Superdirective Leaky Radiation from a PT-Synthetic Metachannel

Spectral singularities appearing in parity-time (PT)-symmetric non-Hermitian optical systems have aroused a growing interest due to their new, exhilarating applications, such as bifurcation effects at exceptional points and the coexistence of coherent perfect absorber and laser (so-called CPAL point). We introduce here how the concept of CPAL action provoked in PT-symmetric metasurfaces can be translated into practical implementation of a low-loss zero- or low-index channel supporting a nearly undamped fast-wave propagation. Such a PT-synthetic metachannel shows the capability to produce a high-directivity leaky radiation, with a beam angle that can be altered by varying the gain-loss parameter. The proposed structure may enable new kinds of super-directivity antennas working in different regions of the electromagnetic spectrum, as well as various applications that demand extreme dielectric properties, such as epsilon-near-zero (ENZ).

In the Letter, we will propose new types of electromagnetic medium formed by PTsymmetric metasurfaces operating at the CPAL point.This system is composed of a pair of active and passive metasurfaces, with the spatial dependence of surface impedance given by ( ) where ±R is the surface resistance, d is the spacing between two metasurfaces, and is the Kronecker delta function.The spatiallydistributed balanced gain ( R ) and loss ( R ) form the basis of a PT-symmetric system .
While scattering from PT-symmetric metasurfaces has been studied for different purposes, wave propagation and leakage in a PT-synthetic metachannel sketched in Fig. 1(a) (e.g., when excited by a waveguide port or point source) is yet to be explored.Understanding waveguiding characteristics and effective medium properties of a PT-synthetic metachannel may bring out new physical phenomena and applications underlying them.In the following, we will demonstrate that such a low-profile and relatively unsophisticated metachannel can exhibit extreme effective dielectric properties, such as epsilon-near-zero (ENZ), existing in a dispersive medium [6], [7] or waveguide operating at its cutoff frequency [23], [24].More interestingly, by varying the dimensionless gain-loss parameter or non-Hermiticity, / R ( / is the impedance of the background medium), the propagation constant of the guided transverse electric (TE) mode can be continuously tuned from zero to the background wavenumber ( k ).From the perspective of effective medium theory [23], [24], the effective permittivity of the PT-synthetic metachannel can vary from ENZ to that of the background medium (i.e., 0 Re[ ] eff ).Unlike conventional ENZ and low-index media, the calculated Im[ ] eff related to the power attenuation rate or path loss could be vanishingly small.As a result, the almost undamped fast wave propagating in the PT-synthetic metachannel will produce coherent radiation leakage and form a superdirective radiation pattern, owning to an uniform and large radiating aperture.Additionally, by adjusting the gain-loss parameter ( ), the radiating angle can be reconfigured to any direction between broadside and end-fire.
In order to understand singularities in PT-symmetric metasurfaces, we first consider scattering of the TE-polarized plane wave from this composite structure [see Fig. , where t and r are transmission bottom ( ) and top (+) incidences [25].Figure 1(c) shows contours of two eigenvalues of S as a function of and the angle of incidence , with the electrical length between the two metasurfaces / 2.

y =k d
We find that at the CPAL point, which exists when 1/ 2 cos and / 2 = , the two eigenvalues are zero and infinity.
The exceptional point is also observed in Fig. 1(c).The two eigenvalues coalesce at this branch point singularity, dividing the system into the exact symmetry phase with unimodular eigenvalues and the broken symmetry phases with non-unimodular ones [10].The CPAL point occurs in the broken symmetry phase, as is in most PT optical systems.
Next, we will discuss the use of PT-symmetric metasurfaces as a guided propagation 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.The eigenmodal solutions of a PTsynthetic metachannel can be derived from the transverse-resonance relation that considers an equivalent TLN model similar to that used for the scattering event [the inset of Fig. 1

