Abstract

Observation of exceptional points (EPs) in non-Hermitian parity-time (PT) symmetric systems has led to various nontrivial physics and exotic phenomena. Here, a metal-graphene hybrid non-Hermitian metasurface is proposed in the terahertz regime, whose unit cell is composed of two orthogonally oriented split-ring resonators (SRRs) with identical dimensions but only one SRR containing a graphene patch at the gap. An EP in polarization space is theoretically observed at a certain Fermi level of the graphene patch, where the induced asymmetric loss and the coupling strength between the two SRRs match a certain relation predicted by a coupled mode theory. The numerical fittings using the coupled mode theory agree well with the simulations. Besides, an abrupt phase flip around the EP frequency is observed in the transmission in circular polarization basis, which can be very promising in ultra-sensitive sensing applications.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Most operators in quantum mechanics are Hermitian, which guarantee the observables are real. In 1998 Bender and Boettcher showed that non-Hermitian Hamiltonians can also exhibit real spectra if they commute with the parity-time (PT) operator [1]. Since then, PT symmetry has become an active research area, and a tremendous progress has been achieved in both the theory [26] and experimental implementations in optics [710], acoustics [1113], electronics [1416], and quantum gases [17], etc. Interestingly, it is found that the transition point between the unbroken and broken (complex spectra) PT symmetric regions is an exceptional point (EP), which is a branch point singularity in the parameter space of a system whose eigenvalues and eigenvectors become simultaneously degenerate [18]. Nontrivial physics and exotic phenomena could arise at, near, and encircling an EP [19]. In optics, drawing inspiration from such conception of PT symmetry and EPs in quantum mechanics, intriguing effects have been observed, such as loss-induced transparency [20], unidirectional invisibility [21,22], nonreciprocal light propagation [2326], reversal of pump dependence of a laser [27], and unconventional beam refraction [28].

With unprecedented controlling ability over electromagnetic waves, metasurfaces provide a fertile platform to exploit PT symmetric concepts and related applications. One particular example is a metasurface system constructed by two orthogonally arranged coupled resonators with similar resonance frequency, which can be described by a Hamiltonian in a 2 × 2 matrix form. By tuning the resonance frequencies, dissipation loss rates and radiation loss rates of the two resonators, as well as their coupling strength, PT symmetric phase transition effect can be observed in the polarization space at a certain relation of these parameters [2932]. M. Lawrence et al. observed such a PT symmetric phase transition in coupled split-ring resonators (SRRs) with anisotropic absorption by passively varying their coupling distance [29]. At the EP, the two eigen polarization states merged into a single circular polarized state, even though the structure lacks of threefold or larger rotational symmetry. Later, D. Wang et al. adopted a similar concept but a different controlling strategy, where they used type II superconductor niobium nitride to compose one SRR and tuned its dissipation loss rate to meet the EP condition by temperature [30]. Instead of directly manipulating the conductivity of the SRR, T. Cao et al. tune the dissipation loss rate of one SRR by putting it on a phase transition layer GeTe whose conductivity can be controlled by electrical heating [31]. In these works, the resonance frequencies and the radiation loss rates of the two SRRs are designed to be the same in order to guarantee a dispersionless PT symmetry. Very recently, S. H. Park et al. investigated a more general non-Hermitian metasurface system consisted of two SRRs with different scattering rates and radiation efficiencies [32]. Such system is only in PT symmetry at a certain frequency. The EP is observed in the parameter space formed by frequency and coupling strength, as well as by frequency and incident angle, at which an abrupt phase jump and magnitude decrease emerged in one cross-polarized transmission spectra in circular polarization basis.

To observe such EPs in polarization space, early studies mainly used resonators composed by different metals for providing the asymmetric loss [29,31,33], which are difficult to be fabricated and hard to achieve the EP through controlling their coupling distance. Integrating functional materials to control the asymmetric loss is one good way to overcome these difficulties [30,31]. Graphene, as a single layer of carbon atoms in a two-dimensional hexagonal lattice [34], possesses many superior properties [35,36], which make graphene a promising candidate for various applications, such as high-frequency transistors [37], label-free sensing [38], optical spatial computing [39], and terahertz generation [40,41], etc. One particular and important property of graphene is the fact that the Fermi level EF can be controlled by external stimuli, such as electric bias, optical pump, and chemical doping [4244], making graphene very promising in composing active electromagnetic modulators [42,45]. Such controllable feature is also very suitable to be applied to tune the asymmetric loss in such PT metasurface platform, which is so far seldom investigated.

In this article, a metal-graphene hybrid non-Hermitian metasurface is theoretically proposed in the terahertz regime. The metasurface is composed of two orthogonally arranged SRRs with the same dimensions, where one SRR contains a graphene patch at the gap. Thus, the dissipation loss rate of this graphene-integrated SRR can be actively tuned by changing the conductivity of the graphene, namely, tunable asymmetric loss compared to the other SRR. An EP can be observed in the parameter space formed by frequency and this dissipation loss rate. Such controlled parameter through graphene provides a convenient path to explore the EP. A coupled-mode theory is applied to describe the Hamiltonian and analyze the evolution of the eigenvalues and eigenstates in the polarization space, which agrees well with the simulation. Besides, the corresponding transmission spectra in circular polarization basis is also investigated, a giant phase change is observed around the EP, which can be utilized in reconfigurable sensing applications.

2. Theoretical model

SRRs are widely applied in metamaterial design which support strong LC resonances. The resonance frequency and the loss rate of the LC mode can be tailored by the geometry and the constituted material. It has been shown that when the gap is covered by conductive materials, the dissipation loss rate increases, resulting in reduced resonance strength [45]. As shown in Fig. 1(a), two identical but orthogonally oriented SRRs made from 200 nm-thickness aluminum are placed near to each other in a unit cell on a sapphire substrate, and only the gap of the left SRR is covered by a square graphene patch, whose conductivity could adjust the dissipation loss rate of this SRR. The left and right SRRs can be excited by x-polarized and y-polarized waves, respectively. Here, the two SRRs can interact with each other through magneto-coupling effect [46,47]. Coupled-mode theory is applied to study the transmission response of the metasurface. Figure 1(b) shows the corresponding schematic of this system, which has two ports and two coupled orthogonally resonated modes ax and ay. Here, only port 1 is exited.

 figure: Fig. 1.

Fig. 1. (a) Schematic of the unit cell of the metasurface composed by two orthogonally arranged SRRs with one SRR containing a square graphene patch on the gap. The conductivity of the graphene can be tuned by optical pump and chemical doping by surface adsorbates (e.g. reactive gas, represented by purple dots). The geometric parameters are P = 150 µm, L = 40 µm, w = 5 µm, h = 10 µm, u = 5 µm and g = 5 µm, respectively. (b) A schematic representation of a system composed of two coupled resonators and two ports. Here, the system is only excited at port 1.

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According to the coupled-mode theory with time dependence exp(jωt) [4850], the dynamic equations for the two resonances and the output at port 2 can be expressed as:

$$\frac{{d{a_x}}}{{dt}} = (j{\omega _x} - {\gamma _x} - {\Gamma _x}){a_x} + j\kappa {a_y} + j\sqrt {{\gamma _x}} {E_x}^{in},$$
$$\frac{{d{a_y}}}{{dt}} = (j{\omega _y} - {\gamma _y} - {\Gamma _y}){a_y} + j\kappa {a_x} + j\sqrt {{\gamma _y}} {E_y}^{in},$$
$$\left( {\begin{array}{c} {{E_x}^{out}}\\ {{E_y}^{out}} \end{array}} \right) = \left( {\begin{array}{c} {{E_x}^{in}}\\ {{E_y}^{in}} \end{array}} \right) + \left( {\begin{array}{cc} {j\sqrt {{\gamma_x}} }&0\\ 0&{j\sqrt {{\gamma_y}} } \end{array}} \right)\left( {\begin{array}{c} {{a_x}}\\ {{a_y}} \end{array}} \right) = T\left( {\begin{array}{c} {{E_x}^{in}}\\ {{E_y}^{in}} \end{array}} \right),$$
respectively, where ωx(ωy), γx(γy), Γxy) are the resonance frequency, radiation loss rate, dissipative loss rate of the resonance ax(ay), respectively; κ denotes the coupling between the two resonances; Exout(Eyout) and Exin(Eyin) are the transmitted and incident x(y)-polarized electric fields, respectively; T is the transmission matrix. Here, instead of denoting the resonances [ax ay]T as the state of the system like in the previous studies [51], the scattering field ${\left[ {\begin{array}{cc} {j\sqrt {{\gamma_x}} {a_x}}&{j\sqrt {{\gamma_y}} {a_y}} \end{array}} \right]^\textrm{T}}$ is used due to the close analogy between the Schrödinger equation  =  and the scattering equation  = λψ. Equations (1) and (2) can be rewritten as
$$\left( {\begin{array}{cc} {{\gamma_y}({\omega_x} - \omega ) + j{\gamma_y}({\gamma_x} + {\Gamma _x})}&{\kappa \sqrt {{\gamma_x}} \sqrt {{\gamma_y}} }\\ {\kappa \sqrt {{\gamma_x}} \sqrt {{\gamma_y}} }&{{\gamma_x}({\omega_y} - \omega ) + j{\gamma_x}({\gamma_y} + {\Gamma _y})} \end{array}} \right)\left( {\begin{array}{c} {j\sqrt {{\gamma_x}} {a_x}}\\ {j\sqrt {{\gamma_y}} {a_y}} \end{array}} \right) ={-} j{\gamma _x}{\gamma _y}\left( {\begin{array}{c} {{E_x}^{in}}\\ {{E_y}^{in}} \end{array}} \right).$$
The matrix in the left side of Eq. (4) can be denoted as the Hamiltonian matrix H of the system, which can be further expressed as
$$H = \left( {\begin{array}{cc} {{\gamma_y}({\omega_x} - \omega ) + j{\gamma_y}({\gamma_x} + {\Gamma _x})}&{\kappa \sqrt {{\gamma_x}} \sqrt {{\gamma_y}} }\\ {\kappa \sqrt {{\gamma_x}} \sqrt {{\gamma_y}} }&{{\gamma_x}({\omega_y} - \omega ) + j{\gamma_x}({\gamma_y} + {\Gamma _y})} \end{array}} \right) = j\chi I + {H_0},$$
where I is the identity matrix, and
$$\chi = {\gamma _y}{\gamma _x} + ({\gamma _y}{\Gamma _x} + {\gamma _x}{\Gamma _y})/2,$$
$${H_0} = \left( {\begin{array}{cc} {{\gamma_y}({\omega_x} - \omega ) + j({\gamma_y}{\Gamma _x} - {\gamma_x}{\Gamma _y})/2}&{\kappa \sqrt {{\gamma_x}} \sqrt {{\gamma_y}} }\\ {\kappa \sqrt {{\gamma_x}} \sqrt {{\gamma_y}} }&{{\gamma_x}({\omega_y} - \omega ) + j({\gamma_x}{\Gamma _y} - {\gamma_y}{\Gamma _x})/2} \end{array}} \right).$$
Here, the frequency ω is just a parameter of H, but not the eigenvalues of the Hamiltonian matrix in the usual sense. The eigenvalues of H in our case are related to the eigen transmissions of the proposed metasurface system.

