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Abstract
Parity-time-reciprocal scaling (PTX)-symmetry has been recently proposed to tailor the resonance linewidth and gain threshold of non-Hermitian systems with new exhilarating applications, such as coherent perfect absorber-laser (CPAL) and exceptional point (EP)-based devices. Here, we put forward a nearly-lossless, low-index metachannel formed by PTX-symmetric metasurfaces operating at the CPAL point, supporting the undamped weakly-guided fast wave (leaky mode) and thus achieving ultradirective leaky-wave radiation. Moreover, this structure allows for a reconfigurable and tunable radiation angle as well as beamwidth determined by the reciprocally scaled gain-loss parameter. We envision that the proposed PTX-symmetric metasurfaces will shed light on the design of antennas and emitters with ultrahigh directionality, as well as emerging applications enabled by extreme material properties, such as epsilon-near-zero (ENZ) and beyond.
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1. Introduction
In the past decade, there has been increasing research interest in non-Hermitian wave systems, which can exhibit the exotic spectral degeneracy (i.e., exceptional point [1–8]) and the self-dual singularity that allows co-existence of the coherent perfect absorber (CPA) and the laser (i.e., CPAL point [9–13]). These isolated singular points, at which Taylor series expansion fail to converge, are missed in generic situations and may require special engineering of a quantum [14], optical [1–13], acoustic [15], elastic [16,17], and electronic [18,19] systems. Perhaps, one of the most interesting fields emerging in non-Hermitian physics is parity-time (PT)-symmetry, since it provides an experimentally-accessible platform to study phase transitions and EPs in the eigenspectrum of non-Hermitian Hamiltonians [43]. In this context, many potential applications have been proposed, including unidirectional reflectionless propagation [4,20], negative refraction [21], and sub-diffraction focusing [22], unidirectional invisibility [23], non-reciprocal devices [8], high-contrast optical amplitude modulators enabled by the CPAL effect, and high-performance sensors based on the EP and CPAL singularities [24–27]. The CPA-laser self-dual device is of particular interest. In general, a laser oscillator emits coherent outgoing radiations, while its time-reversed counterpart, CPA, is a dark medium absorbing incoming radiation from all directions. Nonetheless, a PT–symmetric CPAL device can act as a CPA and a laser, which are switchable at a given wavelength, for which the two eigenvalues of its scattering matrix approach zero and infinity, corresponding respectively to the CPA mode and the laser mode. The CPAL action has been theoretically proposed [9,10] and later experimentally demonstrated using optical waveguides and coupled resonators [11,12]. Furthermore, we have recently proposed a simple, low-profile CPAL devices based on PT-symmetric metasurfaces and have experimentally demonstrated this concept in the radio-frequency region by using a transmission line-based equivalent circuit [28,29]. In this context, a new generalized PT-symmetry, termed PTX-symmetry, has been proposed to reduce the gain threshold and tailor the resonance linewidth of the PT-symmetric system without affecting its eigenvalues, since the two system share the same eigenvalues, but different eigenstates and physical structures [2,30,31]. This enables breaking the balance between gain and loss coefficients in PT-symmetric non-Hermitian systems, thus increasing the design flexibility.
In the paper, we will present a tunable and ultradirective antenna or beamshaper based on the PTX-symmetric metasurface operating at the CPAL point. This device consists of a pair of active and passive metasurfaces with surface resistances of $- Z/\kappa _{}^{}$ and $\kappa Z$, respectively [see Fig. 1(a)], where Z is the characteristic impedance of the dielectrics of thickness d between the two metasurfaces, and $\kappa$ is the dimensionless scaling factor. The spatially distributed gain and loss, sourced respectively from the active metasurface with negative surface resistance and the passive metasurface with positive surface resistance, form the PTX-symmetric optical system [1]. In contrary to typical PT-symmetric systems that require balanced gain and loss (i.e., $\kappa = 1$), the PTX-symmetric systems comprise reciprocally scaled and flipped-sign surface resistances and host media, given by:
Fig. 1. (a) Schematics of PTX-symmetric metasurfaces composed of an active metasurface (-R2) and a passive metasurface (R1). The PTX-symmetric metasurfaces can be regarded as a metachannel having a longitudinal propagation constant $\beta $, varied between zero and the wavenumber of the background medium 1, resulting in an effective permittivity, $0 \le {\varepsilon _{{eff}}} \le \varepsilon .$ (b) Transmission-line network model for scattering of plane waves by PTX-symmetric metasurfaces, and (c) radiation from a line source inside PTX-symmetric metasurface.
