Exceptional point in a metal-graphene hybrid metasurface with tunable asymmetric loss

Shaoxian Li; Xueqian Zhang; Xueqian Zhang; Quan Xu; Quan Xu; Meng Liu; Ming Kang; Jiaguang Han; Weili Zhang; Weili Zhang

doi:10.1364/OE.391917

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 [2–6] and experimental implementations in optics [7–10], acoustics [11–13], electronics [14–16], 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 [23–26], 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 [29–32]. 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 E_F can be controlled by external stimuli, such as electric bias, optical pump, and chemical doping [42–44], 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 a_x and a_y. Here, only port 1 is exited.

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) [48–50], the dynamic equations for the two resonances and the output at port 2 can be expressed as:

(1)$$\frac{{d{a_x}}}{{dt}} = (j{\omega _x} - {\gamma _x} - {\Gamma _x}){a_x} + j\kappa {a_y} + j\sqrt {{\gamma _x}} {E_x}^{in},$$

(2)$$\frac{{d{a_y}}}{{dt}} = (j{\omega _y} - {\gamma _y} - {\Gamma _y}){a_y} + j\kappa {a_x} + j\sqrt {{\gamma _y}} {E_y}^{in},$$

(3)$$\left( {\begin{array}{c} {{E_x}^{out}}\\ {{E_y}^{out}} \end{array}} \right) = \left( {\begin{array}{c} {{E_x}^{in}}\\ {{E_y}^{in}} \end{array}} \right) + \left( {\begin{array}{cc} {j\sqrt {{\gamma_x}} }&0\\ 0&{j\sqrt {{\gamma_y}} } \end{array}} \right)\left( {\begin{array}{c} {{a_x}}\\ {{a_y}} \end{array}} \right) = T\left( {\begin{array}{c} {{E_x}^{in}}\\ {{E_y}^{in}} \end{array}} \right),$$

respectively, where ω_x(ω_y), γ_x(γ_y), Γ_x(Γ_y) are the resonance frequency, radiation loss rate, dissipative loss rate of the resonance a_x(a_y), respectively; κ denotes the coupling between the two resonances; E_x^out(E_y^out) and E_xⁱⁿ(E_yⁱⁿ) are the transmitted and incident x(y)-polarized electric fields, respectively; T is the transmission matrix. Here, instead of denoting the resonances [a_x a_y]^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 Hψ = Eψ and the scattering equation Sψ = λψ. Equations (1) and (2) can be rewritten as

(4)$$\left( {\begin{array}{cc} {{\gamma_y}({\omega_x} - \omega ) + j{\gamma_y}({\gamma_x} + {\Gamma _x})}&{\kappa \sqrt {{\gamma_x}} \sqrt {{\gamma_y}} }\\ {\kappa \sqrt {{\gamma_x}} \sqrt {{\gamma_y}} }&{{\gamma_x}({\omega_y} - \omega ) + j{\gamma_x}({\gamma_y} + {\Gamma _y})} \end{array}} \right)\left( {\begin{array}{c} {j\sqrt {{\gamma_x}} {a_x}}\\ {j\sqrt {{\gamma_y}} {a_y}} \end{array}} \right) ={-} j{\gamma _x}{\gamma _y}\left( {\begin{array}{c} {{E_x}^{in}}\\ {{E_y}^{in}} \end{array}} \right).$$

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

(5)$$H = \left( {\begin{array}{cc} {{\gamma_y}({\omega_x} - \omega ) + j{\gamma_y}({\gamma_x} + {\Gamma _x})}&{\kappa \sqrt {{\gamma_x}} \sqrt {{\gamma_y}} }\\ {\kappa \sqrt {{\gamma_x}} \sqrt {{\gamma_y}} }&{{\gamma_x}({\omega_y} - \omega ) + j{\gamma_x}({\gamma_y} + {\Gamma _y})} \end{array}} \right) = j\chi I + {H_0},$$

where I is the identity matrix, and

(6)$$\chi = {\gamma _y}{\gamma _x} + ({\gamma _y}{\Gamma _x} + {\gamma _x}{\Gamma _y})/2,$$