As mentioned above, a system should commute with the PT operator to be PT symmetric. Note that the parity P operator corresponds to the Pauli operator $\left( {\begin{array}{cc} \textrm{0}&1\\ 1&0 \end{array}} \right)$, and the time-reversal T operator corresponds to complex conjugation. By making H0PT = PTH0 [9], it can be found that H0 is only PT symmetric when its two diagonal elements are complex conjugate to each other, which requires

$${\gamma _x}{\delta _y} = {\gamma _y}{\delta _x},$$
where δx = ωωx and δy = ωωy are the detuning frequencies of the two resonance modes, respectively. From the Eqs. (3)–(5), the transmission matrix T can be calculated as
$$T = I - j{\gamma _x}{\gamma _y}{H^{ - 1}}\textrm{ = }I - \frac{{j{\gamma _x}{\gamma _y}}}{{\det (H)}}({\det ({H_0}){H_0}^{ - 1} + j\chi I} ),$$
The eigenstates of T are the same with those of the matrix H and H0. Thus, we can study the eigenvalue problems of transmission matrix T to reveal the EP and PT symmetry of the system. The eigenvalues and eigenstates of T are calculated as
$${\lambda _T} = 1 - A({\lambda _{{H_0}}} + j\chi ),$$
$$v = \left( {1,\left\{ {[{ - {\lambda_{{H_\textrm{0}}}} - {\gamma_x}{\delta_y} - j({\gamma_y}{\Gamma _x} - {\gamma_x}{\Gamma _y})/2} ]/\left( {\sqrt {{\gamma_x}{\gamma_y}} \kappa } \right)} \right\}} \right),$$
respectively, where
$$A = j{\gamma _x}{\gamma _y}\textrm{/}\det (H) = j{[{({{\delta_x} - j({\gamma_x} + {\Gamma _x})} )({{\delta_y} - j({\gamma_y} + {\Gamma _y})} )- {\kappa^2}} ]^{ - 1}},$$
$${\lambda _{{H_\textrm{0}}}} = \left( { - {\gamma_x}{\delta_y} - {\gamma_y}{\delta_x} \pm \sqrt {{{[{{\gamma_x}{\delta_y} - {\gamma_y}{\delta_x} + j({{\gamma_y}{\Gamma _x} - {\gamma_x}{\Gamma _y}} )} ]}^2} + 4{\gamma_x}{\gamma_y}{\kappa^2}} } \right)/2.$$
Equation (8) has to be fulfilled to make sure the system is PT symmetric and the EP can be observed. One possible solution of Eq. (8) is that ωx = ωy and γx = γy, where H0 is PT symmetric for all the frequencies [2931,33]. In this case, if the radiation loss rates and the coupling fulfill the relation of |Γx – Γy| = 2|κ|, according to Eqs. (10), (11), and (13), eigenvalues and eigenvectors of H0 coalesce, results in the emergence of the EP. The second solution of Eq. (8) is that ωx = ωy and γxγy [32]. Then, the system is only PT symmetric at a single frequency ω = ωx = ωy. However, the condition of exactly ωx = ωy is quite difficult to realize, especially when the two SRRs are made of different materials. Here, we focus on a more general case, i.e. ωxωy and γxγy, where H0 is found to be PT symmetric only at a specific frequency of
$${\omega _0} = ({\gamma _x}{\omega _y} - {\gamma _y}{\omega _x})/({\gamma _x} - {\gamma _y}).$$
At ω0, according to Eq. (13), the condition of PT symmetric phase transition can be analyzed. When $|{{\gamma_y}{\Gamma _x} - {\gamma_x}{\Gamma _y}} |\;{ < }\;\textrm{2}\sqrt {{\gamma _x}{\gamma _y}} |\kappa |$, the system is in unbroken PT symmetric state. The eigen polarization states are given by $({\textrm{1,} \pm \exp ({\pm} j\theta )} )$, where $\theta = {\sin ^{ - 1}}[({\gamma _x}{\Gamma _y} - {\gamma _y}{\Gamma _x})/(\textrm{2}\sqrt {{\gamma _x}{\gamma _y}} \kappa )]$. The major axes of the eigen polarization states orient along ±45°. When $|{{\gamma_y}{\Gamma _x} - {\gamma_x}{\Gamma _y}} |{ > 2}\sqrt {{\gamma _x}{\gamma _y}} |\kappa |$, the system is in broken PT symmetric state. The eigen polarization states are given by $\left( {\textrm{1, sgn[(}{\gamma_x}{\Gamma _y} - {\gamma_y}{\Gamma _x}\textrm{)/(2}\sqrt {{\gamma_x}{\gamma_y}} \kappa )]j\textrm{exp(} \pm \theta \textrm{)}} \right)$, where sgn is the sign function and $\theta = {\cosh ^{ - 1}}$$(\left|{({\gamma_x}{\Gamma _y} - {\gamma_y}{\Gamma _x})/(\textrm{2}\sqrt {{\gamma_x}{\gamma_y}} \kappa )} \right|)$. In this case, the major axes of the eigen polarization states orient along 0° and 90°. Only when $|{{\gamma_y}{\Gamma _x} - {\gamma_x}{\Gamma _y}} |\textrm{ = 2}\sqrt {{\gamma _x}{\gamma _y}} |\kappa |$, the eigenvalues of T, H, and H0 degenerate to one value, and the corresponding eigen polarization states degenerate to one certain circular polarization state given by $\left( {1,j{\mathop{\rm sgn}} [({\gamma_x}{\Gamma _y} - {\gamma_y}{\Gamma _x})/(\textrm{2}\sqrt {{\gamma_x}{\gamma_y}} \kappa )]} \right)$. Therefore, the EP can only be found at ω0 under the condition of
$$|{{\gamma_y}{\Gamma _x} - {\gamma_x}{\Gamma _y}} |= \textrm{2}\sqrt {{\gamma _x}{\gamma _y}} |\kappa |,$$
where a sudden 45° rotation of the major axes occurs when the EP is passed.

3. Simulation and discussion

We first study the linear transmissions of the proposed metasurface at various EF of the graphene using CST Microwave Studio, as shown in Fig. 2(a). Here, tij represents the i-polarized transmission under j-polarized incidence with {i, j}${\in} ${x, y}. In the simulation, the graphene patch was modeled as a 2D material and described by Kubo model. The relaxation time was fixed at 0.03 ps while EF was varied from 0 eV to 0.2 eV. Correspondingly, the experimentally feasible DC surface conductivity was varied from 0.15 mS to 0.71 mS [45,52]. The dielectric constant of sapphire substrate was 9.67, and the conductivity of aluminum was σAl = 3.56 × 107 S/m which was described by surface impedance of Z ≈ (µ0ω/σAl)1/2exp(/4) with µ0 and ω being the vacuum permeability and angular frequency, respectively. Finite-element frequency-domain method was used to obtain the transmission spectra, where unit-cell boundary conditions were applied along the x and y directions while open boundary condition was applied along the z direction. The whole structure was excited and detected by Floquet TM00 and TE00 modes, corresponding to the x- and y-polarized waves, respectively. The mesh was set as tetrahedral mesh which was adaptively refined at 0.7 THz during the simulations. The convergence criteria of all s-parameters were set to be 0.0005. With the above settings, full s-parameters of the metasurface were obtained. As for the s-parameters of the reference, they were obtained by replacing the materials of the resonators and the graphene with vacuum. It can be seen from Fig. 2(a) that the amplitudes of the resonances in txx and txy (= tyx) gradually decrease while the amplitude of the resonance in tyy increases due to the increased dissipation loss rate of the left SRR as EF increases. The two resonance dips in txx and the cross-polarized output txy (tyx) as well as the converse changing manner of the resonance amplitude in tyy all indicate there is strong coupling effect between the two SRRs. Figure 2(b) illustrates the fitted transmission spectra using Eq. (9), which agree well with the simulation results in Fig. 2(a). The corresponding fitting parameters are shown in Fig. 3. Since the graphene only covers the gap of the left SRR, the conductivity change of graphene only affects the resonance of the left SRR. The dissipation loss rate Γx of the left SRR increases as EF increases owing to the shorting effect at the capacitance gap. The resonance frequency fx first slightly increases and then increases quickly. This behavior can be explained using perturbation theory [5355], where the complex resonance shift Δ$ \mathop{\omega}\limits^\sim $ of a metamaterial resonator with graphene in the gap can be calculated by Δ$ \mathop{\omega}\limits^\sim $ = −/W0 × ∫S|Exy|2dS with σ = Re[σ] + jIm[σ] being the complex conductivity of graphene, W0 being the stored electromagnetic energy in the uncovered resonator, S being the area, and Exy being the in-plane electric field, respectively. Since the real and imaginary parts of the conductivity of graphene studied here are both positive and increase as EF increases, the resonance frequency increases accordingly. The radiative loss rates γx and γy is also found to be different from each other. All of these fitting parameters indicate the proposed system fulfills the above-mentioned conditions of ωxωy, γxγy. Since all the geometric parameters of the structure are fixed, κ, γx, γy, fy and Γy are nearly constant during the whole process. Figure 3 also plots the parameter $({\gamma _\textrm{x}}{\Gamma _y} - {\gamma _y}{\Gamma _x})/2\sqrt {{\gamma _\textrm{x}}{\gamma _y}} $ as a function of EF, which crosses with κ around EF = 0.1322 eV, indicating the location area of the EP according to Eq. (15). From Fig. 2, it can be seen that the spectra of txx is slightly asymmetric, and the resonance dip of tyy is also not at the center of the two dips of txx. This is just owing to the small frequency detuning between ωx and ωy, which can be checked by observing the transmission spectra with respect to the corresponding fitted values of fx and fy in Fig. 3. Such phenomena gradually decrease as the frequency detuning decreases, which is consistent with the previous report [51].

 figure: Fig. 2.

