1. Introduction

Unusual points in the spectra of non-Hermitian physical systems, such as the exceptional point [1–8] and the merging point of coherent perfect absorber (CPA) and laser (i.e., CPAL point) [9–12], have attracted substantial attention in the past few years. At these points, Taylor series expansion become singular and fail to converge. These isolated singular points are generally missed in a generic situation and may require special engineering of a quantum [13], optical [1–12], acoustic [14] or electronic systems [15,16]. The CPAL systems with parity-time (PT) symmetry is of particular interest because they allow realization of laser and CPA within a single component [9–12]. Traditionally, a laser oscillator emits coherent outgoing radiations, whereas a CPA is its time-reversed counterpart that acts as a dark medium absorbing all incoming radiation. At a given CPAL wavelength, the two eigenvalues of the scattering matrix become infinity and zero, which corresponds to laser and CPA modes, respectively. These two modes with dramatically different scattering properties can be switched by altering the initial amplitude/phase offset of two counter-propagating incoming beams. The CPAL action has been theoretically proposed [9,10] and later experimentally demonstrated [11,12] using waveguides and coupled resonators with PT symmetry. In the context, PT-symmetric metasurfaces have been recently proposed to make a simplified, low-profile alternative to CPAL devices [17], negative-index media [18], optical sensing and imaging devices [19,20], as well as the unidirectional reflectionless channel biased at the exceptional point [21].

In this paper, we will propose new types of electromagnetic medium formed by PT-symmetric metasurfaces operating at the CPAL point. This system comprises a pair of active and passive metasurfaces with the spatial dependency of surface impedance given by ${{Z}_{s}}({y}) ={-} {R}\,\delta ({y} - {d}/2) + {R}\,\delta ({y} + {d}/2)$ (see Fig. 1(a)), where ± R are the surface resistances, d is the spacing between two metasurfaces, and $\delta$ is the Kronecker delta function. The spatially-distributed balanced gain ($- {R}$) and loss ($+ {R}$) form the basis of a PT-symmetric optical system [1]. While scattering from PT-symmetric metasurfaces has been studied for the above-mentioned applications [17–21], the guided-wave and leakage characteristics of a parallel-plate waveguide formed by PT-symmetric metasurfaces (Fig. 1(a)) is yet to be explored. Understanding basic characteristics and effective medium properties of such a PT-synthetic metachannel may lead to new physical phenomena and applications underlying them. In the following, we will show that this low-profile and unsophisticated metachannel can exhibit an extreme (effective) dielectric properties, such as epsilon-near-zero (ENZ) phenomena found in a dispersive lossy medium [6,7] or in a waveguide operating at its cutoff frequency [22,23]. Interestingly, the propagation constant of the guided transverse electric (TE) mode can be continuously varied from nearly zero to that of the background wavenumber (${k} = \omega \sqrt {\mu \varepsilon }$) by changing the dimensionless gain-loss parameter $\gamma = {R}/\eta$ ($\eta = \sqrt {\mu /\varepsilon }$ is the impedance of background medium). From the effective-medium perspective [22,23], the effective permittivity of the PT-synthetic metachannel can vary from ENZ to that of the background medium, i.e., $0 \le {\textrm{Re}} [{\varepsilon _{{eff}}}] \le \varepsilon$. However, unlike other ENZ and low-index media [6,7,22,23], the calculated ${\mathop{\rm Im}\nolimits} [{\varepsilon _{{eff}}}]$ related to the power attenuation rate or propagation loss can be vanishingly small. Additionally, the almost “undamped” fast wave propagating in the PT-synthetic metachannel can produce the coherent radiation leakage and form a highly directional beam, thanks to the large radiating aperture size of the metachannel. More interestingly, the beam angle can be reconfigured to any direction between broadside and end-fire by altering the gain-loss parameter (namely, the surface impedance profile of metasurfaces).

 

Fig. 1. (a) Schematics of PT-synthetic metachannel composed of an active metasurface (-R) and a passive metasurface (R). The metachannel has a longitudinal propagation constant $\beta$ that can be varied between zero and the wavenumber of the background medium, corresponding to an effective permittivity, $0 \le {\varepsilon _{{eff}}} \le \varepsilon .$ (b) Scattering of plane waves by PT-symmetric metasurfaces and its corresponding transmission-line network model. (c) Contours of two eigenvalues of the scattering matrix for the PT scattering system in (b), as a function of the gain-loss parameter $\gamma$ and the angle of incidence $\alpha .$

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2. Results and discussion

In order to understand singularities in PT-symmetric metasurfaces, we first consider scattering of the TE-polarized plane wave from this composite structure [Fig. 1(b)], which can be described by the two-port transmission-line network (TLN) shown in the inset of Fig. 1(b) [17–20]. In the TLN model, the background medium has a tangential wavenumber and a characteristic impedance given by${{k}_{y}} = {k}\cos \alpha$ and ${Z} = \eta {k}/{{k}_{y}},$ where $\alpha$ is angle of incidence. The outgoing scattered waves and the incoming waves can be related by the scattering matrix (see Appendix A). Figure 1(c) presents the evolution of the two eigenvalues of scattering matrix as a function of $\gamma$ and $\alpha ,$ with the electrical length between the two metasurfaces $\Phi { = }{{k}_{y}}{d} = \pi /2.$ From Fig. 1(c), we find that CPAL points can exist when$\gamma = 1/(\sqrt 2 \cos \alpha )$, which makes eigenvalues become zero and infinity. Exceptional points are also observed in Fig. 1(c). Such branch point singularities divide the system into the exact symmetry phase with unimodular eigenvalues and the broken symmetry phases with non-unimodular ones. Moreover, eigenvalues coalesce at exceptional points [10].

