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Abstract
Exceptional points (EPs), the critical phase transition points of non-Hermitian parity-time (PT) systems, exhibit many novel physical properties and associated applications, such as ultra-sensitive detection of perturbations. Here, a bilayer metasurface with two orthogonally oriented split-ring resonators (SRRs) is proposed and a phase transition of the eigenpolarization states is introduced via changing the conductivity of vanadium dioxide (VO2) patch integrated into the gap of one SRR. The metasurface possesses a passive PT symmetry and an EP in polarization space is observed at a certain conductivity of the VO2. Two sensing schemes with the metasurface are proposed to achieve high-sensitivity sensing of temperature and refractive index in the terahertz (THz) range. The metasurface is promising for applications in THz biosensing and polarization manipulation.
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1. Introduction
There has been tremendous interest in non-Hermitian physical systems since Bender and Boettcher revealed that the Hamiltonian of non-Hermitian systems with parity-time (PT) symmetry may exhibit real eigenvalues [1,2–5]. Much of the interest has focused on exceptional points (EPs) [5], where the eigenvalues and eigenvectors of such PT symmetric system coalesce, in contrast to the Hermitian system [6]. The transition from PT symmetric phase to PT-broken phase can also be investigated near the EP [7]. Various interesting phenomena associated with EP have been observed, such as unidirectional invisibility [8,9], sensitivity enhancement [10,11], and non-Hermitian topological phenomena [12,13].
Metasurfaces are artificial two-dimensional electromagnetic structures consisting of periodic subwavelength elements. Metasurfaces can achieve unusual electromagnetic functionalities for controlling the wavefront and polarization state of light. The non-Hermitian system facilitates the design of metasurfaces and enriches the metasurface concept. It allows researchers to fully utilize the relationship between the real and imaginary parts of the refractive index [14]. PT symmetry metasurfaces could realize various important applications including coherent perfect absorption [15], phase modulation [16], and polarization manipulation [17,18]. Previous studies show that in a two-coalescing-level system, the eigenfrequency splitting follows the square root of the perturbation strength [10]. Furthermore, at an EPn, the eigenfrequency splittings are generally proportional to the nth root of the perturbation strength which leads to extreme sensitivity to sufficiently small perturbations [6]. This new ultrasensitive sensing scheme has been proved to be a possible way to detect tiny environment changes [19,20], as a result it could be employed in the label-free detection of biomolecules, viruses, and proteins [11,21,22].
In previous studies, a series of tunable materials have been applied to the designs of non-Hermitian metasurfaces, such as type II superconductor niobium nitride [23], GeTe [24], graphene [25], and photoactive silicon [26]. Vanadium dioxide (VO2) can experience insulator-to-metal phase transition with thermal stimuli, and correspondingly its conductivity can go through a significant variation, up to 3-4 orders of magnitude [27]. Therefore, VO2 material can be used to design tunable metasurfaces and realize promising applications such as switching [27,28], absorption [29], beam steering [30–32], holography [33] and temperature sensors [34].
In this work, a bilayer non-Hermitian metasurface is proposed and investigated. The metasurface consists of two orthogonally oriented SRRs with the same dimensions on either side of the dielectric layer, but one SRR contains a VO2 patch within its gap. The dissipative loss of the metasurface could be adjusted and an EP can be achieved by varying the conductivity of the VO2. The coupled-mode theory is used to explain the evolution of the eigenvalues and eigenstates in the polarization space. Moreover, by analyzing the asymmetric transmission phenomenon at the EP, two sensing schemes are designed to realize bifunctional terahertz sensing of temperature and refractive index.