Fig. 2. (a) Simulated and (b) fitted transmission spectra of the metasurface at different EF.

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 figure: Fig. 3.

Fig. 3. Fitted parameters as functions of EF.

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To find the EP and study the PT symmetric phase transition effect, the eigen transmission amplitude and phase spectra, namely, the eigenvalue spectra λT, under different EF are calculated from the simulated transmission matrix T, as shown in Figs. 4(a)–(e) and Figs. 4(f)–(j), respectively. The corresponding eigen polarization states at the special frequency fEP = 0.4977 THz according to Eq. (14) are shown in Figs. 4(k)–(o). The eigen transmission spectra of eigen polarization state 1 and eigen polarization state 2 show two resonance dips at different frequencies. As EF increases, the two resonance frequencies gradually shift to be overlap. Notice that the eigen polarization states are different at different frequencies in our case, as shown in Fig. 5(a). The EP is almost reached at 0.4977 THz when EF = 0.1322 eV since both the amplitudes and phases of the two eigenvalues are almost identical (the transmission amplitude and phase spectra of the two eigenvalues cross simultaneously) and the two eigenstates nearly degenerate to the right circular polarization as shown in Figs. 4(c), (h) and (m). The electric field of this circular polarization rotates clockwise, i.e. from + y to + x. The orientations of the major axes of the eigen polarization states at 0.4977 THz exhibit a sudden change from ±45° to 0° and 90° when the EP is crossed, as shown in Figs. 4(k)–(o).

 figure: Fig. 4.

Fig. 4. The eigen transmission (a-e) amplitude and (f-j) phase spectra at different EF obtained from simulations. The dashed lines indicate the specific frequency of 0.4977 THz, where the EP can occur. (k-o) The corresponding eigen polarization states at 0.4977 THz at different EF.

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 figure: Fig. 5.

Fig. 5. Eigen polarization states at seven selected frequencies around the EP frequency fEP obtained from (a) simulations and (b) theoretical calculations plotted on the Poincaré spheres in viewed from the north pole. The arrows indicate the changing directions of the eigen polarization states with increasing EF (dissipative loss rate Γx in calculation). Theoretical calculated surfaces of (c) the amplitude and (d) phase of the eigenvalues λT in (ω, Γx) parameter space. The yellow points indicate the EP with fEP = 0.4977 THz and Γx = 0.0475 THz.

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Figure 5(a) shows the simulated eigen polarization state evolving traces with frequencies near fEP on Poincaré sphere when EF increases from 0 to 0.2 eV. The arrow in each quadrant indicates the evolving trends with increasing EF. The scatters correspond to the five cases in Fig. 4. It can be seen that only the traces at fEP can almost reach the pole of the Poincaré sphere, which agrees with the previous analysis that H0 is only PT symmetric at a specific frequency ω0 where the EP can emerge. Meanwhile, the two traces lie along the 0°, 90°, 180° and 270° longitudes of the Poincaré sphere, which are twice of the corresponding orientation of the eigen polarization states, indicating the orientations of the major axes of the polarization states get an abrupt change of 45°. At the other frequencies, the traces also show 90° turns on the Poincaré sphere as those at fEP, but they do not follow longitude lines. For frequencies higher than fEP, the eigenstates evolve in the second and fourth quadrants. However, for frequencies lower than fEP, the eigenstates evolve in the first and third quadrants. This indicates a sudden cross happens when the frequency crosses the EP. Figure 5(b) shows the calculated eigenstates based on the above theoretical model. Since the dynamics of the system is mainly driven by the dissipative loss rate Γx controlled by the Fermi level of graphene, to simplify the calculation and meanwhile guarantee the effectiveness of the model, the parameters γy, Γy, κ, γx, ωx, ωy are fixed to those fitted at EF = 0.1322 eV in Fig. 3, while Γx is varied from 0.0171 THz to 0.0653 THz. The calculated results reproduce the features of the simulation results very well. Figures 5(c) and (d) show the corresponding calculated surfaces of the amplitude and phase of the complex eigenvalues λT in (ω, Γx) parameter space. The yellow dots represent the EP, where fEP = 0.4977 THz and Γx = 0.0475 THz. The splitting, intersecting, and fitting together of the two surfaces are result from the square-root parameter dependence according to Eq. (13).

4. Sensing applications

In our design, the two eigen polarization states coalesce to the right-handed circular polarization at fEP = 0.4977 THz and EF_EP = 0.1322 eV. This indicates that the transmission of the left-handed circularly polarized wave under the right-handed circularly polarized incidence should be zero. The simulated transmission spectra of the metasurface in the circular polarization basis are shown in Fig. 6(a). The subscripts r and l represent right-handed and left-handed circular polarizations, respectively. It is seen that the cross-polarized transmissions tlr at EF = 0.134 eV and EF = 0.132 eV are nearly the same, where a dip occurs at 0.4977 THz with almost zero transmission, indicating the EP condition is almost satisfied (near the actual EP point from two sides). However, it is interesting to see that an abrupt phase change of π is observed at 0.4977 THz, as shown in Fig. 6(b), although the change in EF is quite slight (ΔEF = 2 meV). Such an abrupt phase change of π accompanying spectral singularities has previously been observed at the EP in PT symmetric Fano coupled disk resonators [56,57]. It is also found from Fig. 6(b) that the phase spectra at EF < EF_EP cross at one point while those at EF > EF_EP cross at another point around 0.5 THz. This feature can be explained by investigating the theoretical expression of

$${t_{lr}} = [A({\gamma _x}{\delta _y} - {\gamma _y}{\delta _x} + j( - {\gamma _x}{\Gamma _y} + {\gamma _y}{\Gamma _x} + 2\sqrt {{\gamma _x}{\gamma _y}} \kappa ))]/2,$$
which can be calculated from the transmission matrix in Eq. (9). At the EP frequency, γxδy = γyδx and tlr gets a π phase jump when the value of $- {\gamma _x}{\Gamma _y} + {\gamma _y}{\Gamma _x} + 2\sqrt {{\gamma _x}{\gamma _y}} \kappa $ changes from negative to positive as Γx increases. The transition point tlr = 0 is exactly the EP since Eq. (8) and Eq. (15) are both fulfilled.

 figure: Fig. 6.

Fig. 6. Transmission (a) amplitude and (b) phase spectra of the metasurface in circular polarization basis at different EF. (c) Extracted phase values as a function of EF at three frequencies indicated by the dash line in (b).

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Taking advantage of such an abrupt π phase jump, the metasurface could function as an ultra-sensitive sensor for surface adsorbates (e.g. reactive gas, molecular, bacterial and chemical species), since they can induce a minor change in EF through doping effect [44,58]. Such sensing manner by an abrupt phase change is different from previous studies using graphene-based PT systems where the sensing is based on the intensity and resonance frequency changes of the scattering fields [59]. Besides monitoring the phase, one can also check the abrupt polarization rotation to accomplish the sensing according to Figs. 4 and 5. Figure 6(c) shows the phase values of tlr at three frequencies indicated by the dash lines in Fig. 6(b) as a function of EF. The phase changes slowly when EF is far below and above 0.132 eV, while changes rapidly around 0.132 eV. There is a trade-off between the sensitivity and the sensing range. At the EP frequency of 0.497 THz, the phase change slope is quite sharp around 0.132 eV within 0.01 eV. However, at 0.495 THz and 0.490 THz, the phase changes more slowly when crossing 0.132 eV. Thus, the EF of graphene should be set near the EF_EP in advance to get such high sensitivity. In reality, this can be challenging since the fabricated metasurface may deviate from the design owing to the mismatched dispersion relations of the real metal and graphene as well as the fabrication error. To overcome these shortcomings, one can separately fabricated several metasurfaces with different coupling distance u (corresponding to different values of κ). For example, in our design, EF_EP = 0.229 eV at u = 1 µm, and EF_EP = 0.071 eV at u = 12 µm, indicating a large tuning range of EF_EP by u. Therefore, one can always find the metasurface with certain u whose EF_EP is in the tuning range under the optical pump. Alternatively, one can process the graphene by proper plasma treatment to tune its EF around the EF_EP [60]. One particular advantage of such design is the reconfigurable sensing ability. Once the EP is passed due to the surface adsorbates after one sensing process, the metasurface sensor can be revived by setting a new pump intensity to drive the state of the metasurface back to the initial sensing state, as schematically illustrated in Fig. 1(a). Therefore, the metasurface is reusable with high sensitivity during the sensing process.