Next, we will discuss the use of PT-symmetric metasurfaces as a waveguiding channel and will show that the CPAL point found in scattering events can shed light on tailoring effective medium properties of a PT-synthetic metachannel. According to the Lorentz reciprocity theorem [24–26], if a current density J₁ placed at point r₁ produces an electric field E₁ at point r₂, then by switching the position of source and observation, their product remains constant. In our case, we assume that the tangential electric field E_s on PT-symmetric metasurfaces is induced by an incident plane wave sustained by the current density J_FF placed in the volume V_FF. Under the lasing condition, a source (J_s) of arbitrary input amplitude can produce a huge E_s. We can now apply reciprocity considerations to evaluate the radiated field E_FF(x,y) produced by the equivalent surface current densities ${{\mathbf J}_{s}}({x})\delta ({y} \pm {d}/2)$ on metasurfaces (which is defined the ratio of the tangential electric field to the surface impedance) enclosed by the volume V_MTS. The reciprocity formula yields

The eigenmodal solutions of this PT metachannel can be derived using the transverse-resonance relation that considers the TLN model of the transverse cross section of the waveguide, analogous to the one used for studying scattering of plane waves by PT-symmetric metasurfaces [the inset of Fig. 1(b)] [27]. In this case, the line has a transverse propagation constant ${{k}_{y}} = \sqrt {{{k}^2} - {\beta ^2}}$ and a characteristic impedance for the TE mode given by ${Z} = \eta {k}/{{k}_{y}},$ where $\beta$ is the longitudinal propagation constant. If the transverse resonance condition is satisfied, at any point along the y-axis, the sum of the input impedances seen looking to either side must be zero, i.e., ${Z}_{{in}}^{( + )} + {Z}_{{in}}^{( - )} = 0,$ where ${Z}_{{in}}^{( + )}$ and ${Z}_{{in}}^{( - )}$ are respectively input impedances seen looking to $+ \hat{{\boldsymbol {y}}}$ and $- \hat{{\boldsymbol {y}}}$ at any point on the resonant line, $- {d}/2 \le {y} \le {d}/2$ (see Appendix B). This yields the following dispersion equation:

 

Fig. 2. Dispersion diagram for the PT-symmetric metachannel in Fig. 1 with (a) $(\gamma ,{d}) = (1/\sqrt 2 \,,\,\,{\lambda _0}/4)$ and (b) $(\gamma ,{d}) = (1/2\,,\,\,{\lambda _0}/2\sqrt 2 )$, which lead to $\beta = 0$ (${\varepsilon _{{eff}}} = 0$) and $\beta = {k}/\sqrt 2$ (${\varepsilon _{{eff}}} = \varepsilon /2$) at the frequency of operation ${{f}_0}$, respectively.

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We note further that a PT-synthetic metachannel locked at the CPAL point exhibits a fast-wave propagation behavior, i.e., $\beta < {k}$, and, thus, has a low effective permittivity given by ${\varepsilon _{{eff}}}/\varepsilon { } = { }si{n^2}\alpha .$ Fast waves propagating in the unbounded channel formed by metasurfaces will result in the radiation leakage and the beam angle measured from broadside $\alpha = {\sin ^{ - 1}}(\beta /{k}).$ The gain-loss parameter that controls the CPAL at a certain angle of incidence (scattering events) also governs the beam angle in the leaky-wave mode. We first consider a metachannel composed of PT-symmetric metasurfaces with $\gamma = 1/\sqrt 2$ and a height of one-quarter wavelength, which forms a CPAL for normally-incident waves at frequency f₀. In accordance with the discussion above, when a PT-synthetic metachannel is excited by a waveguide port at f₀, one can expect that $\beta \approx 0,$ and, thus, an ENZ medium with infinite phase velocity is achieved. Figures 3(a) and 3(b) show the calculated radiation pattern (see Appendix C) and electrical field distributions [28] for this unbounded metachannel at ${f} = {{f}_0} - \delta {f}$ (here, $\delta {f} = { }{10^{ - 4}}{{f}_0}\;$ which leads to $\beta /{k} = 0.005 - {j}0.010$). It is seen from Fig. 3(b) that inside the channel, a nearly constant phase distribution can be obtained due to the ENZ-like characteristics. Moreover, the nearly undamped fast-wave property with $\beta \sim 0$ results in a highly directive broadside radiation, as can be seen in Fig. 3(a). In the far (Fraunhofer) zone, the directivity of 2-D radiative apertures can be defined quantitively as the ratio of the maximum radiation intensity of the main lobe (${{U}_{\max }}$) to the average radiation intensity over all space [29]:

 

Fig. 3. (a) Far-field (Fraunhofer) radiation patterns for PT-symmetric metachannels with $(\gamma ,\Phi ) = (1/\sqrt 2 {\kern 1pt} \,,\,\,\pi /2)$ and different lengths (L). The structure is excited by a waveguide port on the left and is terminated by a match load or perfect absorber on the right. Due to the channel’s ENZ characteristics, broadside radiations are observed. (b) Snapshots of electric field distributions for CPAL-locked metachannels with $(\gamma ,\Phi ) = \left( {1/(\sqrt 2 \cos \alpha ){\kern 1pt} \,,\,\,\pi /2} \right)$ and $\Phi = \pi /2,$ where $\alpha$ is the beam angle. The results demonstrate a steerable beam angle. (c) Radiation patterns for PT metachannels in (b), with the channel length fixed to 20 λ₀. Results in (a) and (c) were obtained using the analytical method, while results in (b) were obtained from full-wave simulations.