(b)]
[26], [27]; here, the line has a transverse propagation constant and that looking into the y side, ( ) , in Z must be zero.This yields a dispersion equation given by: ), solving Eq. ( 1) leads to a purely real propagation constant given by sin .Interestingly, the seemingly unrelated scattering and guided propagation problems can be correlated at the singular point.In the (although Z and ky are defined differently), if the system is locked at the CPAL point, the longitudinal propagation constant, , is identical to the tangential wavenumber in the scattering event, sin .
x k k This result is consistent with the eigenmodal solution obtained from Eq. ( 1).
We note further that a PT-synthetic metachannel locked at the CPAL point exhibits a fast-wave propagation behavior (i.e., ) and, thus, has a low effective permittivity given by 2 / sin .
eff Fast waves propagating in the unbounded PT channel corresponds to the leaky-wave mode induce 1 sin ( ).More interestingly, the gain-loss parameter governs the radiation direction, analogous to how it controls the CPAL action at a certain angle of incidence, , in the scattering event.We first consider a metachannel composed of PT-symmetric metasurfaces with 1/ 2 = and height of one-quarter wavelength, which makes a CPA-laser for normally-incident waves at frequency f0.Based on the above discussions, when a PTsynthetic metachannel is excited by a waveguide port at f0, one can expect that 0 and, thus, an ENZ medium with infinite phase velocity is achieved.Figure 2(a) and 2(b) show the calculated radiation pattern [25] and electrical field distributions [29] for this unbounded PT channel at 0 ; f f f here, results in a highly directive broadside radiation, as can be seen in Fig. 2(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 ( max U ) to the average radiation intensity over all space [28]: where rad P is the total radiated power.Our calculations show that the directivity of beam increases with increasing the channel length L. For example, Dmax is 10.89 (10.37 dB) for 2 L , and is increased to 51.86 (17.15 dB) for 10 L and 138.84 (21.43 dB) for an infinitely long structure.Given that Im[ ] ~0 , in light of the contactless gain-loss interaction, the PT leaky-wave structure can have a very large effective aperture [25] and superdirectivity.
Moreover, changing the gain-loss parameter will alter the beam angle, as can be seen in far-field radiation patterns in Fig. 2(c) and contour plots of electric field distributions in Fig. 2(b).For different targeted beam angles 0 o , 30 o , 45 o , and 60 o , surface resistances of two metasurfaces and the spacing between them must be changed accordingly ( 1 / 2 cos ), in order to lock the system at the CPAL point.The radiation pattern is somehow bidirectional, as a result of unidirectional scattering responses of PT systems [22].
We also analyze radiation from an electric line source (  here, the operating frequency has an 1% offset from the CPAL point.The electric field in the far zone can be obtained as an inverse Fourier transform [28]: where z E is the spectral electric field on the metasurface.From Fig. 3(b), we find that the agreement between analytical (lines) and numerical (dots) results is excellent, and that radiation from the line source can be reshaped into a directive beam and can be steered towards a specific direction in the far field.The beam angle that depends on the gain-loss parameter can be continuously tuned from broadside towards end-fire direction, as can be seen in Figs.3(b) and 3(c).Our results demonstrate that a highly directive and reconfigurable antenna or emitter can be realized by exploiting the CPAL singularity, at which the transverse resonance relation is satisfied at any point of arbitrary cross sections of a metachannel [18].Finally, we also briefly discuss the practical implementation of PT-symmetric metasurfaces.The positive surface resistance R can be readily achieved by a resistive sheet or passive metasurface made of lossy materials.In the optical region, an active metasurface could be a (patterned) thin layer of material with negative conductivity (e.g., optically-pumped 2D materials [30], organic dyes, or semiconductors).The active metasurface working at microwave frequencies could be a metasurface loaded with negative-resistance elements [17], [31].
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 locked at the CPAL point is fed by a waveguide port or line source, the leaky-wave mode can couple the guided fast wave into the background medium, resulting in a highly directive radiation leakage.Additionally, 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 electromagnetic spectra, including the high-directivity antenna or emitter with tunable radiating angles, as well as low-attenuation ENZ or low-index media.
This work was partially supported by NSF ECCS-CCSS CAREER No.1914420.

Z=
Using the transfer matrix formalism, and assuming time-harmonic fields e j , the scattering parameters, involving transmission (t r bottom ( ) and top (+) incidences are obtained as:  [2] , where the parity operator 0 1 , 1 0 the timereversal operator 0 1 , 1 0 and is the complex conjugation operator.
A condition of special interest resides in the exceptional point, when 1/ (2cos ), the unidirectional reflectionless propagation can be achieved.In addition to this branch point singularity, the CPAL action is achieved when 1/ ( 2 cos ) and / 2. 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: .
The CPAL system based on PT-symmetric metasurfaces can operate in the laser mode when E =E even for non-zero input fields ( , 0

S2. 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 .The electric surface current density on a metasurface is induced by discontinuity of magnetic fields.For the PT-symmetric metasurface channel sketched in Fig. S1, surface current densities are given by: .
In the far-field (Fraunhofer) region, the electric and magnetic fields due to s J is given by 1 ( ) , 1 , where the magnetic vector potential A due to s J is The time-averaged Poynting vector is therefore written as:  Similarly, the spectral electric field at the background-passive metasurface interface is: The maximum radiated power is obtained if the slab thickness is equal to one quarterwavelength.
1(b)], which can be described by the two-port transmission-line network (TLN) in the inset of Fig.1(b).In the TLN model, the background medium has a tangential wavenumber and a characteristic .The outgoing scattered waves and the incoming waves can be related by the scattering matrix, transverse resonance condition means that at any point along the y-axis (transverse direction), the sum of the input impedance looking to the y side,( ) , in Z scattering event [Fig.1(b)], the laser mode of CPAL action is achieved when ( ) the y-axis, such that the scattering coefficients in S become infinite.Since the scattering and guided propagation problems share a similar TLN model shown in Fig.1(b)

Fig. 2 (
Fig. 2(b) that inside the PT channel, a nearly constant phase distribution can be obtained due to center of the PT-synthetic metachannel, as schematically shown in Fig.3(a).

Figures 3 (
Figures 3(b) and 3(c) show the far-field radiation pattern and contours of electric field
y and = xx yy ( is the radial distance and cos sin x y ) are the position vectors of the source and the observer, respectively, and(2)  0 ( )His 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 z and components in the cylindrical coordinates.Those constitute a transverse electromagnetic (TEM) wave propagating in the direction, given by: 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 from a PT-synthetic channels under excitation of an electric line sourceThe 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.3(a).The transverse-equivalent network[6]-[9]  as in Fig.S2can be used to model such an antenna.The electric field in the background produced by a unit amplitude electric line source can be represented as an inverse Fourier transform, satisfy the radiation condition at infinity.The spectral electric field at the background-active metasurface interface is given by:

Figure S2 .
Figure S2.Transmission-line network model for 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