5. Conclusion

A general metal-graphene hybrid metasurface exhibiting the EP in polarization space is theoretically proposed and investigated by simulation and coupled-mode theory. The graphene here is for inducing a loss difference between the two SRRs. The balance between the loss difference and the coupling between the two SRRs leads to the EP. Before and after crossing the EP, an abrupt phase change can be monitored in tlr, making the metasurface a promising method for sensing applications.

Funding

National Key Research and Development Program of China (2017YFA0701004); National Natural Science Foundation of China (11974259, 61735012, 61875150, 61935015); Tianjin Municipal Fund for Distinguished Young Scholars (18JCJQJC45600).

Disclosures

The authors declare no conflicts of interest.

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5. Q. Zhong and R. El-Ganainy, “Crossing exceptional points without phase transition,” Sci. Rep. 9(1), 134 (2019). [CrossRef]  

6. V. V. Konotop, J. Yang, and D. A. Zezyulin, “Nonlinear waves in PT-symmetric systems,” Rev. Mod. Phys. 88(3), 035002 (2016). [CrossRef]  

7. S. Droulias, I. Katsantonis, M. Kafesaki, C. M. Soukoulis, and E. N. Economou, “Chiral metamaterials with PT symmetry and beyond,” Phys. Rev. Lett. 122(21), 213201 (2019). [CrossRef]  

8. H. Zhao and L. Feng, “Parity–time symmetric photonics,” Natl. Sci. Rev. 5(2), 183–199 (2018). [CrossRef]  

9. K. Özdemir, S. Rotter, F. Nori, and L. Yang, “Parity–time symmetry and exceptional points in photonics,” Nat. Mater. 18(8), 783–798 (2019). [CrossRef]  

10. B. Qi, H.-Z. Chen, L. Ge, P. Berini, and R.-M. Ma, “Parity–time symmetry synthetic lasers: Physics and devices,” Adv. Opt. Mater. 7(22), 1900694 (2019). [CrossRef]  

11. X. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, “PT-symmetric acoustics,” Phys. Rev. X 4(3), 031042 (2014). [CrossRef]  

12. R. Fleury, D. Sounas, and A. Alù, “An invisible acoustic sensor based on parity-time symmetry,” Nat. Commun. 6(1), 5905 (2015). [CrossRef]  

13. C. Shi, M. Dubois, Y. Chen, L. Cheng, H. Ramezani, Y. Wang, and X. Zhang, “Accessing the exceptional points of parity-time symmetric acoustics,” Nat. Commun. 7(1), 11110 (2016). [CrossRef]  

14. J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with PT symmetries,” Phys. Rev. A 84(4), 040101 (2011). [CrossRef]  

15. Z. Dong, Z. Li, F. Yang, C.-W. Qiu, and J. S. Ho, “Sensitive readout of implantable microsensors using a wireless system locked to an exceptional point,” Nat. Electron. 2(8), 335–342 (2019). [CrossRef]  

16. P.-Y. Chen, M. Sakhdari, M. Hajizadegan, Q. Cui, M. M.-C. Cheng, R. El-Ganainy, and A. Alù, “Generalized parity–time symmetry condition for enhanced sensor telemetry,” Nat. Electron. 1(5), 297–304 (2018). [CrossRef]  

17. J. Li, A. K. Harter, J. Liu, L. de Melo, Y. N. Joglekar, and L. Luo, “Observation of parity-time symmetry breaking transitions in a dissipative floquet system of ultracold atoms,” Nat. Commun. 10(1), 855 (2019). [CrossRef]  

18. M.-A. Miri and A. Alù, “Exceptional points in optics and photonics,” Science 363(6422), eaar7709 (2019). [CrossRef]  

19. X.-L. Zhang, T. Jiang, and C. T. Chan, “Dynamically encircling an exceptional point in anti-parity-time symmetric systems: Asymmetric mode switching for symmetry-broken modes,” Light: Sci. Appl. 8(1), 88 (2019). [CrossRef]  

20. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009). [CrossRef]  

21. A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012). [CrossRef]  

22. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011). [CrossRef]  

23. H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82(4), 043803 (2010). [CrossRef]  

24. N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110(23), 234101 (2013). [CrossRef]  

25. F. Nazari, N. Bender, H. Ramezani, M. K. Moravvej-Farshi, D. N. Christodoulides, and T. Kottos, “Optical isolation via PT -symmetric nonlinear fano resonances,” Opt. Express 22(8), 9574–9584 (2014). [CrossRef]  

26. L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity–time symmetry and variable optical isolation in active–passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014). [CrossRef]  

27. M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5(1), 4034 (2014). [CrossRef]  

28. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008). [CrossRef]  

29. M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of PT symmetry breaking in polarization space with terahertz metasurfaces,” Phys. Rev. Lett. 113(9), 093901 (2014). [CrossRef]  

30. D. Wang, C. Li, C. Zhang, M. Kang, X. Zhang, B. Jin, Z. Tian, Y. Li, S. Zhang, J. Han, and W. Zhang, “Superconductive PT-symmetry phase transition in metasurfaces,” Appl. Phys. Lett. 110(2), 021104 (2017). [CrossRef]  

31. T. Cao, Y. Cao, and L. Fang, “Reconfigurable parity-time symmetry transition in phase change metamaterials,” Nanoscale 11(34), 15828–15835 (2019). [CrossRef]  

32. P. Sang Hyun, L. Sung-Gyu, B. Soojeong, H. Taewoo, L. Sanghyub, M. Bumki, Z. Shuang, L. Mark, and K. Teun-Teun, “Observation of an exceptional point in a non-Hermitian metasurface,” Nanophotonics20190489 (2020). [CrossRef]  

33. B. Jin, W. Tan, C. Zhang, J. Wu, J. Chen, S. Zhang, and P. Wu, “High-performance terahertz sensing at exceptional points in a bilayer structure,” Adv. Theory Simul. 1(9), 1800070 (2018). [CrossRef]  

34. D. Li and R. B. Kaner, “Graphene-based materials,” Science 320(5880), 1170–1171 (2008). [CrossRef]  

35. A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007). [CrossRef]  

36. P. Avouris and F. Xia, “Graphene applications in electronics and photonics,” MRS Bull. 37(12), 1225–1234 (2012). [CrossRef]  

37. F. Schwierz, “Graphene transistors,” Nat. Nanotechnol. 5(7), 487–496 (2010). [CrossRef]  

38. F. Zangeneh-Nejad and R. Safian, “A graphene-based THz ring resonator for label-free sensing,” IEEE Sens. J. 16(11), 4338–4344 (2016). [CrossRef]  

39. F. Z.-N. a and A. Khavasi, “Spatial integration by a dielectric slab and its planar graphene-based counterpart,” Opt. Lett. 42(10), 1954–1957 (2017). [CrossRef]  

40. Z.-N. Farzad and S. Reza, “Significant enhancement in the efficiency of photoconductive antennas using a hybrid graphene molybdenum disulphide structure,” J. Nanophotonics 10(3), 036005 (2016). [CrossRef]  

41. Z. W. Ullah, G. Nawi, I. Tansu, N. Irfan Khattak, and M. Junaid, “A review on the development of tunable graphene nanoantennas for terahertz optoelectronic and plasmonic applications,” Sensors 20(5), 1401 (2020). [CrossRef]  

42. T.-T. Kim, H.-D. Kim, R. Zhao, S. S. Oh, T. Ha, D. S. Chung, Y. H. Lee, B. Min, and S. Zhang, “Electrically tunable slow light using graphene metamaterials,” ACS Photonics 5(5), 1800–1807 (2018). [CrossRef]  

43. A. Tomadin, S. M. Hornett, H. I. Wang, E. M. Alexeev, A. Candini, C. Coletti, D. Turchinovich, M. Kläui, M. Bonn, F. H. L. Koppens, E. Hendry, M. Polini, and K.-J. Tielrooij, “The ultrafast dynamics and conductivity of photoexcited graphene at different Fermi energies,” Sci. Adv. 4(5), eaar5313 (2018). [CrossRef]  

44. Z. Zhang, T. Lin, X. Xing, X. Lin, X. Meng, Z. Cheng, Z. Jin, and G. Ma, “Photoexcited terahertz conductivity dynamics of graphene tuned by oxygen-adsorption,” Appl. Phys. Lett. 110(11), 111108 (2017). [CrossRef]  

45. S. J. Kindness, N. W. Almond, B. Wei, R. Wallis, W. Michailow, V. S. Kamboj, P. Braeuninger-Weimer, S. Hofmann, H. E. Beere, D. A. Ritchie, and R. Degl’Innocenti, “Active control of electromagnetically induced transparency in a terahertz metamaterial array with graphene for continuous resonance frequency tuning,” Adv. Opt. Mater. 6(21), 1800570 (2018). [CrossRef]  

46. N. Liu, S. Kaiser, and H. Giessen, “Magnetoinductive and electroinductive coupling in plasmonic metamaterial molecules,” Adv. Mater. 20(23), 4521–4525 (2008). [CrossRef]  

47. N. Liu and H. Giessen, “Coupling effects in optical metamaterials,” Angew. Chem., Int. Ed. 49(51), 9838–9852 (2010). [CrossRef]  

48. S. Wonjoo, W. Zheng, and F. Shanhui, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40(10), 1511–1518 (2004). [CrossRef]  

49. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the fano resonance in optical resonators,” J. Opt. Soc. Am. A 20(3), 569–572 (2003). [CrossRef]  

50. W. Tan, Y. Sun, Z.-G. Wang, and H. Chen, “Manipulating electromagnetic responses of metal wires at the deep subwavelength scale via both near- and far-field couplings,” Appl. Phys. Lett. 104(9), 091107 (2014). [CrossRef]  

51. K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6(2), 021007 (2016). [CrossRef]  

52. T.-T. Kim, S. S. Oh, H.-D. Kim, H. S. Park, O. Hess, B. Min, and S. Zhang, “Electrical access to critical coupling of circularly polarized waves in graphene chiral metamaterials,” Sci. Adv. 3(9), e1701377 (2017). [CrossRef]  

