinear effects at lower frequency regimes, such as in microwaves.

 

Fig. 2. (a) Analytical results of the transmission as a function of ${\varepsilon _\textrm{d}}$ and $\gamma $. (b) Transmission as a function of ${\varepsilon _\textrm{d}}$ for $\gamma = 0$ (blue) and $\gamma = 0.05$ (red). The solid lines represent analytical results, and the circles represent simulated results. (c) Simulated magnetic field the transmission dip one (${\varepsilon _\textrm{d}} = 4.767$) in (b) . The right panel is field patterns inside ZIM regions with an enlarged color bar. (d) and (e) are the same as (c) but for the transmission peak (${\varepsilon _\textrm{d}} = 4.818$) and the transmission dip two (${\varepsilon _\textrm{d}} = 4.867$), respectively. In simulations and calculations, $d = 60\textrm{ mm}$, $h = 44\textrm{ mm}$, $R = 8\textrm{ mm}$, and $f = 15\textrm{ GHz}$. The amplitude of the incident magnetic field is $1\textrm{ }{\textrm{A} / \textrm{m}}$. The permittivity and permeability of the ZIM host are ${\varepsilon _{\textrm{ZIM}}} = {\mu _{\textrm{ZIM}}} = {10^{ - 4}}$.

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Moreover, the magnetic flux of each cylinder was analyzed to further reveal the physics, with the obtained results shown in Fig. 3. Figure 3(a) depicts the $\gamma = 0$ case, and Fig. 3(b) shows the $\gamma = 0.05$ case. The sign of ${\textrm{Re}} ({{\Phi _i}} )$ (see the solid red line) changes when it crosses the resonant point of ${\varepsilon _\textrm{d}} = 4.818$. Moreover, we have ${\textrm{Re}} ({{\Phi _1}} )= {\textrm{Re}} ({{\Phi _2}} )$ and ${\mathop{\rm Im}\nolimits} ({{\Phi _1}} )={-} {\mathop{\rm Im}\nolimits} ({{\Phi _2}} )$. In fact, due to ${\varepsilon _{\textrm{d,}i}}(r )= \varepsilon _{\textrm{d,}i}^ \ast ({ - r} )$, the magnetic flux also satisfies PT symmetry, i.e., ${\Phi _1}(r )= \Phi _2^ \ast ({ - r} )$, so that ${\Phi _1} + {\Phi _2} = 2\,{\textrm{Re}} ({{\Phi _1}} )$. Then, Eq. (10) becomes $\Im = {1 / {[{1 - ({{{2i\pi } / h}} ){\textrm{Re}} ({{\Phi _1}} )} ]}}$. Consequently, when ${\textrm{Re}} ({{\Phi _1}} )= 0$, total transmission occurs, which is consistent with the results in Fig. 3(b). Physically, the zero total magnetic flux in the two cylinders results in the fact that the ZIM embedded with the PT cylinder behaves as a uniform and impedance-matched ZIM; thus, the incident wave can completely pass through it.

 

Fig. 3. The analytical ${\Phi _i}$ as a function of ${\varepsilon _\textrm{d}}$. (a) Real part (solid red line) and imaginary part (dashed blue line) of ${\Phi _i}$ for $\gamma = 0$. (b) The real and imaginary parts of ${\Phi _i}$ for $\gamma = 0.05$. Notably, the signs of ${\mathop{\rm Im}\nolimits} ({{\Phi _1}} )$ and ${\mathop{\rm Im}\nolimits} ({{\Phi _2}} )$ are opposite, and they are shown only by a dashed blue line. The three vertical dashed lines manifest the locations of the two transmission dips and the transmission peak in Fig. 2(b).