Fig. 2. Evolution of two eigenvalues of the scattering matrix of the PTX-symmetric metasurfaces shown in Fig. 1(b).
2. Concept of the PTX-symmetry CPAL system and generation of a superdirectional beam
Figure 1(b) illustrates the PTX-symmetric metasurfaces, of which the surface resistance of the passive metasurfaces and the characteristic impedance of its host substrate are scaled by the dimensionless scaling factor $\kappa$, while those of the active sheet and its substrate are scaled by $1/\kappa .$ We first consider scattering of the TE-polarized plane wave from The PTX-symmetric metasurfaces [Fig. 1(b)], which can be modeled using the two-port transmission-line network (TLN) shown in Fig. 1(b). In the TLN model, the tangential wavenumber and characteristic impedance of the i-th medium are: ${{k}_{{i},{y}}} = \sqrt {{k}_{i}^2 - {{({{k}_1}\sin \alpha )}^2}}$ and ${{Z}_{i}} = {\eta _{i}}{{k}_{i}}/{{k}_{{i},{y}}},$ where $\alpha$ is angle of incidence and ${\eta _{i}} = \sqrt {{\mu _{i}}/{\varepsilon _{i}}}$ is the intrinsic impedance of the material. Active and passive metasurfaces are separated by a dielectric slab with impedance $\eta = \sqrt {\mu /\varepsilon }$ and wavenumber ${k} = \omega \sqrt {\mu \varepsilon }$. To satisfy the PTX-symmetry, the required surface impedances of metasurfaces and the permittivity of each layer are given by:
Figure 2 presents the evolution of the two eigenvalues ${\lambda _ \pm }$, given in [24], of scattering matrix as a function of $\gamma _{}^{}$; here $\alpha $ and $\kappa $ can be any arbitrary value provided that the condition in Eq. (1) is valid. From Fig. 2, we find that there exists an exceptional point dividing the system into the exact symmetry phase with unimodular eigenvalues and the broken symmetry phases with non-unimodular ones. In the broken symmetry phase, a self-dual singularity, CPAL point, appears at $\gamma = 1/\sqrt 2$, where the eigenvalues approach zero and infinity.
Let’s now consider PTX-symmetric metasurfaces in Fig. 1 as a parallel-plate waveguiding structure operating at the CPAL point. The seemingly unrelated scattering properties at the CPAL point may provide a useful guideline for manipulation of the effective medium property of this waveguiding structure. In general, the transverse-resonance method, considering the TLN model of the transverse cross section of the waveguide, is commonly used in analyzing the complex propagation constant of a waveguide. The same technique can be applied to study the PTX waveguide. Further, in this case, the transverse TLN model [Fig. 1(c)] is similar to that used in the scattering problem [Fig. 1(b)] [25,28]; however, each transmission line section has a transverse propagation constant ${{k}_{{i,y}}} = \sqrt {{k}_{i}^2 - {\beta ^2}}$ and a characteristic impedance ${{Z}_{i}} = \eta {{k}_{i}}/{{k}_{{i,y}}}$ (TE mode), where $\beta$ is the longitudinal propagation constant. We note that once the transverse resonance relation is satisfied, at any point along the y-axis, the sum of the input impedances seen looking to either side is zero, namely ${Z}_{{in}}^{( + )} + {Z}_{{in}}^{( - )} = 0,$ where ${Z}_{{in}}^{( + )}$ and ${Z}_{{in}}^{( - )}$ are input impedances seen looking to $+ \hat{\boldsymbol{y}}$ and $- \hat{\boldsymbol{y}}$ at any point along the resonant line, $- {d}/2 \le {y} \le {d}/2$ [28]. The dispersion equation derived based on the transverse resonance relation is given by:
When CPAL conditions (Eq. (2) with $\gamma = 1/\sqrt 2$ and Eq. (3)) are met, solving Eq. (1) leads to a propagation constant $\beta = {{k}_1}\sin \alpha$. It is worthwhile mentioning that the laser mode also exists when ${Z}_{{in}}^{( + )} + {Z}_{{in}}^{( - )} = 0$ everywhere along the y direction, for which transmission and reflection coefficients are infinite. To better illustrate this idea, let’s consider the interface between the active metasurface and the host medium as an example. At the CPAL point, ${Z}_{{in}}^{( - )}|{_{{y} ={-} {d}/2}} = \eta \sqrt {{{x}^2} + {{\tan }^2}\alpha } /{{x}^2}$ and ${Z}_{{in}}^{( + )}|{_{{y} ={-} {d}/2}} ={-} \eta \sqrt {{{x}^2} + {{\tan }^2}\alpha } /{{x}^2},$ such that the reflection coefficient for oblique TE, ${{r}^ + }{ } = ({ Z}_{{in}}^{( + )} - {Z}_{{in}}^{( - )})/({Z}_{{in}}^{( + )} + {Z}_{{in}}^{( - )})$, is infinity. The CPA mode, which is of less interest in antenna and emitter applications, is achieved with the same setup, but requiring a different initial phase offset between two incident waves [17]. As a result, the CPAL point that achieves the lasing/amplification effect in the scattering event, also implies the validity of the transverse resonance relation in the waveguiding scenario. In other words, if the CPAL condition is satisfied for an TE-polarized plane wave incident at angle $\alpha ,$ the longitudinal propagation constant of the guided TE mode in the PTX waveguide is given by: ${\beta _{}} = {{k}_1}\sin \alpha .$ Further, the fast wave exists (i.e., $0 \le \beta \le {k}$), provided that the conditions in Eq. (2) are satisfied, leading to ${{k}_1}\sin \alpha = {k/}\sqrt {1 + {\kappa ^2}{{\cot }^2}\alpha } \le {k}$.
Figure 3 shows the dispersion diagram for the PTX waveguide with its geometric and material profiles described by Eqs. (1) and (2), and $\gamma = 1/\sqrt 2$ (i.e., the system is operated around the CPAL point). It can be evidently seen that at the CPAL frequency f0, $\beta = 0$, k/2, and $\sqrt 3 k/2$ can be obtained by setting (a)$\alpha = 0$, (b) $\alpha = \pi /6$, and (c) $\alpha = \pi /3$. The results presented in Fig. 3 validate the proposed concept, showing that the fast-wave propagation mode with $\beta < {{k}_1}$ can exist at the CPAL frequency, which infers a low effective permittivity ${\varepsilon _{{eff}}}/{\varepsilon _1}{ } = { }\sin^{2}\alpha .$ We should emphasize that when compared with traditional ENZ medium made of metamaterials or Drude-dispersion semiconductors/metals, the proposed low-index metachannel may not only ease manufacturing complexity, but also address the long-standing high loss issue at the narrow ENZ band, so as to facilitate the practice of ENZ-enabled applications, including leaky-wave antennas [7,34], superluminal effect, energy squeezing, and enhanced nonlinear wave mixing [6,7,32,33]. We should also point out from Fig. 3 that radiation (i.e., leaky mode) may occurs in this unbounded fast-wave medium, and the beam angle with respect to the broadside direction is simply $\alpha = {\sin ^{ - 1}}(\beta /{{k}_1})$ in the medium 1 and $\alpha ^{\prime} = {\sin ^{ - 1}}(\beta /{{k}_2}) = {\sin ^{ - 1}}({{k}_1}\sin \alpha /{{k}_2})$ in the medium 2 [Fig. 1(b)]. Ideally, the nearly lossless leaky mode obtained here (see Fig. 3 for ${\mathop{\rm Im}\nolimits} [\beta ]$ at the CPAL point) can lead to a huge radiating aperture and thus an ultrahigh directivity of the radiation pattern. Further, the beam angle can be tuned over a wide range from broadside to end-fire, depending on the metasurface impedance and the dielectric constant of the host medium.