53. C. Kottke, A. Farjadpour, and S. G. Johnson, “Perturbation theory for anisotropic dielectric interfaces, and application to subpixel smoothing of discretized numerical methods,” Phys. Rev. E 77(3), 036611 (2008). [CrossRef]  

54. S. H. Mousavi, I. Kholmanov, K. B. Alici, D. Purtseladze, N. Arju, K. Tatar, D. Y. Fozdar, J. W. Suk, Y. Hao, A. B. Khanikaev, R. S. Ruoff, and G. Shvets, “Inductive tuning of fano-resonant metasurfaces using plasmonic response of graphene in the mid-infrared,” Nano Lett. 13(3), 1111–1117 (2013). [CrossRef]  

55. J. Chen, X. Li, X. Shi, C. Fan, M. Tuhtasun, X. He, W. Shi, and F. Liu, “Active control of light slowing enabled by coupling electromagnetic metamaterials with low-lossy graphene,” Opt. Lett. 43(20), 4891–4894 (2018). [CrossRef]  

56. H. Ramezani, H.-K. Li, Y. Wang, and X. Zhang, “Unidirectional spectral singularities,” Phys. Rev. Lett. 113(26), 263905 (2014). [CrossRef]  

57. H. Ramezani, Y. Wang, E. Yablonovitch, and X. Zhang, “Unidirectional perfect absorber,” IEEE J. Sel. Top. Quantum Electron. 22(5), 115–120 (2016). [CrossRef]  

58. L. Zhang, K. Khan, J. Zou, H. Zhang, and Y. Li, “Recent advances in emerging 2d material-based gas sensors: Potential in disease diagnosis,” Adv. Mater. Interfaces 6(22), 1901329 (2019). [CrossRef]  

59. P.-Y. Chen and J. Jung, “PT symmetry and singularity-enhanced sensing based on photoexcited graphene metasurfaces,” Phys. Rev. Appl. 5(6), 064018 (2016). [CrossRef]  

60. A. Dey, A. Chroneos, N. S. J. Braithwaite, R. P. Gandhiraman, and S. Krishnamurthy, “Plasma engineering of graphene,” Appl. Phys. Rev. 3(2), 021301 (2016). [CrossRef]  

References

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  7. S. Droulias, I. Katsantonis, M. Kafesaki, C. M. Soukoulis, and E. N. Economou, “Chiral metamaterials with PT symmetry and beyond,” Phys. Rev. Lett. 122(21), 213201 (2019).
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  8. H. Zhao and L. Feng, “Parity–time symmetric photonics,” Natl. Sci. Rev. 5(2), 183–199 (2018).
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  16. P.-Y. Chen, M. Sakhdari, M. Hajizadegan, Q. Cui, M. M.-C. Cheng, R. El-Ganainy, and A. Alù, “Generalized parity–time symmetry condition for enhanced sensor telemetry,” Nat. Electron. 1(5), 297–304 (2018).
    [Crossref]
  17. J. Li, A. K. Harter, J. Liu, L. de Melo, Y. N. Joglekar, and L. Luo, “Observation of parity-time symmetry breaking transitions in a dissipative floquet system of ultracold atoms,” Nat. Commun. 10(1), 855 (2019).
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    [Crossref]
  20. A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
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  21. A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
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  22. Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
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  23. H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82(4), 043803 (2010).
    [Crossref]
  24. N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110(23), 234101 (2013).
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    [Crossref]
  26. L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity–time symmetry and variable optical isolation in active–passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
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  27. M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5(1), 4034 (2014).
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  28. K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
    [Crossref]
  29. M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of PT symmetry breaking in polarization space with terahertz metasurfaces,” Phys. Rev. Lett. 113(9), 093901 (2014).
    [Crossref]
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    [Crossref]
  31. T. Cao, Y. Cao, and L. Fang, “Reconfigurable parity-time symmetry transition in phase change metamaterials,” Nanoscale 11(34), 15828–15835 (2019).
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  32. P. Sang Hyun, L. Sung-Gyu, B. Soojeong, H. Taewoo, L. Sanghyub, M. Bumki, Z. Shuang, L. Mark, and K. Teun-Teun, “Observation of an exceptional point in a non-Hermitian metasurface,” Nanophotonics20190489 (2020).
    [Crossref]
  33. B. Jin, W. Tan, C. Zhang, J. Wu, J. Chen, S. Zhang, and P. Wu, “High-performance terahertz sensing at exceptional points in a bilayer structure,” Adv. Theory Simul. 1(9), 1800070 (2018).
    [Crossref]
  34. D. Li and R. B. Kaner, “Graphene-based materials,” Science 320(5880), 1170–1171 (2008).
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  35. A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
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  36. P. Avouris and F. Xia, “Graphene applications in electronics and photonics,” MRS Bull. 37(12), 1225–1234 (2012).
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  37. F. Schwierz, “Graphene transistors,” Nat. Nanotechnol. 5(7), 487–496 (2010).
    [Crossref]
  38. F. Zangeneh-Nejad and R. Safian, “A graphene-based THz ring resonator for label-free sensing,” IEEE Sens. J. 16(11), 4338–4344 (2016).
    [Crossref]
  39. F. Z.-N. a and A. Khavasi, “Spatial integration by a dielectric slab and its planar graphene-based counterpart,” Opt. Lett. 42(10), 1954–1957 (2017).
    [Crossref]
  40. Z.-N. Farzad and S. Reza, “Significant enhancement in the efficiency of photoconductive antennas using a hybrid graphene molybdenum disulphide structure,” J. Nanophotonics 10(3), 036005 (2016).
    [Crossref]
  41. Z. W. Ullah, G. Nawi, I. Tansu, N. Irfan Khattak, and M. Junaid, “A review on the development of tunable graphene nanoantennas for terahertz optoelectronic and plasmonic applications,” Sensors 20(5), 1401 (2020).
    [Crossref]
  42. T.-T. Kim, H.-D. Kim, R. Zhao, S. S. Oh, T. Ha, D. S. Chung, Y. H. Lee, B. Min, and S. Zhang, “Electrically tunable slow light using graphene metamaterials,” ACS Photonics 5(5), 1800–1807 (2018).
    [Crossref]
  43. A. Tomadin, S. M. Hornett, H. I. Wang, E. M. Alexeev, A. Candini, C. Coletti, D. Turchinovich, M. Kläui, M. Bonn, F. H. L. Koppens, E. Hendry, M. Polini, and K.-J. Tielrooij, “The ultrafast dynamics and conductivity of photoexcited graphene at different Fermi energies,” Sci. Adv. 4(5), eaar5313 (2018).
    [Crossref]
  44. Z. Zhang, T. Lin, X. Xing, X. Lin, X. Meng, Z. Cheng, Z. Jin, and G. Ma, “Photoexcited terahertz conductivity dynamics of graphene tuned by oxygen-adsorption,” Appl. Phys. Lett. 110(11), 111108 (2017).
    [Crossref]
  45. S. J. Kindness, N. W. Almond, B. Wei, R. Wallis, W. Michailow, V. S. Kamboj, P. Braeuninger-Weimer, S. Hofmann, H. E. Beere, D. A. Ritchie, and R. Degl’Innocenti, “Active control of electromagnetically induced transparency in a terahertz metamaterial array with graphene for continuous resonance frequency tuning,” Adv. Opt. Mater. 6(21), 1800570 (2018).
    [Crossref]
  46. N. Liu, S. Kaiser, and H. Giessen, “Magnetoinductive and electroinductive coupling in plasmonic metamaterial molecules,” Adv. Mater. 20(23), 4521–4525 (2008).
    [Crossref]
  47. N. Liu and H. Giessen, “Coupling effects in optical metamaterials,” Angew. Chem., Int. Ed. 49(51), 9838–9852 (2010).
    [Crossref]
  48. S. Wonjoo, W. Zheng, and F. Shanhui, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40(10), 1511–1518 (2004).
    [Crossref]
  49. S. Fan, W. Suh, and J. D. Joannopoulos, “Temporal coupled-mode theory for the fano resonance in optical resonators,” J. Opt. Soc. Am. A 20(3), 569–572 (2003).
    [Crossref]
  50. W. Tan, Y. Sun, Z.-G. Wang, and H. Chen, “Manipulating electromagnetic responses of metal wires at the deep subwavelength scale via both near- and far-field couplings,” Appl. Phys. Lett. 104(9), 091107 (2014).
    [Crossref]
  51. K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6(2), 021007 (2016).
    [Crossref]
  52. T.-T. Kim, S. S. Oh, H.-D. Kim, H. S. Park, O. Hess, B. Min, and S. Zhang, “Electrical access to critical coupling of circularly polarized waves in graphene chiral metamaterials,” Sci. Adv. 3(9), e1701377 (2017).
    [Crossref]
  53. C. Kottke, A. Farjadpour, and S. G. Johnson, “Perturbation theory for anisotropic dielectric interfaces, and application to subpixel smoothing of discretized numerical methods,” Phys. Rev. E 77(3), 036611 (2008).
    [Crossref]
  54. S. H. Mousavi, I. Kholmanov, K. B. Alici, D. Purtseladze, N. Arju, K. Tatar, D. Y. Fozdar, J. W. Suk, Y. Hao, A. B. Khanikaev, R. S. Ruoff, and G. Shvets, “Inductive tuning of fano-resonant metasurfaces using plasmonic response of graphene in the mid-infrared,” Nano Lett. 13(3), 1111–1117 (2013).
    [Crossref]
  55. J. Chen, X. Li, X. Shi, C. Fan, M. Tuhtasun, X. He, W. Shi, and F. Liu, “Active control of light slowing enabled by coupling electromagnetic metamaterials with low-lossy graphene,” Opt. Lett. 43(20), 4891–4894 (2018).
    [Crossref]
  56. H. Ramezani, H.-K. Li, Y. Wang, and X. Zhang, “Unidirectional spectral singularities,” Phys. Rev. Lett. 113(26), 263905 (2014).
    [Crossref]
  57. H. Ramezani, Y. Wang, E. Yablonovitch, and X. Zhang, “Unidirectional perfect absorber,” IEEE J. Sel. Top. Quantum Electron. 22(5), 115–120 (2016).
    [Crossref]
  58. L. Zhang, K. Khan, J. Zou, H. Zhang, and Y. Li, “Recent advances in emerging 2d material-based gas sensors: Potential in disease diagnosis,” Adv. Mater. Interfaces 6(22), 1901329 (2019).
    [Crossref]
  59. P.-Y. Chen and J. Jung, “PT symmetry and singularity-enhanced sensing based on photoexcited graphene metasurfaces,” Phys. Rev. Appl. 5(6), 064018 (2016).
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2020 (1)