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4. Location-dependent features

In previous studies of ZIM hosts embedded with dielectric defects, it is widely believed that all ZIM-induced anomalous phenomena, such as the tunneling effect, are independent of the location of the defects. Here, we will show that this concept fails in the currently considered ZIM system. For discussion, an energy loss coefficient is defined by $\xi = 1 - {|\Im |^2} - {|\Re |^2}$, where $\Re $ is the reflection coefficient of the whole system. When $\xi > 0$, the system absorbs EM waves and behaves as an absorber; conversely, when $\xi < 0$, the system radiates energy outward and behaves as a source. Figure 4(a) shows $\xi $ as a function of ${\varepsilon _\textrm{d}}$ for different ZIMs when $\gamma = 0.05$. Clearly, due to the excitation of quasi-BICs, a resonance similar to a Fano-like line shape is observed in the energy loss coefficient spectra [51]. Such Fano-like resonance is preserved even if the refractive index of the ZIM host is as low as ${10^{ - 4}}$. Figure 4(b) shows the energy loss coefficient $\xi $ as a function of ${\varepsilon _\textrm{d}}$ and $\gamma $ when ${n_{\textrm{ZIM}}} = {10^{ - 3}}$. The energy loss coefficient is zero only for $\gamma = 0$ (the dashed red line). For $\gamma > 0$, the system absorbs EM waves ($\xi > 0$) before resonance and emits EM waves ($\xi < 0$) after resonance. However, for $\gamma < 0$, the system emits (absorbs) energy before (after) resonance. Obviously, $\xi $ is a function of $\gamma $. However, for a fixed ${\varepsilon _\textrm{d}}$, $\xi (\gamma )$ is different from $\xi ({ - \gamma } )$ (here, we assume $\gamma > 0$). For the case of $\xi (\gamma )$, the cylinder at $({{x_1},{y_1}} )$ is gain, and the cylinder at $({{x_2},{y_2}} )$ is loss. However, for the case of $\xi ({ - \gamma } )$, the cylinder at $({{x_1},{y_1}} )$ is considered the loss and the cylinder at $({{x_2},{y_2}} )$ is considered the gain. Therefore, exchanging the locations of the gain and loss cylinders will affect the energy loss coefficient $\xi $. In other words, the energy loss coefficient $\xi $ is dependent on the locations of the two cylinders. This finding breaks the common wisdom in ZIM properties that the transport of EM waves passing through the ZIM junction is independent of the locations of the embedded defects.

 

Fig. 4. (a) The energy loss coefficient $\xi $ as a function of ${\varepsilon _\textrm{d}}$ for $\gamma = 0.05$ in different ZIMs. The ZIM is an impedance-matched material (i.e., ${\varepsilon _{\textrm{ZIM}}} = {\mu _{\textrm{ZIM}}}$), and ${n_{\textrm{ZIM}}} = \sqrt {{\varepsilon _{\textrm{ZIM}}}{\mu _{\textrm{ZIM}}}} $ is the refractive index of the ZIM. The red, blue, and green lines are the results of ${n_{\textrm{ZIM}}} = 1 \times {10^{ - 3}}$, ${n_{\textrm{ZIM}}} = 5 \times {10^{ - 4}}$, and ${n_{\textrm{ZIM}}} = 1 \times {10^{ - 4}}$, respectively. The gray area represents the resonance region for ${n_{\textrm{ZIM}}} = 1 \times {10^{ - 3}}$. (b) Numerically calculated $\xi $ as a function of ${\varepsilon _\textrm{d}}$ and $\gamma $ for ${n_{\textrm{ZIM}}} = {10^{ - 3}}$.