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Moreover, changing the gain-loss parameter will alter the beam angle, as can be seen in the far-field radiation patterns in Fig. 3(c) and contour plots of electric field distributions in Fig. 3(b). For different targeted beam angles $\alpha =$ 0°, 30°, 45°, and 60°, surface resistances of two metasurfaces ($\gamma = 1/\sqrt 2 \cos \alpha$) and the spacing between them ($\Phi { = kd}\cos \alpha = \pi /2$) must be changed accordingly, in order to lock the system at the CPAL point. The radiation pattern is somehow bidirectional, due to the nature of unidirectional scattering in PT systems [21]. Compared with other ENZ medium made of metamaterials or Drude-dispersion materials, the proposed low-index metachannel may not only ease manufacturing complexity, but also greatly reduce the attenuation rate. The proposed structure may therefore facilitate the practice of ENZ-allowed applications (e.g., supercoupling and superluminal effect, energy squeezing, and enhanced nonlinear wave mixing [6,7,22,23], as well as leaky-wave emitters [7,27]). Leaky-wave antennas based on guided-wave devices with periodic grids/slots have been enormously studied in different spectral ranges. However, their effective aperture size is generally limited by the non-negligible attenuation rate, especially for optical applications. Besides, the occurrence of higher-order (Floquet) spatial harmonics could produce unintended grating lobes. These long-standing challenges may be addressed by the PT-synthetic leaky-wave structures, with homogeneous non-graded surfaces and contactless gain-loss interactions.

We also analyze radiation from an electric line source ($\overline {\textrm J} = {\boldsymbol {\hat{z}} }{{I}_0}\delta ({x})\delta ({y})\;\;[A/{m^2}]$) placed at the center of a PT-synthetic metachannel, as schematically shown in Fig. 4(a). Figures 4(b) and 4(c) show the far-field radiation pattern and contours of electric field distributions for the metachannel in Fig. 4(c), under excitation of a line source; here $\delta {f} = { }{10^{ - 1}}{{f}_0}.$ The electric field in the far zone can be obtained as an inverse Fourier transform [29]:

 

Fig. 4. (a) Schematics, (b) far-field radiation patterns, and (c) snapshots of electric field distributions for CPAL-locked PT metachannels excited by a line source. Solid lines and dots in (b) represent analytical and simulated results, respectively. Results in (c) were obtained from full-wave simulations.

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3. Conclusion

In conclusion, we have proposed the concept of a PT-synthetic metachannel exhibiting zero or low effective permittivity, for which the CPAL point offers a comprehensive guidance on tailoring the extreme effective permittivity. When this metachannel working at the CPAL point is fed by a waveguide port or line source, the leaky-wave mode can couple the (nearly undamped) guided fast wave into the background medium, resulting in an ultrahigh-directivity radiative leakage. In addition, the beam can be steered from broadside towards end-fire direction by controlling the gain-loss parameter. We envision that the proposed active component may be applied to many applications of interest in different spectra, including ultrahigh-directivity antennas or emitters with tunable radiating angles, as well as low-attenuation ENZ/low-index media with leaky-wave properties.

Appendix A. Coherent perfect absorber-laser using PT-symmetric metasurface

Consider scattering of a plane wave (wave vector ${\mathbf k}{ = }\hat{{\boldsymbol {x}}}{{k}_{x}} + \hat{{\boldsymbol {y}}}{{k}_{y}}$) from PT-symmetric metasurfaces, the elements of the scattering matrix can be calculated using the two-port transmission-line network (TLN) model in Fig. 1(b). The background medium has a characteristic impedance Z, and the two shunt surface resistances are separated by a portion of transmission line with a characteristic impedance Z and an electrical length $\Phi { = }{{k}_{y}}{d}{.}$ The surface resistances have opposite values, ±R. In this system, the dimensionless gain-loss parameter (or non-Hermiticity parameter) can be defined as $\gamma = {R}/{Z}{.}$ For the transverse electric (TE) plane wave incident at an arbitrary angle $\alpha$, the wave impedance is given by ${Z = }\eta /\cos \alpha$ and the propagation constant is ${{k}_{y}}{ = k}\cos \alpha ,$ where η is the characteristic impedance of background medium. Similar considerations apply to the transverse magnetic (TM) incidence, but with ${Z = }\eta \cos \alpha .$ Using the transfer matrix formalism, and assuming time-harmonic fields e^jωt, the scattering parameters, involving transmission (t) and reflection (r) coefficients for bottom ($-$) and top (+) incidences are obtained as:

(5)$$\begin{aligned} {\mathbf S} &= \left( {\begin{array}{cc} {t}&{{{r}^ + }}\\ {{{r}^ - }}&{t} \end{array}} \right)\\ &= \left( {\begin{array}{cc} {\frac{1}{{{{e}^{{j}\Phi }} - {j}\sin (\Phi )/(2{\gamma^2})}}}&{\frac{{(1 + 2\gamma )\sin (\Phi )}}{{ - \sin (\Phi ) - 2{j}{\gamma^2}{{e}^{{j}\Phi }}}}}\\ {\frac{{(1 - 2\gamma )\sin (\Phi )}}{{ - \sin (\Phi ) - 2{j}{\gamma^2}{{e}^{{j}\Phi }}}}}&{\frac{1}{{{{e}^{{j}\Phi }} - {j}\sin (\Phi )/(2{\gamma^2})}}} \end{array}} \right). \end{aligned}$$

The validity of PT-symmetry imposes a generalized conservation relation on the scattering matrix: ${{\mathbf S}^\ast }(\omega ) = \mathcal{PT}{\mathbf S}(\omega )\mathcal{PT} = {{\mathbf S}^{ - 1}}(\omega )$ [10,33], where the parity operator $\mathcal{P} = \left( {\begin{array}{cc} 0&1\\ 1&0 \end{array}} \right),$ the time-reversal operator $\mathcal{T} = \left( {\begin{array}{cc} 0&1\\ 1&0 \end{array}} \right)\mathcal{K},$ and $\mathcal{K}$ is the complex conjugation operator. The transition between the exact and broken symmetry can be known from tracing the evolution of eigenvalues of ${\mathbf S}$, given by:

(6)$${\lambda _{1,2}} = \frac{{2{\gamma ^2} \pm \sqrt {(4{\gamma ^2} - 1){{\sin }^2}(\Phi )} }}{{2{\gamma ^2}{{e}^{{j}\Phi }} - {j}\sin (\Phi )}}.$$

Contours of eigenvalues as a function of $\gamma$ and $\alpha$ are presented in Fig. 1(c). The two eigenvalues collapse into one at the exceptional point, $\gamma = 1/(2\cos \alpha ),$ dividing the system into two distinct phases. In the exact PT-symmetry phase, the eigenvalues are nondegenerate and unimodular ($|{\lambda _{1,2}}|= 1$), whereas in the broken symmetry phase, the eigenvalues are non-unimodular (${\lambda _1} = {(\lambda _2^\ast )^{ - 1}}$). In addition to this branch point singularity, the CPAL action is achieved when $\gamma = 1/(\sqrt 2 \cos \alpha )$ and $\Phi = \mathrm{\pi }/2.$ From Fig. 3(c), we find that if $\gamma { = }\sqrt 2 ,$ a self-dual spectral singularity is obtained, with eigenvalue become $0$ and $\infty$; such values stand for the CPA and lasing states, respectively. Electric fields on bottom ($-$) and top (+) sides can be decomposed into forward (f)- and backward (b)-propagating waves, whose relations can be described by the transfer matrix M as: $\left( {\begin{array}{c} {{E}_{f}^ + }\\ {{E}_{b}^ + } \end{array}} \right) = {\mathbf M}\left( {\begin{array}{c} {{E}_{f}^ - }\\ {{E}_{b}^ - } \end{array}} \right).$ The CPAL system based on PT-symmetric metasurfaces can operate in the laser mode when ${E}_{b}^ +{/}{E}_{f}^ - \ne {{M}_{21}},$ or in the CPA mode when ${E}_{b}^ +{/}{E}_{f}^ -{=} {{M}_{21}}.$The lasing oscillator mode provides output fields ${E}_{b}^ - {,E}_{f}^ + \ne 0$ even for zero input fields (${E}_{f}^ - ,{E}_{b}^ +{\approx} 0$), while the CPA mode makes ${E}_{b}^ - { = E}_{f}^ +{=} 0$ even for non-zero input fields (${E}_{f}^ - ,{E}_{b}^ + \ne 0$) [9,17]. In physical systems, the conditions for a laser/oscillator and a CPA are ${{M}_{22}} = 0$ and ${{M}_{11}} = 0$, respectively. Nevertheless, an PT-symmetric system allows ${{M}_{22}}$ and ${M}_{11}^{}$ to be simultaneously zero at the CPAL point. By varying the initial condition, such as the complex amplitude ratio between the two incoming waves, one can switch the operation mode from lasing to CPA, and vice versa.

Appendix B. Eigenmodes in a PT-synthetic channel

Consider first the eigenmodes of the PT-synthetic channel in Fig. 1(a), a guided wave propagates along the x-axis with a factor${{e}^{ - {j}\mathrm{\beta }{x}}}.$ Electromagnetic fields can be separated into transverse electric (TE) and transverse magnetic (TM) fields with respect to a lateral coordinate. The TE mode has the following electric field distributions:

(7)$$\begin{array}{l} {\mathbf E}{ = }{\boldsymbol {\hat{z}} }{{E}_{z}}({y}){{e}^{{j}(\omega \,{t} - \beta \,{x})}}\\ {{E}_{z}}({y}) = \left\{ \begin{array}{l} {c}_1^{TE}{{e}^{ - {j}\sqrt {{k} - \beta {\,^2}} ({y} - {d}/2)}}\quad {\textrm{if}}\;{y} \ge {d}/2\\ {c}_2^{TE}{{e}^{{j}\sqrt {{k} - \beta {\,^2}} {y}}} + {c}_3^{TE}{{e}^{ - {j}\sqrt {{k} - \beta {\,^2}} {y}}}\quad {\textrm{if}}\; - {d}/2 \le {y} \le {d}/2\\ {c}_4^{TE}{{e}^{{j}\sqrt {{k} - \beta {\,^2}} ({y + d}/2)}}\quad {\textrm{if}}\;{y} \le - {d}/2 \end{array} \right. \end{array}$$

where $\beta$ is the (longitudinal) propagation constant, the transverse propagation constant ${{k}_{y}} = \sqrt {{{k}^2} - {\beta ^2}} , {k} = \omega \sqrt {\mu \varepsilon }$, $\omega$ is the angular frequency, $\mu$ and $\varepsilon$ are the wavenumber, permeability and permittivity of the background medium, respectively. Electric and magnetic fields (E, H) for the TE mode in each region can be obtained from source-free Maxwell’s equations. The complex coefficients ${c}_{i}^{TE}$ may be determined by matching the boundary conditions enforced on the metasurface: ${{\mathbf J}_{\mathbf s}} = \hat{{\mathbf n}} \times ({{{\mathbf H}^ + } - {{\mathbf H}^ - }} )= {{\mathbf E}_{\tan }}{\mathbf /}{{Z}_{s}}$ and $\hat{{\mathbf n}} \times ({{{\mathbf E}^ + } - {{\mathbf E}^ - }} )= 0,$ where $\hat{{\mathbf n}}$ is the surface normal vector, and the surface impedance has a PT-symmetric profile: ${{Z}_{s}}({y}) ={-} {R}\delta ({y} - {d}/2) + {R}\,\delta ({y} + {d}/2),$ ${R}$ and $- {R}$ are the surface resistances for the passive and active metasurfaces, respectively. The resulting dispersion equation for the complex eigenmodal solution$\beta$ is given by:

(8)$$\tan \left( {\sqrt {{k}_{}^2 - {\beta^2}} d} \right) = \frac{{{j}({k}_{}^2 - {\beta ^2})}}{{({k}_{}^2 - {\beta ^2}) - {\omega ^2}{\mu ^2}/2{R}_{}^2}}.$$

The dispersion equation can also be solved by using the transverse resonance technique [17], which employs a transmission-line model of the transverse cross section of a waveguiding structure. Eigenmodes are obtained in a resonant line, if the sum of the input impedances seen looking to either size of an arbitrary point ${y}^{\prime}$ is zero, namely:

(9)$${Z}_{{in}}^{( + )}({y}^{\prime}) + {Z}_{{in}}^{( - )}({y}^{\prime}) = 0.$$

To find the eigenmodes for the TE mode, the equivalent transverse resonance circuit shown in the inset of Fig. 1(b) can be used. The line for $- {d}/2 \le {y} \le {d}/2$ represents the PT-synthetic channel and has a transverse propagation constant ${k}_{t}^{}$ and a characteristic impedance for TE modes given by ${Z} = {k}\eta /{k}_{y}^{}$ and $\eta = \sqrt {\mu /\varepsilon } .$ Due to the fact that the longitudinal propagation constant, β, must be the same in both regions for phase matching of the tangential fields at the interface. For ${y < } - {d}/2$ and ${y > d}/2,$ the transverse line is terminated with an impedance given by ${Z} = {k}\eta /{k}_{y}^{}.$ Applying the transverse resonance condition [Eq. (9)] will lead to the dispersion equation in Eq. (8).

Appendix C. Radiation from a PT-synthetic channel under excitation of a waveguide port

The electric surface current density on a metasurface is induced by discontinuity of magnetic fields. For the PT-symmetric metasurface channel sketched in Fig. 5, surface current densities are given by:

(10)$$\begin{array}{l} {{\mathbf J}_{s,1}} = {\boldsymbol {\hat{z}} }{{J}_{{z,}1}} = \hat{y} \times [{{{\mathbf H}_ + } - {{\mathbf H}_ - }} ]|{_{{y} = {d}/2}} = \frac{{{\boldsymbol {\hat{z}} }({{{E}_{z}}|{_{{y} = {d}/2}} } )}}{{ - {R}}};\\ {{\mathbf J}_{s,2}} = {\boldsymbol {\hat{z}} }{{J}_{{z},2}} ={-} \hat{y} \times [{{{\mathbf H}_ + } - {{\mathbf H}_ - }} ]|{_{{y} ={-} {d}/2}} = \frac{{{\boldsymbol {\hat{z}} }({{{E}_{z}}|{_{{y} ={-} {d}/2}} } )}}{{R}}. \end{array}$$

In the far-field (Fraunhofer) region, the electric and magnetic fields due to ${{\mathbf J}_s}$ is given by

(11)$$\begin{array}{l} {\mathbf E} ={-} {j}\omega {\mathbf A} + \frac{1}{{{j}\omega \varepsilon \mu }}\nabla (\nabla \cdot {\mathbf A}) \approx{-} {j}\omega {\mathbf A},\\ {\mathbf H} = \frac{1}{\mu }\nabla \times {\mathbf A}, \end{array}$$

where the magnetic vector potential A due to ${{\mathbf J}_s}$ is given in terms of the Green’s function:

 

Fig. 5. PT-synthetic metachannel geometry and far-field approximations.