Z. W. Ullah, G. Nawi, I. Tansu, N. Irfan Khattak, and M. Junaid, “A review on the development of tunable graphene nanoantennas for terahertz optoelectronic and plasmonic applications,” Sensors 20(5), 1401 (2020).
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2019 (10)

L. Zhang, K. Khan, J. Zou, H. Zhang, and Y. Li, “Recent advances in emerging 2d material-based gas sensors: Potential in disease diagnosis,” Adv. Mater. Interfaces 6(22), 1901329 (2019).
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Q. Zhong and R. El-Ganainy, “Crossing exceptional points without phase transition,” Sci. Rep. 9(1), 134 (2019).
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S. Droulias, I. Katsantonis, M. Kafesaki, C. M. Soukoulis, and E. N. Economou, “Chiral metamaterials with PT symmetry and beyond,” Phys. Rev. Lett. 122(21), 213201 (2019).
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K. Özdemir, S. Rotter, F. Nori, and L. Yang, “Parity–time symmetry and exceptional points in photonics,” Nat. Mater. 18(8), 783–798 (2019).
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B. Qi, H.-Z. Chen, L. Ge, P. Berini, and R.-M. Ma, “Parity–time symmetry synthetic lasers: Physics and devices,” Adv. Opt. Mater. 7(22), 1900694 (2019).
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J. Li, A. K. Harter, J. Liu, L. de Melo, Y. N. Joglekar, and L. Luo, “Observation of parity-time symmetry breaking transitions in a dissipative floquet system of ultracold atoms,” Nat. Commun. 10(1), 855 (2019).
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M.-A. Miri and A. Alù, “Exceptional points in optics and photonics,” Science 363(6422), eaar7709 (2019).
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X.-L. Zhang, T. Jiang, and C. T. Chan, “Dynamically encircling an exceptional point in anti-parity-time symmetric systems: Asymmetric mode switching for symmetry-broken modes,” Light: Sci. Appl. 8(1), 88 (2019).
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Z. Dong, Z. Li, F. Yang, C.-W. Qiu, and J. S. Ho, “Sensitive readout of implantable microsensors using a wireless system locked to an exceptional point,” Nat. Electron. 2(8), 335–342 (2019).
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T. Cao, Y. Cao, and L. Fang, “Reconfigurable parity-time symmetry transition in phase change metamaterials,” Nanoscale 11(34), 15828–15835 (2019).
[Crossref]

2018 (7)

B. Jin, W. Tan, C. Zhang, J. Wu, J. Chen, S. Zhang, and P. Wu, “High-performance terahertz sensing at exceptional points in a bilayer structure,” Adv. Theory Simul. 1(9), 1800070 (2018).
[Crossref]

P.-Y. Chen, M. Sakhdari, M. Hajizadegan, Q. Cui, M. M.-C. Cheng, R. El-Ganainy, and A. Alù, “Generalized parity–time symmetry condition for enhanced sensor telemetry,” Nat. Electron. 1(5), 297–304 (2018).
[Crossref]

H. Zhao and L. Feng, “Parity–time symmetric photonics,” Natl. Sci. Rev. 5(2), 183–199 (2018).
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J. Chen, X. Li, X. Shi, C. Fan, M. Tuhtasun, X. He, W. Shi, and F. Liu, “Active control of light slowing enabled by coupling electromagnetic metamaterials with low-lossy graphene,” Opt. Lett. 43(20), 4891–4894 (2018).
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T.-T. Kim, H.-D. Kim, R. Zhao, S. S. Oh, T. Ha, D. S. Chung, Y. H. Lee, B. Min, and S. Zhang, “Electrically tunable slow light using graphene metamaterials,” ACS Photonics 5(5), 1800–1807 (2018).
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A. Tomadin, S. M. Hornett, H. I. Wang, E. M. Alexeev, A. Candini, C. Coletti, D. Turchinovich, M. Kläui, M. Bonn, F. H. L. Koppens, E. Hendry, M. Polini, and K.-J. Tielrooij, “The ultrafast dynamics and conductivity of photoexcited graphene at different Fermi energies,” Sci. Adv. 4(5), eaar5313 (2018).
[Crossref]

S. J. Kindness, N. W. Almond, B. Wei, R. Wallis, W. Michailow, V. S. Kamboj, P. Braeuninger-Weimer, S. Hofmann, H. E. Beere, D. A. Ritchie, and R. Degl’Innocenti, “Active control of electromagnetically induced transparency in a terahertz metamaterial array with graphene for continuous resonance frequency tuning,” Adv. Opt. Mater. 6(21), 1800570 (2018).
[Crossref]

2017 (4)

T.-T. Kim, S. S. Oh, H.-D. Kim, H. S. Park, O. Hess, B. Min, and S. Zhang, “Electrical access to critical coupling of circularly polarized waves in graphene chiral metamaterials,” Sci. Adv. 3(9), e1701377 (2017).
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Z. Zhang, T. Lin, X. Xing, X. Lin, X. Meng, Z. Cheng, Z. Jin, and G. Ma, “Photoexcited terahertz conductivity dynamics of graphene tuned by oxygen-adsorption,” Appl. Phys. Lett. 110(11), 111108 (2017).
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F. Z.-N. a and A. Khavasi, “Spatial integration by a dielectric slab and its planar graphene-based counterpart,” Opt. Lett. 42(10), 1954–1957 (2017).
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D. Wang, C. Li, C. Zhang, M. Kang, X. Zhang, B. Jin, Z. Tian, Y. Li, S. Zhang, J. Han, and W. Zhang, “Superconductive PT-symmetry phase transition in metasurfaces,” Appl. Phys. Lett. 110(2), 021104 (2017).
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2016 (8)

V. V. Konotop, J. Yang, and D. A. Zezyulin, “Nonlinear waves in PT-symmetric systems,” Rev. Mod. Phys. 88(3), 035002 (2016).
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C. Shi, M. Dubois, Y. Chen, L. Cheng, H. Ramezani, Y. Wang, and X. Zhang, “Accessing the exceptional points of parity-time symmetric acoustics,” Nat. Commun. 7(1), 11110 (2016).
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Z.-N. Farzad and S. Reza, “Significant enhancement in the efficiency of photoconductive antennas using a hybrid graphene molybdenum disulphide structure,” J. Nanophotonics 10(3), 036005 (2016).
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F. Zangeneh-Nejad and R. Safian, “A graphene-based THz ring resonator for label-free sensing,” IEEE Sens. J. 16(11), 4338–4344 (2016).
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K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6(2), 021007 (2016).
[Crossref]

P.-Y. Chen and J. Jung, “PT symmetry and singularity-enhanced sensing based on photoexcited graphene metasurfaces,” Phys. Rev. Appl. 5(6), 064018 (2016).
[Crossref]

A. Dey, A. Chroneos, N. S. J. Braithwaite, R. P. Gandhiraman, and S. Krishnamurthy, “Plasma engineering of graphene,” Appl. Phys. Rev. 3(2), 021301 (2016).
[Crossref]

H. Ramezani, Y. Wang, E. Yablonovitch, and X. Zhang, “Unidirectional perfect absorber,” IEEE J. Sel. Top. Quantum Electron. 22(5), 115–120 (2016).
[Crossref]

2015 (1)

R. Fleury, D. Sounas, and A. Alù, “An invisible acoustic sensor based on parity-time symmetry,” Nat. Commun. 6(1), 5905 (2015).
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2014 (7)

M. Lawrence, N. Xu, X. Zhang, L. Cong, J. Han, W. Zhang, and S. Zhang, “Manifestation of PT symmetry breaking in polarization space with terahertz metasurfaces,” Phys. Rev. Lett. 113(9), 093901 (2014).
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F. Nazari, N. Bender, H. Ramezani, M. K. Moravvej-Farshi, D. N. Christodoulides, and T. Kottos, “Optical isolation via PT -symmetric nonlinear fano resonances,” Opt. Express 22(8), 9574–9584 (2014).
[Crossref]

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity–time symmetry and variable optical isolation in active–passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
[Crossref]

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5(1), 4034 (2014).
[Crossref]

X. Zhu, H. Ramezani, C. Shi, J. Zhu, and X. Zhang, “PT-symmetric acoustics,” Phys. Rev. X 4(3), 031042 (2014).
[Crossref]

H. Ramezani, H.-K. Li, Y. Wang, and X. Zhang, “Unidirectional spectral singularities,” Phys. Rev. Lett. 113(26), 263905 (2014).
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W. Tan, Y. Sun, Z.-G. Wang, and H. Chen, “Manipulating electromagnetic responses of metal wires at the deep subwavelength scale via both near- and far-field couplings,” Appl. Phys. Lett. 104(9), 091107 (2014).
[Crossref]

2013 (2)

S. H. Mousavi, I. Kholmanov, K. B. Alici, D. Purtseladze, N. Arju, K. Tatar, D. Y. Fozdar, J. W. Suk, Y. Hao, A. B. Khanikaev, R. S. Ruoff, and G. Shvets, “Inductive tuning of fano-resonant metasurfaces using plasmonic response of graphene in the mid-infrared,” Nano Lett. 13(3), 1111–1117 (2013).
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N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110(23), 234101 (2013).
[Crossref]