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To further illustrate the effect of the location dependence, we move two cylinders in the x-direction (see Fig. 5(a)) and in the y-direction (see Fig. 5(b)) to explore their impact on $\xi $, with the corresponding obtained results shown in Figs. 5(c) and 5(d), respectively. One can see that the Fano-like resonance line shape is completely independent of $\Delta y$. In contrast, $\Delta x$ has a strong influence on $\xi $, and the change in the sign of $\Delta x$ leads to a reversal of the resonance. In particular, when $\Delta x = 0$, the energy loss coefficient is zero for all ${\varepsilon _\textrm{d}}$, even at the position corresponding to the quasi-BIC. This means that only in this specific case are the gains and losses perfectly balanced. The underlying physics is attributed to the phase shift between two cylinders. In fact, although the refractive index of the ZIM is very small, there is still a tiny phase shift as the EM wave propagates in the ZIM. The phase shift between the two cylinders is ${\phi _x} = {k_0}{n_{\textrm{ZIM}}}\Delta x$ in the x-direction and ${\phi _y} = 0$ in the y-direction. This location-dependent effect induced by the quasi-BIC can be understood with the interaction between the scattering fields of the two cylinders. The scattering fields of the cylinders can be written as ${S_\textrm{1}}\exp ({i{\varphi_1}} )$ and ${S_2}\exp ({i{\varphi_2}} )$, where ${S_i}$ and ${\varphi _i}$ are the amplitude and phase of the scattering fields of the i-th cylinder, respectively. If the refractive index of the ZIM is a small value, the total scattering field is ${S_\textrm{1}}\exp [{i({{\varphi_1} + {n_{\textrm{ZIM}}}{k_0}\Delta x} )} ]+ {S_2}\exp ({i{\varphi_2}} )$. When the quasi-BIC is excited, ${\varphi _1} = {\varphi _2} + \pi $ and ${S_1} = {S_2}$, resulting in a total scattering field of ${S_1}\exp ({i{\varphi_1}} )[{1 + i{n_{\textrm{ZIM}}}{k_0}\Delta x - 1} ]= iC{n_{\textrm{ZIM}}}{k_0}\Delta x$, where $C = |{{S_1}\exp ({i{\varphi_1}} )} |$. Therefore, C, ${n_{\textrm{ZIM}}}$, and $\Delta x$ can all affect the total scattering field, which in turn influences the electromagnetic response of the system. As a result, when the scattering fields of the cylinders is greatly enhanced by the quasi-BIC, the phase shift induced by the ZIM environment becomes non-negligible. This result provides a way to study and reveal new interesting effects of ZIM.

 

Fig. 5. Schematic of the two cylinders moving (a) in the x-direction and (b) in the y-direction. The center of the red cylinder is at $({{x_1},{y_1}} )$, and the center of the blue cylinder is at $({{x_2},{y_2}} )$. The lateral deviation of the two cylinders is $\Delta x = {x_1} - {x_2}$, and the longitudinal deviation of the two cylinders is $\Delta y = {y_1} - {y_2}$. (c) The energy loss coefficient $\xi $ changes as ${\varepsilon _\textrm{d}}$ for different $\Delta x$, i.e., $\Delta x ={\pm} 20$, $\Delta x ={\pm} 10$, and $\Delta x = 0$. (d) $\xi $ changes as ${\varepsilon _\textrm{d}}$ for different $\varDelta y$. In all calculations, $\gamma = 0.05$, ${n_{\textrm{ZIM}}} = {10^{ - 3}}$, and the initial coordinates of the red and blue cylinders are $({ - 10,10} )$ and $({10, - 10} )$, respectively. The coordinates are in millimeters.

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5. Conclusions

We have studied the EM wave transport behavior in a non-Hermitian doping ZIM with PT symmetry in a waveguide system. We have shown that the introduced PT breaks the spatial symmetry of the doped ZIM junction, which makes the ideal Friedrich-Wintgen BICs in the Hermitian system quasi-BICs, producing perfect resonance transmission in the perfect reflection background. Since the quasi-BIC modes in non-Hermitian doping ZIM are severely constrained by loss/gain, the Q factor of perfect transmission resonance is loss/gain-controlled. In addition, we have also demonstrated that the EM transport behavior stemming from the excitation of BIC modes depends on the locations of the two cylinders in the ZIM. This is because although the refractive index of the ZIM background is very small, the BIC resonance greatly amplifies the effect of the small phase difference between the scatterers. Our results provide a unique route for the study of non-Hermitian physics and BICs in ZIM, and the obtained results have great potential applications in optical communications, sensing, and boosting nonlinear effects.