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${\mathbf A}(\bar{\rho }) = \int\!\!\!\int\limits_{s^{\prime}} {{{\mathbf J}_s}\frac{\mu }{{4{j}}}} {H}_0^{(2)}({{k|}\bar{\rho } - \bar{\rho }^{\prime}{|}} ){ds}^{\prime},$ where ${g}({\bar{\rho },\bar{\rho }^{\prime}} )= \frac{1}{{4{j}}}{H}_0^{(2)}({{k|}\bar{\rho } - \bar{\rho }^{\prime}{|}} )$ is the two-dimensional Green’s function, $\bar{\rho }^{\prime} = {x}^{\prime}\hat{\textrm x} + {y}^{\prime}\hat{\textrm y}$ and $\bar{\rho }{ = }\rho \hat{\rho } = {x}\hat{\textrm x} + {y}\hat{\textrm y}$ (ρ is the radial distance and $\hat{\rho } = \cos \phi {\hat{x}} + \sin \phi {\hat{y}}$) are the position vectors of the source and the observer, respectively, and${H}_0^{(2)}({\cdot} )$ is the Hankel function of the second kind. In the far zone, the electric and magnetic fields produced by sheet currents induced on the metasurfaces only have${\hat{z}}$ and $\hat{\phi }$ components in the cylindrical coordinates. Those constitute a transverse electromagnetic (TEM) wave propagating in the $\hat{\rho } -$ direction, given by:

(12)$$\begin{array}{l} {\mathbf E} = {\hat{z}}{{E}_{z}} \simeq{-} {\hat{z} j}\omega {{A}_{z}};\\ {\mathbf H} = \hat{\phi }{{H}_\phi } \simeq \hat{\phi }{ }{{E}_{z}}/\eta . \end{array}$$

The time-averaged Poynting vector is therefore written as:

(13)$${\mathbf W} = \frac{1}{2}{\textrm{Re}} [{{\hat{z}}{{E}_{z}} \times \hat{\phi }{H}_\phi^\ast } ]= \hat{\rho }\frac{1}{{2\eta }}{|{{{E}_{z}}} |^2}.$$

Approximations can be made, especially for the far-field region that is usually the one of most practical interest, to simplify the formulation of fields radiated by a PT metachannel with length L and an infinite width, as sketched in Fig. 5. In the far zone, the distance from any point on the active metasurface ($({x}^{\prime},{d}/2)$ for $- {L}/2 \le {x}^{\prime} \le {L}/2$) to the observation point can be approximately expressed as:

(14)$$\begin{array}{l} {{r}_1} = |{\bar{\rho } - \bar{\rho }^{\prime}} |= \sqrt {{{({x} - {x}^{\prime})}^2} + {{({y} - {d}/2)}^2}} \\ = \sqrt {{\rho ^2} - 2{x}^{\prime}\rho \cos \phi - \rho {d}\sin \phi { + }{{({x}^{\prime})}^2} + {{({d}/2)}^2}} \\ \approx \rho - {x}^{\prime}\cos \phi - {d}\sin \phi /2\quad \quad {\textrm{for}}\;{\textrm{the}}\;{\textrm{phase}}\;{\textrm{term}}\;{\textrm{in}}\;{\textrm{the}}\;{\textrm{far}}\;{\textrm{zone}}\\ \approx \rho \quad \quad {\textrm{for}}\;{\textrm{the}}\;{\textrm{amplitude}}\;{\textrm{term}}\;{\textrm{in}}\;{\textrm{the}}\;{\textrm{far}}\;{\textrm{zone}}, \end{array}$$

where $\phi$ is the angle in cylindrical coordinates measured from the x-axis. Similarly, the distance from any point on the passive metasurface ($({x}^{\prime}, - {d}/2)$ for $- {L}/2 \le {x}^{\prime} \le {L}/2$) to the observation point can be written as:

(15)$$\begin{array}{l} {{r}_2} = |{\bar{\rho } - \bar{\rho }^{\prime\prime}} |= \sqrt {{{({x} - {x}^{\prime})}^2} + {{({y} + {d}/2)}^2}} \\ \approx \rho - {x}^{\prime}\cos \phi + {d}\sin \phi /2\\ \approx \rho . \end{array}$$

In the far zone where $\rho \gg \rho ^{\prime}$ and ${k}\rho \gg 1,$ the vector potentials for the two metasurfaces are approximately given by:

(16)$$\begin{array}{l} {{\mathbf A}_1} = \hat{{\mathbf z}}{{A}_{{z},1}} \approx \hat{{\mathbf z}}\frac{\mu }{{4{j}}}\int_{ - {L}/2}^{{L}/2} {{{J}_{{z},1}}({x}^{\prime})} \frac{{{{e}^{ - {jk}\rho }}}}{{\sqrt {{8j}\mathrm{\pi }{k}\rho } }}{{e}^{{j}({{k}\cos \phi \cdot {x}^{\prime} + {k}\sin \phi \cdot {d}/2} )}}{dx}^{\prime};\\ {{\mathbf A}_2} = \hat{{\mathbf z}}{{A}_{{z},2}} \approx \hat{{\mathbf z}}\frac{\mu }{{4{j}}}\int_{ - {L}/2}^{{L}/2} {{{J}_{{z},2}}({x}^{\prime})} \frac{{{{e}^{ - {jk}\rho }}}}{{\sqrt {{8j}\mathrm{\pi }{k}\rho } }}{{e}^{{j}({{k}\cos \phi \cdot {x}^{\prime} - {k}\sin \phi \cdot {d}/2} )}}{dx}^{\prime}{.} \end{array}$$

Since Maxwell's equations are linear, superposition applies and, therefore, the electromagnetic fields produced by the two currents sheets induced on the active and passive metasurfaces can be expressed as ${\mathbf E} = {{\mathbf E}_1}{\mathbf + }{{\mathbf E}_2},$ where ${{\mathbf E}_1} \approx{-} {\hat{z} j}\omega {{A}_{{z},1}}$ and ${{\mathbf E}_2} \approx{-} {\hat{z} j}\omega {{A}_{{z},2}}$ are electric fields produced by ${{\mathbf J}_{{s},1}}$ and ${{\mathbf J}_{{s},2}},$ respectively.