2012 (2)

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[Crossref]

P. Avouris and F. Xia, “Graphene applications in electronics and photonics,” MRS Bull. 37(12), 1225–1234 (2012).
[Crossref]

2011 (2)

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
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J. Schindler, A. Li, M. C. Zheng, F. M. Ellis, and T. Kottos, “Experimental study of active LRC circuits with PT symmetries,” Phys. Rev. A 84(4), 040101 (2011).
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2010 (3)

H. Ramezani, T. Kottos, R. El-Ganainy, and D. N. Christodoulides, “Unidirectional nonlinear PT-symmetric optical structures,” Phys. Rev. A 82(4), 043803 (2010).
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F. Schwierz, “Graphene transistors,” Nat. Nanotechnol. 5(7), 487–496 (2010).
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N. Liu and H. Giessen, “Coupling effects in optical metamaterials,” Angew. Chem., Int. Ed. 49(51), 9838–9852 (2010).
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2009 (1)

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
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2008 (4)

K. G. Makris, R. El-Ganainy, D. N. Christodoulides, and Z. H. Musslimani, “Beam dynamics in PT symmetric optical lattices,” Phys. Rev. Lett. 100(10), 103904 (2008).
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D. Li and R. B. Kaner, “Graphene-based materials,” Science 320(5880), 1170–1171 (2008).
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N. Liu, S. Kaiser, and H. Giessen, “Magnetoinductive and electroinductive coupling in plasmonic metamaterial molecules,” Adv. Mater. 20(23), 4521–4525 (2008).
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C. Kottke, A. Farjadpour, and S. G. Johnson, “Perturbation theory for anisotropic dielectric interfaces, and application to subpixel smoothing of discretized numerical methods,” Phys. Rev. E 77(3), 036611 (2008).
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2007 (2)

A. K. Geim and K. S. Novoselov, “The rise of graphene,” Nat. Mater. 6(3), 183–191 (2007).
[Crossref]

C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70(6), 947–1018 (2007).
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2004 (1)

S. Wonjoo, W. Zheng, and F. Shanhui, “Temporal coupled-mode theory and the presence of non-orthogonal modes in lossless multimode cavities,” IEEE J. Quantum Electron. 40(10), 1511–1518 (2004).
[Crossref]

2003 (1)

2002 (1)

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
[Crossref]

2000 (1)

G. Lévai and M. Znojil, “Systematic search for PT-symmetric potentials with real energy spectra,” J. Phys. A: Math. Gen. 33(40), 7165–7180 (2000).
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1998 (1)

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
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a, F. Z.-N.

Aimez, V.

A. Guo, G. J. Salamo, D. Duchesne, R. Morandotti, M. Volatier-Ravat, V. Aimez, G. A. Siviloglou, and D. N. Christodoulides, “Observation of PT-symmetry breaking in complex optical potentials,” Phys. Rev. Lett. 103(9), 093902 (2009).
[Crossref]

Alexeev, E. M.

A. Tomadin, S. M. Hornett, H. I. Wang, E. M. Alexeev, A. Candini, C. Coletti, D. Turchinovich, M. Kläui, M. Bonn, F. H. L. Koppens, E. Hendry, M. Polini, and K.-J. Tielrooij, “The ultrafast dynamics and conductivity of photoexcited graphene at different Fermi energies,” Sci. Adv. 4(5), eaar5313 (2018).
[Crossref]

Alici, K. B.

S. H. Mousavi, I. Kholmanov, K. B. Alici, D. Purtseladze, N. Arju, K. Tatar, D. Y. Fozdar, J. W. Suk, Y. Hao, A. B. Khanikaev, R. S. Ruoff, and G. Shvets, “Inductive tuning of fano-resonant metasurfaces using plasmonic response of graphene in the mid-infrared,” Nano Lett. 13(3), 1111–1117 (2013).
[Crossref]

Almond, N. W.

S. J. Kindness, N. W. Almond, B. Wei, R. Wallis, W. Michailow, V. S. Kamboj, P. Braeuninger-Weimer, S. Hofmann, H. E. Beere, D. A. Ritchie, and R. Degl’Innocenti, “Active control of electromagnetically induced transparency in a terahertz metamaterial array with graphene for continuous resonance frequency tuning,” Adv. Opt. Mater. 6(21), 1800570 (2018).
[Crossref]

Alù, A.

M.-A. Miri and A. Alù, “Exceptional points in optics and photonics,” Science 363(6422), eaar7709 (2019).
[Crossref]

P.-Y. Chen, M. Sakhdari, M. Hajizadegan, Q. Cui, M. M.-C. Cheng, R. El-Ganainy, and A. Alù, “Generalized parity–time symmetry condition for enhanced sensor telemetry,” Nat. Electron. 1(5), 297–304 (2018).
[Crossref]

R. Fleury, D. Sounas, and A. Alù, “An invisible acoustic sensor based on parity-time symmetry,” Nat. Commun. 6(1), 5905 (2015).
[Crossref]

Arju, N.

S. H. Mousavi, I. Kholmanov, K. B. Alici, D. Purtseladze, N. Arju, K. Tatar, D. Y. Fozdar, J. W. Suk, Y. Hao, A. B. Khanikaev, R. S. Ruoff, and G. Shvets, “Inductive tuning of fano-resonant metasurfaces using plasmonic response of graphene in the mid-infrared,” Nano Lett. 13(3), 1111–1117 (2013).
[Crossref]

Avouris, P.

P. Avouris and F. Xia, “Graphene applications in electronics and photonics,” MRS Bull. 37(12), 1225–1234 (2012).
[Crossref]

Beere, H. E.

S. J. Kindness, N. W. Almond, B. Wei, R. Wallis, W. Michailow, V. S. Kamboj, P. Braeuninger-Weimer, S. Hofmann, H. E. Beere, D. A. Ritchie, and R. Degl’Innocenti, “Active control of electromagnetically induced transparency in a terahertz metamaterial array with graphene for continuous resonance frequency tuning,” Adv. Opt. Mater. 6(21), 1800570 (2018).
[Crossref]

Bender, C. M.

C. M. Bender, “Making sense of non-Hermitian Hamiltonians,” Rep. Prog. Phys. 70(6), 947–1018 (2007).
[Crossref]

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
[Crossref]

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

Bender, N.

F. Nazari, N. Bender, H. Ramezani, M. K. Moravvej-Farshi, D. N. Christodoulides, and T. Kottos, “Optical isolation via PT -symmetric nonlinear fano resonances,” Opt. Express 22(8), 9574–9584 (2014).
[Crossref]

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110(23), 234101 (2013).
[Crossref]

Berini, P.

B. Qi, H.-Z. Chen, L. Ge, P. Berini, and R.-M. Ma, “Parity–time symmetry synthetic lasers: Physics and devices,” Adv. Opt. Mater. 7(22), 1900694 (2019).
[Crossref]

Bersch, C.

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[Crossref]

Bodyfelt, J. D.

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110(23), 234101 (2013).
[Crossref]

Boettcher, S.

C. M. Bender and S. Boettcher, “Real spectra in non-Hermitian Hamiltonians having PT symmetry,” Phys. Rev. Lett. 80(24), 5243–5246 (1998).
[Crossref]

Bonn, M.

A. Tomadin, S. M. Hornett, H. I. Wang, E. M. Alexeev, A. Candini, C. Coletti, D. Turchinovich, M. Kläui, M. Bonn, F. H. L. Koppens, E. Hendry, M. Polini, and K.-J. Tielrooij, “The ultrafast dynamics and conductivity of photoexcited graphene at different Fermi energies,” Sci. Adv. 4(5), eaar5313 (2018).
[Crossref]

Braeuninger-Weimer, P.

S. J. Kindness, N. W. Almond, B. Wei, R. Wallis, W. Michailow, V. S. Kamboj, P. Braeuninger-Weimer, S. Hofmann, H. E. Beere, D. A. Ritchie, and R. Degl’Innocenti, “Active control of electromagnetically induced transparency in a terahertz metamaterial array with graphene for continuous resonance frequency tuning,” Adv. Opt. Mater. 6(21), 1800570 (2018).
[Crossref]

Braithwaite, N. S. J.

A. Dey, A. Chroneos, N. S. J. Braithwaite, R. P. Gandhiraman, and S. Krishnamurthy, “Plasma engineering of graphene,” Appl. Phys. Rev. 3(2), 021301 (2016).
[Crossref]

Brandstetter, M.

M. Brandstetter, M. Liertzer, C. Deutsch, P. Klang, J. Schöberl, H. E. Türeci, G. Strasser, K. Unterrainer, and S. Rotter, “Reversing the pump dependence of a laser at an exceptional point,” Nat. Commun. 5(1), 4034 (2014).
[Crossref]

Brody, D. C.

C. M. Bender, D. C. Brody, and H. F. Jones, “Complex extension of quantum mechanics,” Phys. Rev. Lett. 89(27), 270401 (2002).
[Crossref]

Bumki, M.

P. Sang Hyun, L. Sung-Gyu, B. Soojeong, H. Taewoo, L. Sanghyub, M. Bumki, Z. Shuang, L. Mark, and K. Teun-Teun, “Observation of an exceptional point in a non-Hermitian metasurface,” Nanophotonics20190489 (2020).
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Candini, A.

A. Tomadin, S. M. Hornett, H. I. Wang, E. M. Alexeev, A. Candini, C. Coletti, D. Turchinovich, M. Kläui, M. Bonn, F. H. L. Koppens, E. Hendry, M. Polini, and K.-J. Tielrooij, “The ultrafast dynamics and conductivity of photoexcited graphene at different Fermi energies,” Sci. Adv. 4(5), eaar5313 (2018).
[Crossref]

Cao, H.

Z. Lin, H. Ramezani, T. Eichelkraut, T. Kottos, H. Cao, and D. N. Christodoulides, “Unidirectional invisibility induced by PT-symmetric periodic structures,” Phys. Rev. Lett. 106(21), 213901 (2011).
[Crossref]

Cao, T.