Fig. 3. Dispersion relations for the PTX-symmetric metachannel when the electrical length between two metasurface is x = 2 (black), x = 1 (red), and x = 0.5 (green) in Fig. 1 under the CPAL condition (Eqs. (2) and (3) with $\gamma = 1/\sqrt 2$) with (a)$\alpha = 0$ (b)$\alpha = \pi /6$ and (c) $\alpha = \pi /3$; here, the solid and dashed lines represents the real and imaginary parts, respectively.
According to the Lorentz reciprocity theorem [35,36], if a current density J1 placed at point r1 can produce an electric field E1 at point r2, their product remains constant even when the position of source and observer are switched. In the same vein, one can assume that the tangential electric field Es on PTX-symmetric metasurfaces is induced by an incident plane wave sustained by the current density JFF placed in the volume VFF. The CPAL point implies that a single source (Js) of arbitrary input amplitude can produce a huge Es. In fact, with the same structure, the CPA effect requires placing two sources with appropriate phase offsets on each side of the PTX-symmetric metasurfaces. Now, we apply reciprocity considerations to evaluate the radiated field EFF(x,y) produced by the equivalent surface current densities ${\mathbf{J}_{s}}({x})\delta ({y} \pm {d}/2)$ on metasurfaces enclosed by the volume VMTS. The reciprocity formula, $\int\limits_{{{V}_{{MTS}}}} {{\mathbf{J}_{s}}} \cdot {\mathbf{E}_{s}}{dV} = \int\limits_{{{V}_{{FF}}}} {{\mathbf{J}_{{FF}}}} \cdot {\mathbf{E}_{{FF}}}{dV}$, states that when the PTX metasurfaces operated at the CPAL point is excited by a source placed within it or in its proximity, strong and angle-selective radiated fields can be produced in the far (Fraunhofer) field. We also analyze radiation from an electric line source ($\overline {J} = \boldsymbol{\hat{z} }{{I}_0}\delta ({x})\delta ({y})\;\;[\rm{A}/{\rm{m}^2}]$) placed in the middle of a PTX-symmetric metasurfaces, as shown in Fig. 1(c). The radiation problem can be described by the transverse-equivalent network in Fig. 1(c). The electric field in the background produced by a unit amplitude electric line source can be represented as an inverse Fourier transform [37],
The TLN model can be used to determine fields radiated by a line source by means of application of the Lorentz reciprocity theorem. Subsequently, the far-zone electric field ($\rho \gg {d}$) can be obtained through an asymptotic evaluation of Eq. (5) [34–38]. The result for the upper half-plane can be written as:
The radiated power density in the upper and lower half-planes are given by:
Figures 4(a)-(c) show the theoretical and simulated results of far-field radiation patterns for the PTX-symmetric metasurfaces with $\kappa = 0.5$, 1, and 2, respectively. The PTX-symmetric metasurfaces operated near the CPAL point is excited by a line source [Fig. 1(c)]; here, the operating frequency ${f} = {{f}_0} - \delta {f}$ where ${{f}_0}$ is the CPAL frequency and $\delta {f} = { }{10^{ - 1}}{{f}_0}.$ Figs. 4(d)-(f) are the corresponding simulated contours of electric field distributions with $\kappa = 1$. It can be clearly seen from Fig. 4 that analytical (lines) and numerical (dots) results are in excellent agreement, and that radiation from the line source can be reshaped into a directive beam towards the desired direction, regardless of the value of $\kappa .$ The beam angle can be continuously tuned from broadside towards end-fire direction ($\theta = {0^o},{30^o},{45^o}$ and ${60^o}$) by varying the metasurface impedances (Eq. (2) with $\gamma = 1/\sqrt 2$ and Eq. (3)). The radiation pattern is somewhat asymmetric, due to the nature of unidirectional scattering in the generalized PT systems [19] and the reciprocal scaling operation. The beam angles on both sides are $\alpha$ (medium 1) and $\alpha ^{\prime}$ (medium 2); $\alpha = \alpha ^{\prime}$ when $\kappa = 1$, which makes the system degenerate into the PT-symmetric one. We note that at a fixed beam angle, the beamwidth increases with increasing the reciprocal scaling factor $\kappa$. From the dispersion diagram in Fig. 3, we find that the CPAL point is in the fast-wave region (leaky mode), with an abnormal zero attenuation constant, i.e., ${\mathop{\rm Im}\nolimits} [\beta ]\sim 0$, owning to the contactless gain-loss interaction. As a result, the proposed leaky-wave structure can have an ultralarge effective radiating aperture, which in turn leads to ultrahigh directivity. The simulated results also show that when $\alpha = 0,$ the electric fields inside the waveguiding structure have a nearly constant phase distribution, namely ENZ-like characteristics with $\textrm{Re} [\beta ] \approx 0$ can be obtained. Also, the amplitude contours reveals that this fast-wave propagation mode is nearly undamped, i.e., ${\mathop{\rm Im}\nolimits} [\beta ]\sim 0$.