Appendix D. Radiation from a PT-synthetic channels under excitation of an electric line source

The structure considered here is PT-symmetric metasurfaces excited by an electric line source along z with a time-harmonic dependence, embedded in the middle of the PT-symmetric metasurfaces, as sketched in Fig. 5. The transverse-equivalent network as in Fig. 6 can be used to model such an antenna [27,34–36]. The electric field in the background produced by a unit amplitude electric line source can be represented as an inverse Fourier transform [27,36]:

(17)$${{E}_z}({x},{y}) = \frac{1}{{2\mathrm{\pi }}}\int\limits_{ - \infty }^{ + \infty } {{\tilde{E}}_z^{}} ({{k}_{x}}){{e}^{ - {j}({{k}_{x}}{x} + {{k}_{y}}{y})}}{d}{{k}_{x}},$$

where the sign of ${\mathop{\rm Im}\nolimits} [{{k}_{y}}]$ is carefully chosen in order to remove unwanted unphysical solutions at infinity. The spectral electric field at the background-active metasurface interface is given by:

(18)$${\tilde{E}}_z^{( + )}({{k}_{x}}) = \frac{{{{e}^{{j}\frac{{\beta {d}}}{2}}}{RZ}[{{Z} - {{e}^{{j}\beta {d}}}({Z} + 2{R})} ]}}{{{{e}^{{j}2\beta {d}}}({Z}_{}^2 - 4{R}_{}^2) - {Z}_{}^2}}.$$

Similarly, the spectral electric field at the background-passive metasurface interface is:

(19)$${\tilde{E}}_z^{( - )}({{k}_{x}}) = \frac{{{{e}^{{j}\frac{{\beta {d}}}{2}}}{ZR}[{{{e}^{{j}\beta {d}}}({Z} - 2{R}) - {Z}} ]}}{{{{e}^{{j}2\beta {d}}}({Z}_{}^2 - 4{R}_{}^2) - {Z}_{}^2}}. $$

 

Fig. 6. Transmission-line network model for the PT metachannel excited by a line source.

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The characteristic impedance, Z, for the TE and TM polarizations have the following expressions:

(20)$$\begin{array}{l} {Z}_{}^{TE} = ({{k}/{{k}_{y}}} )\eta ;\\ {Z}_{}^{TM} = ({{{k}_{y}}/{k}} )\eta , \end{array}$$

where the vertical wavenumber k_y depends on the spherical angle $\theta$ as ${{k}_{y}} = \sqrt {{k}_{}^2 - {k}_{x}^2} = {k}\cos \theta .$ The TLN model can be used for determination of fields radiated by a source through an application of the reciprocity theorem. In this case, The far-zone electric field can readily be obtained through an asymptotic evaluation of Eq. (17) for large distances from the origin (i.e., $\rho \gg {d}$) [27,36]. The result for the upper half-plane is given by:

(21)$${E}_z^{( + )}(\rho ,\theta ) = {E}_{z}^{{ff},( + )}(\theta ){{e}^{ - {jk}\rho }}/\sqrt \rho ,$$

where the normalized far-field pattern is

(22)$${E}_{z}^{{ff},({ + })}(\theta ) = \cos \theta \sqrt {\frac{{{jk}}}{{2\pi }}} {\tilde{E}}_{z}^{( + )}({k}\sin \theta ),$$

and $\theta$ is the angle measured from broadside, and ${\pm}$ represents the upper and lower half-planes.

Similarly, the far-zone electric field in the lower half-plane is given by:

(23)$${E}_z^{( - )}(\rho ,\theta ) = {E}_{z}^{{ff},( - )}(\theta ){{e}^{ + {jk}\rho }}/\sqrt \rho ,$$

where

(24)$${E}_{z}^{{ff},( - )}(\theta ) = \cos \theta \sqrt {\frac{{{jk}}}{{2\pi }}} {\tilde{E}}_{z}^{( - )}({k}\sin \theta ).$$

The radiated power density in the upper and lower half-planes are given by:

(25)$${{P}^{({\pm} )}}(\theta ) = \frac{{|{E}_{z}^{{ff},({\pm} )}(\theta ){|^2}}}{{2\eta }}.$$

The maximum radiated power is obtained if the slab thickness is equal to one quarter-wavelength. In this work, full-wave numerical simulations were performed using the commercial software COMSOL Multiphysics. In the simulated domain, open boundaries were adopted for top, bottom and side walls, and an infinite electric line source embedded in a homogeneous media was used to excite the PT-symmetric metasurfaces that have isotropic and homogeneous surface impedances. The electric field intensity and power density of the radiated wave were monitored in cylindrical coordinates, in order to obtain the radiation patterns in Figs. 3 and 4.