T. Cao, Y. Cao, and L. Fang, “Reconfigurable parity-time symmetry transition in phase change metamaterials,” Nanoscale 11(34), 15828–15835 (2019).
[Crossref]

Cao, Y.

T. Cao, Y. Cao, and L. Fang, “Reconfigurable parity-time symmetry transition in phase change metamaterials,” Nanoscale 11(34), 15828–15835 (2019).
[Crossref]

Chan, C. T.

X.-L. Zhang, T. Jiang, and C. T. Chan, “Dynamically encircling an exceptional point in anti-parity-time symmetric systems: Asymmetric mode switching for symmetry-broken modes,” Light: Sci. Appl. 8(1), 88 (2019).
[Crossref]

K. Ding, G. Ma, M. Xiao, Z. Q. Zhang, and C. T. Chan, “Emergence, coalescence, and topological properties of multiple exceptional points and their experimental realization,” Phys. Rev. X 6(2), 021007 (2016).
[Crossref]

Chang, L.

L. Chang, X. Jiang, S. Hua, C. Yang, J. Wen, L. Jiang, G. Li, G. Wang, and M. Xiao, “Parity–time symmetry and variable optical isolation in active–passive-coupled microresonators,” Nat. Photonics 8(7), 524–529 (2014).
[Crossref]

Chen, H.

W. Tan, Y. Sun, Z.-G. Wang, and H. Chen, “Manipulating electromagnetic responses of metal wires at the deep subwavelength scale via both near- and far-field couplings,” Appl. Phys. Lett. 104(9), 091107 (2014).
[Crossref]

Chen, H.-Z.

B. Qi, H.-Z. Chen, L. Ge, P. Berini, and R.-M. Ma, “Parity–time symmetry synthetic lasers: Physics and devices,” Adv. Opt. Mater. 7(22), 1900694 (2019).
[Crossref]

Chen, J.

J. Chen, X. Li, X. Shi, C. Fan, M. Tuhtasun, X. He, W. Shi, and F. Liu, “Active control of light slowing enabled by coupling electromagnetic metamaterials with low-lossy graphene,” Opt. Lett. 43(20), 4891–4894 (2018).
[Crossref]

B. Jin, W. Tan, C. Zhang, J. Wu, J. Chen, S. Zhang, and P. Wu, “High-performance terahertz sensing at exceptional points in a bilayer structure,” Adv. Theory Simul. 1(9), 1800070 (2018).
[Crossref]

Chen, P.-Y.

P.-Y. Chen, M. Sakhdari, M. Hajizadegan, Q. Cui, M. M.-C. Cheng, R. El-Ganainy, and A. Alù, “Generalized parity–time symmetry condition for enhanced sensor telemetry,” Nat. Electron. 1(5), 297–304 (2018).
[Crossref]

P.-Y. Chen and J. Jung, “PT symmetry and singularity-enhanced sensing based on photoexcited graphene metasurfaces,” Phys. Rev. Appl. 5(6), 064018 (2016).
[Crossref]

Chen, Y.

C. Shi, M. Dubois, Y. Chen, L. Cheng, H. Ramezani, Y. Wang, and X. Zhang, “Accessing the exceptional points of parity-time symmetric acoustics,” Nat. Commun. 7(1), 11110 (2016).
[Crossref]

Cheng, L.

C. Shi, M. Dubois, Y. Chen, L. Cheng, H. Ramezani, Y. Wang, and X. Zhang, “Accessing the exceptional points of parity-time symmetric acoustics,” Nat. Commun. 7(1), 11110 (2016).
[Crossref]

Cheng, M. M.-C.

P.-Y. Chen, M. Sakhdari, M. Hajizadegan, Q. Cui, M. M.-C. Cheng, R. El-Ganainy, and A. Alù, “Generalized parity–time symmetry condition for enhanced sensor telemetry,” Nat. Electron. 1(5), 297–304 (2018).
[Crossref]

Cheng, Z.

Z. Zhang, T. Lin, X. Xing, X. Lin, X. Meng, Z. Cheng, Z. Jin, and G. Ma, “Photoexcited terahertz conductivity dynamics of graphene tuned by oxygen-adsorption,” Appl. Phys. Lett. 110(11), 111108 (2017).
[Crossref]

Christodoulides, D. N.

F. Nazari, N. Bender, H. Ramezani, M. K. Moravvej-Farshi, D. N. Christodoulides, and T. Kottos, “Optical isolation via PT -symmetric nonlinear fano resonances,” Opt. Express 22(8), 9574–9584 (2014).
[Crossref]

N. Bender, S. Factor, J. D. Bodyfelt, H. Ramezani, D. N. Christodoulides, F. M. Ellis, and T. Kottos, “Observation of asymmetric transport in structures with active nonlinearities,” Phys. Rev. Lett. 110(23), 234101 (2013).
[Crossref]

A. Regensburger, C. Bersch, M. A. Miri, G. Onishchukov, D. N. Christodoulides, and U. Peschel, “Parity-time synthetic photonic lattices,” Nature 488(7410), 167–171 (2012).
[Crossref]

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Z. W. Ullah, G. Nawi, I. Tansu, N. Irfan Khattak, and M. Junaid, “A review on the development of tunable graphene nanoantennas for terahertz optoelectronic and plasmonic applications,” Sensors 20(5), 1401 (2020).
[Crossref]

Other (1)

P. Sang Hyun, L. Sung-Gyu, B. Soojeong, H. Taewoo, L. Sanghyub, M. Bumki, Z. Shuang, L. Mark, and K. Teun-Teun, “Observation of an exceptional point in a non-Hermitian metasurface,” Nanophotonics20190489 (2020).
[Crossref]

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Figures (6)

Fig. 1.
Fig. 1. (a) Schematic of the unit cell of the metasurface composed by two orthogonally arranged SRRs with one SRR containing a square graphene patch on the gap. The conductivity of the graphene can be tuned by optical pump and chemical doping by surface adsorbates (e.g. reactive gas, represented by purple dots). The geometric parameters are P = 150 µm, L = 40 µm, w = 5 µm, h = 10 µm, u = 5 µm and g = 5 µm, respectively. (b) A schematic representation of a system composed of two coupled resonators and two ports. Here, the system is only excited at port 1.
Fig. 2.
Fig. 2. (a) Simulated and (b) fitted transmission spectra of the metasurface at different EF.
Fig. 3.
Fig. 3. Fitted parameters as functions of EF.
Fig. 4.
Fig. 4. The eigen transmission (a-e) amplitude and (f-j) phase spectra at different EF obtained from simulations. The dashed lines indicate the specific frequency of 0.4977 THz, where the EP can occur. (k-o) The corresponding eigen polarization states at 0.4977 THz at different EF.
Fig. 5.
Fig. 5. Eigen polarization states at seven selected frequencies around the EP frequency fEP obtained from (a) simulations and (b) theoretical calculations plotted on the Poincaré spheres in viewed from the north pole. The arrows indicate the changing directions of the eigen polarization states with increasing EF (dissipative loss rate Γx in calculation). Theoretical calculated surfaces of (c) the amplitude and (d) phase of the eigenvalues λT in (ω, Γx) parameter space. The yellow points indicate the EP with fEP = 0.4977 THz and Γx = 0.0475 THz.
Fig. 6.
Fig. 6. Transmission (a) amplitude and (b) phase spectra of the metasurface in circular polarization basis at different EF. (c) Extracted phase values as a function of EF at three frequencies indicated by the dash line in (b).

Equations (16)

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d a x d t = ( j ω x γ x Γ x ) a x + j κ a y + j γ x E x i n ,
d a y d t = ( j ω y γ y Γ y ) a y + j κ a x + j γ y E y i n ,
( E x o u t E y o u t ) = ( E x i n E y i n ) + ( j γ x 0 0 j γ y ) ( a x a y ) = T ( E x i n E y i n ) ,
( γ y ( ω x ω ) + j γ y ( γ x + Γ x ) κ γ x γ y κ γ x γ y γ x ( ω y ω ) + j γ x ( γ y + Γ y ) ) ( j γ x a x j γ y a y ) = j γ x γ y ( E x i n E y i n ) .
H = ( γ y ( ω x ω ) + j γ y ( γ x + Γ x ) κ γ x γ y κ γ x γ y γ x ( ω y ω ) + j γ x ( γ y + Γ y ) ) = j χ I + H 0 ,
χ = γ y γ x + ( γ y Γ x + γ x Γ y ) / 2 ,
H 0 = ( γ y ( ω x ω ) + j ( γ y Γ x γ x Γ y ) / 2 κ γ x γ y κ γ x γ y γ x ( ω y ω ) + j ( γ x Γ y γ y Γ x ) / 2 ) .
γ x δ y = γ y δ x ,
T = I j γ x γ y H 1  =  I j γ x γ y det ( H ) ( det ( H 0 ) H 0 1 + j χ I ) ,
λ T = 1 A ( λ H 0 + j χ ) ,
v = ( 1 , { [ λ H 0 γ x δ y j ( γ y Γ x γ x Γ y ) / 2 ] / ( γ x γ y κ ) } ) ,
A = j γ x γ y / det ( H ) = j [ ( δ x j ( γ x + Γ x ) ) ( δ y j ( γ y + Γ y ) ) κ 2 ] 1 ,
λ H 0 = ( γ x δ y γ y δ x ± [ γ x δ y γ y δ x + j ( γ y Γ x γ x Γ y ) ] 2 + 4 γ x γ y κ 2 ) / 2.
ω 0 = ( γ x ω y γ y ω x ) / ( γ x γ y ) .
| γ y Γ x γ x Γ y | = 2 γ x γ y | κ | ,
t l r = [ A ( γ x δ y γ y δ x + j ( γ x Γ y + γ y Γ x + 2 γ x γ y κ ) ) ] / 2 ,

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