(7)$${H_0} = \left( {\begin{array}{cc} {{\gamma_y}({\omega_x} - \omega ) + j({\gamma_y}{\Gamma _x} - {\gamma_x}{\Gamma _y})/2}&{\kappa \sqrt {{\gamma_x}} \sqrt {{\gamma_y}} }\\ {\kappa \sqrt {{\gamma_x}} \sqrt {{\gamma_y}} }&{{\gamma_x}({\omega_y} - \omega ) + j({\gamma_x}{\Gamma _y} - {\gamma_y}{\Gamma _x})/2} \end{array}} \right).$$

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

(8)$${\gamma _x}{\delta _y} = {\gamma _y}{\delta _x},$$

where δ_x = ω − ω_x and δ_y = ω − ω_y are the detuning frequencies of the two resonance modes, respectively. From the Eqs. (3)–(5), the transmission matrix T can be calculated as

(9)$$T = I - j{\gamma _x}{\gamma _y}{H^{ - 1}}\textrm{ = }I - \frac{{j{\gamma _x}{\gamma _y}}}{{\det (H)}}({\det ({H_0}){H_0}^{ - 1} + j\chi I} ),$$

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

(10)$${\lambda _T} = 1 - A({\lambda _{{H_0}}} + j\chi ),$$

(11)$$v = \left( {1,\left\{ {[{ - {\lambda_{{H_\textrm{0}}}} - {\gamma_x}{\delta_y} - j({\gamma_y}{\Gamma _x} - {\gamma_x}{\Gamma _y})/2} ]/\left( {\sqrt {{\gamma_x}{\gamma_y}} \kappa } \right)} \right\}} \right),$$

respectively, where

(12)$$A = j{\gamma _x}{\gamma _y}\textrm{/}\det (H) = j{[{({{\delta_x} - j({\gamma_x} + {\Gamma _x})} )({{\delta_y} - j({\gamma_y} + {\Gamma _y})} )- {\kappa^2}} ]^{ - 1}},$$

(13)$${\lambda _{{H_\textrm{0}}}} = \left( { - {\gamma_x}{\delta_y} - {\gamma_y}{\delta_x} \pm \sqrt {{{[{{\gamma_x}{\delta_y} - {\gamma_y}{\delta_x} + j({{\gamma_y}{\Gamma _x} - {\gamma_x}{\Gamma _y}} )} ]}^2} + 4{\gamma_x}{\gamma_y}{\kappa^2}} } \right)/2.$$

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 ω_x = ω_y and γ_x = γ_y, where H₀ is PT symmetric for all the frequencies [29–31,33]. In this case, if the radiation loss rates and the coupling fulfill the relation of |Γ_x – Γ_y| = 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 ω_x = ω_y and γ_x ≠ γ_y [32]. Then, the system is only PT symmetric at a single frequency ω = ω_x = ω_y. However, the condition of exactly ω_x = ω_y 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. ω_x ≠ ω_y and γ_x ≠ γ_y, where H₀ is found to be PT symmetric only at a specific frequency of

(14)$${\omega _0} = ({\gamma _x}{\omega _y} - {\gamma _y}{\omega _x})/({\gamma _x} - {\gamma _y}).$$

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

(15)$$|{{\gamma_y}{\Gamma _x} - {\gamma_x}{\Gamma _y}} |= \textrm{2}\sqrt {{\gamma _x}{\gamma _y}} |\kappa |,$$

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 E_F of the graphene using CST Microwave Studio, as shown in Fig. 2(a). Here, t_ij 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 E_F 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 σ_Al = 3.56 × 10⁷ S/m which was described by surface impedance of Z ≈ (µ₀ω/σ_Al)^1/2exp(jπ/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 t_xx and t_xy (= t_yx) gradually decrease while the amplitude of the resonance in t_yy increases due to the increased dissipation loss rate of the left SRR as E_F increases. The two resonance dips in t_xx and the cross-polarized output t_xy (t_yx) as well as the converse changing manner of the resonance amplitude in t_yy 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 Γ_x of the left SRR increases as E_F increases owing to the shorting effect at the capacitance gap. The resonance frequency f_x first slightly increases and then increases quickly. This behavior can be explained using perturbation theory [53–55], 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 $ = −jσ/W₀ × ∫_S|E_xy|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 E_xy 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 E_F increases, the resonance frequency increases accordingly. The radiative loss rates γ_x and γ_y is also found to be different from each other. All of these fitting parameters indicate the proposed system fulfills the above-mentioned conditions of ω_x ≠ ω_y, γ_x ≠ γ_y. Since all the geometric parameters of the structure are fixed, κ, γ_x, γ_y, f_y and Γ_y 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 E_F, which crosses with κ around E_F = 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 t_xx is slightly asymmetric, and the resonance dip of t_yy is also not at the center of the two dips of t_xx. This is just owing to the small frequency detuning between ω_x and ω_y, which can be checked by observing the transmission spectra with respect to the corresponding fitted values of f_x and f_y in Fig. 3. Such phenomena gradually decrease as the frequency detuning decreases, which is consistent with the previous report [51].