Appendix E. Practical realization of PT-symmetric metasurfaces in the optical regime

According to impedance boundary conditions, a discontinuity on the tangential magnetic field on the metasurface is related to the induced averaged surface current by the surface impedance. For the normally-incident wave on a 2-D array of perfectly-conducting patches, the equivalent surface impedance can be approximately expressed as [37–39]:

(26)$${{Z}_{s}} ={-} {j}\frac{{{\eta _{eff}}}}{{2\xi }},$$

(27)$$\begin{array}{l} \xi = \frac{{{\beta _{eff}}{p}}}{\pi }\left[ {\ln \left( {csc\left( {\frac{{\pi g}}{{2{p}}}} \right)} \right) + \frac{1}{2}\sum\limits_{{n} ={-} \infty ,{n} \ne 0}^\infty {\left( {\frac{{2\pi }}{{\sqrt {{{(2\pi {n})}^2} - \beta_{eff}^2{{p}^2}} }}\frac{1}{{{|n|}}}} \right)} } \right]\quad \\ { } \approx \frac{{{\beta _{eff}}{p}}}{\pi }\ln \left( {csc\left( {\frac{{\pi g}}{{2{p}}}} \right)} \right)\quad if\quad {\beta _{eff}}{p } \ll { }2\mathrm{\pi } \end{array}$$

where ${\eta _{{eff}}} = \sqrt {\mu /{\varepsilon _{e{ff}}}} , {\beta _{{eff}}} = \omega \sqrt {\mu {\varepsilon _{{eff}}}} , {\varepsilon _{{eff}}} = \varepsilon ({\varepsilon _{r}} + 1)/2,$ ${\varepsilon _{r}}$ is the relative permittivity of the host dielectric substrate, p and g are the period and gap of patches of ignorable thickness, respectively. When a dispersive medium is used to constitute the metasurface, the surface impedance becomes complex-valued, ${{Z}_{s}} = {{R}_{s}} + {j}{{X}_{s}}\;[\Omega ],$ where the surface resistance ${{R}_{s}}$ and the surface reactance ${{X}_{s}}$ account for the gain/loss magnitude and the net stored energy in the near field, respectively. When considering a complex-valued sheet conductivity, the surface impedance of metasurface should be modified as [20,40,41]

(28)$$Z_{s}^{} = \frac{1}{{{\sigma _{s}}(1 - {g}/{p})}} - {j}\frac{{{\eta _{{eff}}}}}{{2\xi }}.$$

As an example, the photopumped graphene monolayer and bilayer could exhibit optical gain [20,42,43]. The non-equilibrium conductivity of a graphene monolayer can be modeled using Green’s functions, taking into account both interband and intraband transitions $\sigma = {\sigma _{{intra}}} + {\sigma _{{inter}}}$, given by:

(29)$${\sigma _{{intra}}} = {j}\frac{{{{q}^2}}}{{\pi {\hbar ^2}}}\frac{1}{{\omega - {j}{\tau ^{ - 1}}}}\left[ {\int_0^\infty {\varepsilon \left( {\frac{{\partial {{F}_1}(\varepsilon )}}{{\partial \varepsilon }} - \frac{{\partial {{F}_2}( - \varepsilon )}}{{\partial \varepsilon }}} \right){d}\varepsilon } } \right]$$

(30)$${\sigma _{{inter}}} ={-} {j}\frac{{{{q}^2}}}{{\pi {\hbar ^2}}}(\omega - {j}{\tau ^{ - 1}})\int_0^\infty {\frac{{{{F}_2}({ - \varepsilon } )- {{F}_1}(\varepsilon )}}{{{{(\omega - {j}{\tau ^{ - 1}})}^2} - 4{\varepsilon ^2}/{\hbar ^2}}}{d}\varepsilon } ,$$

where ${{F}_1}(\varepsilon )= {[{1 + {{e}^{(\varepsilon - {{E}_{{Fn}}})/}}^{{{K}_B}{T}}} ]^{ - 1}}, {{F}_2}(\varepsilon )= {[{1 + {{e}^{(\varepsilon - {{E}_{{Fp}}})/}}^{{{K}_B}{T}}} ]^{ - 1}},$ q is the electric charge, $\varepsilon$ is the energy, $\hbar$ is the reduce Planck’s constant, ${{K}_B}$ is the Boltzmann’s constant, T is the temperature, ω is the angular frequency, and τ is the momentum relaxation time of charge carriers. The interband transitions and the cascaded optical-phonon emission could lead to photoexcited electron-hole pairs near the Dirac point, splitting the Fermi level of graphene monolayer into two quasi-Fermi levels${{E}_{{Fn}}},{{E}_{{Fp}}} ={\pm} {\varepsilon _F}.$ Therefore, under optical pumping, ${\textrm{Re}} [{\sigma _{{inter}}}]$ in Eq. (30), could be negative. At sufficiently strong optical pumping, the $\sigma = {\sigma _{{intra}}} + {\sigma _{{inter}}} < 0$ could be achieved in the terahertz and far-infrared regions. Negative conductivity is also observed in other gapped 2D materials and organic dyes [30,44], which pave the way for building PT-symmetric metasurfaces.

Funding

National Science Foundation (1917678).

Disclosures

The authors declare no conflicts of interest.

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