Fig. 4. Far-field radiation pattern and electric field distribution for PTX metasurfaces excited by a line source, which is operated at the CPAL condition with (a)$\alpha = 0$ (b)$\alpha = \pi /6$ and (c) $\alpha = \pi /3$.
Figure 5(a) shows radiation patterns of the CPAL-locked PTX-symmetric metasurfaces fed by a line source, with $\alpha = 0$ and $\kappa$ being varied from 1/5 to 5. It can be seen that the beam angle is locked to the broadside direction, while the beamwidth is tuned by $\kappa $. Here, we note that despite the permittivity ${\varepsilon _1}$ of the dielectric background could differ, the directive pencil-beam can be formed in the desired direction by properly choosing the value of $\kappa$. Figure 5(b) is similar to Fig. 5(a), but for beam angle $\alpha = {45^o}$. Figure 5(c)-(e) shows the corresponding snapshot of electric field distributions. Again, it is evident that the PTX-symmetric metasurfaces can form a pencil-beam in any arbitrary direction, which can be adapted to any background dielectrics. Our results show that a highly directive and reconfigurable RF antenna/lens or optical emitter can be realized using the CPAL singularity, at which the transverse resonance relation is satisfied at any point of arbitrary cross sections of the metachannel. Leaky-wave antennas based on guided-wave devices with periodic grids/slots have been widely studied in RF, microwave, and even optical regions. However, their effective aperture size, especially for optical applications, is limited by the non-negligible attenuation rate. Also, the occurrence of higher-order spatial harmonics could produce unwanted grating lobes. These long-standing challenges may be addressed using the PTX-synthetic leaky-wave structures, which are homogeneous and non-graded surfaces.
Fig. 5. (a) Far-field radiation pattern and electric field distribution for PTX metasurfaces excited by a line source, which is operated at the CPAL condition with $\alpha = 0$ and different values of κ. (b) Similar to (a), but for α = π/4.
We also briefly discuss the practical implementation of PTX-symmetric metasurfaces. The positive surface resistance can be readily achieved by a passive metasurface with suitable loss. In the optical region, an active metasurface could be a patterned thin layer of material with negative conductivity such optically-pumped 2D materials [[39–41].], organic dyes, or semiconductors. In the microwave region, the active metasurface could be a metasurface loaded with negative-impedance converters [41,42]. The low-index host medium, required for PTX-symmetry, can be realized with the Drude-type material, which could be wire- or resonator-based metamaterials in the microwave regions or doped semiconductor in the optical regions.
3. Conclusions
We have presented the superdirective antenna or emitter based on the PTX-symmetric metasurfaces. We have first conducted the eigenmodal analysis for this non-Hermitian open structure, showing that a nearly undamped fast-wave mode (leaky mode) can exist at the CPAL point. Then, we have theoretically and numerically studied radiation from PTX-symmetric metasurfaces fed by a line source, demonstrating that it is possible to produce ultradirective beam, with a controllable beam angle ranging from broadside to end-fire direction, while vanishingly small side lobes. Furthermore, for different background dielectrics (even with extreme permittivity), the focused beam can be locked to a specific direction by varying the reciprocally scaling factor and the impedance profile of metasurfaces. Our results may open up a new pathway for building the adaptive and reconfigurable superdirective metasurface antennas and emitters, as well as the unbounded low-index artificial media.
Funding
National Science Foundation (2210977).
Disclosures
The authors declare no conflict of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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