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

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Fig. 3. Fitted parameters as functions of E_F.

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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 λ_T, under different E_F 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 f_EP = 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 E_F 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 E_F = 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).

Fig. 4. The eigen transmission (a-e) amplitude and (f-j) phase spectra at different E_F 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 E_F.

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Fig. 5. Eigen polarization states at seven selected frequencies around the EP frequency f_EP 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 E_F (dissipative loss rate Γ_x in calculation). Theoretical calculated surfaces of (c) the amplitude and (d) phase of the eigenvalues λ_T in (ω, Γ_x) parameter space. The yellow points indicate the EP with f_EP = 0.4977 THz and Γ_x = 0.0475 THz.

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Figure 5(a) shows the simulated eigen polarization state evolving traces with frequencies near f_EP on Poincaré sphere when E_F increases from 0 to 0.2 eV. The arrow in each quadrant indicates the evolving trends with increasing E_F. The scatters correspond to the five cases in Fig. 4. It can be seen that only the traces at f_EP 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 f_EP, but they do not follow longitude lines. For frequencies higher than f_EP, the eigenstates evolve in the second and fourth quadrants. However, for frequencies lower than f_EP, 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 Γ_x controlled by the Fermi level of graphene, to simplify the calculation and meanwhile guarantee the effectiveness of the model, the parameters γ_y, Γ_y, κ, γ_x, ω_x, ω_y are fixed to those fitted at E_F = 0.1322 eV in Fig. 3, while Γ_x 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 λ_T in (ω, Γ_x) parameter space. The yellow dots represent the EP, where f_EP = 0.4977 THz and Γ_x = 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 f_EP = 0.4977 THz and E_{F_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 t_lr at E_F = 0.134 eV and E_F = 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 E_F is quite slight (ΔE_F = 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 E_F < E_{F_EP} cross at one point while those at E_F > E_{F_EP} cross at another point around 0.5 THz. This feature can be explained by investigating the theoretical expression of

(16)$${t_{lr}} = [A({\gamma _x}{\delta _y} - {\gamma _y}{\delta _x} + j( - {\gamma _x}{\Gamma _y} + {\gamma _y}{\Gamma _x} + 2\sqrt {{\gamma _x}{\gamma _y}} \kappa ))]/2,$$

which can be calculated from the transmission matrix in Eq. (9). At the EP frequency, γ_xδ_y = γ_yδ_x and t_lr 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 Γ_x increases. The transition point t_lr = 0 is exactly the EP since Eq. (8) and Eq. (15) are both fulfilled.

Fig. 6. Transmission (a) amplitude and (b) phase spectra of the metasurface in circular polarization basis at different E_F. (c) Extracted phase values as a function of E_F 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 E_F 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 t_lr at three frequencies indicated by the dash lines in Fig. 6(b) as a function of E_F. The phase changes slowly when E_F 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 E_F of graphene should be set near the E_{F_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, E_{F_EP}= 0.229 eV at u = 1 µm, and E_{F_EP}= 0.071 eV at u = 12 µm, indicating a large tuning range of E_{F_EP} by u. Therefore, one can always find the metasurface with certain u whose E_{F_EP} is in the tuning range under the optical pump. Alternatively, one can process the graphene by proper plasma treatment to tune its E_F around the E_{F_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 t_lr, 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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Exceptional point in a metal-graphene hybrid metasurface with tunable asymmetric loss

Abstract

1. Introduction

2. Theoretical model

3. Simulation and discussion

4. Sensing applications

5. Conclusion

Funding

Disclosures

References

Cited By

Figures (6)

Equations (16)

Optics Express