2. Principle and design
The unit of the designed metasurface consists of two silver (Ag) SRRs of the same size, but rotated 90° with respect to each other. They are placed on the opposite sides of a benzocyclobutene (BCB) [35,36] interlayer, as shown in Figs. 1(a) and 1(b). Each SRR structure is equivalent to an LC resonant model, in which the resonance frequency and the losses can be adjusted by changing the size and material properties of SRRs. The gap of one SRR is integrated with the VO2 microstructure that mainly affects the dissipation loss of its resonant mode. The peak and bandwidth of the resonant spectrum can be adjusted by changing the conductivity of VO2. SRRs on the opposite sides of the interlayer can be respectively excited by x- and y-polarized waves. Coupled-mode theory, as an effective tool, is widely used to describe the coupling between two resonators. The combination of the coupled-mode equation with the transmission matrix is adopted to construct the Hamiltonian matrix for the PT symmetric system. The eigenvalues and the eigenpolarization states of the structure can be subsequently solved. As shown in Fig. 1(c), the resonant modes and transmitted fields in the two orthogonal directions (${\tilde{a}_x} = {a_x}{e^{i\omega t}}$ and ${\tilde{a}_y} = {a_y}{e^{i\omega t}}$) can be written as follows [22]
The left part of Eq. (4) can be considered as the Hamiltonian of the system and can be rewritten as $H = j\chi \mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over I} + {H_0}$, where $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\leftrightarrow$}} \over I} $ is the identity matrix, $\chi = {\gamma _x}{\gamma _y} + ({\gamma _y}{\Gamma _x} + {\gamma _x}{\Gamma _y})/2$,
Fig. 1. (a) Schematic of the unit cell of the metasurface composed of two orthogonally arranged Ag SRRs (yellow). One SRR contains a VO2 patch (red) within the gap. (b) Front view of the metasurface, s denotes the horizontal distance between the two structures as one SRR is projected in the plane with the other SRR is. The orthogonal SRRs can be considered as two resonant modes ${\tilde{a}_y}$ and ${\tilde{a}_x}$. (c) A schematic representation of a system composed of two coupled orientation resonators.
When γx (ωy - ω) = γy (ωx - ω), we have [PT, H0] = 0. Note that the parity P operator is the Pauli matrix σx and the time-reversal T operator corresponds to complex conjugation. This means that the system satisfies PT symmetry.
The previous works mainly consider the EP generation in systems with the same resonant frequency and radiation loss, which place a stringent requirement on the parameters of the structure, and are difficult to achieve in the actual fabrication and measurement [17,18,22,24,26]. In this work, the conditions for the formation of PT symmetry are discussed in the general case ωx ≠ ωy and γx ≠ γy. The Hamiltonian of system can to be PT symmetric at a frequency of ω0 = (γxωy - γyωx) / (γx - γy). According to Eq. (5), when $|{{\gamma_y}{\Gamma _x} - {\gamma_x}{\Gamma _y}} |\lt 2\sqrt {{\gamma _x}{\gamma _y}} |\kappa |$, the system is in PT symmetric phase. The eigenpolarization states are given by $(1, \pm \exp (i{\sin ^{ - 1}}[({\gamma _y}{\Gamma _x} - {\gamma _x}{\Gamma _y})/2\sqrt {{\gamma _x}{\gamma _y}} \kappa ]))$, and the major axes of the eigenpolarization states are oriented along ±45°. When $|{{\gamma_y}{\Gamma _x} - {\gamma_x}{\Gamma _y}} |\lt 2\sqrt {{\gamma _x}{\gamma _y}} |\kappa |$, the PT symmetry is broken. The eigenpolarization states are given by $(1, \pm i\exp ({\cosh ^{ - 1}}[({\gamma _y}{\Gamma _x} - {\gamma _x}{\Gamma _y})/2\sqrt {{\gamma _x}{\gamma _y}} \kappa ]))$, and the major axes are oriented along 0° and 90°, respectively. The EP occurs at $|{{\gamma_y}{\Gamma _x} - {\gamma_x}{\Gamma _y}} |= 2\sqrt {{\gamma _x}{\gamma _y}} |\kappa |$, where the eigenvalues of H0 are degenerate and the corresponding eigenpolarization states coalesce into the right-handed circularly polarization state. In order to obtain the eigenvalues and eigenpolarization states of the system, the four complex transmission spectra, txx, txy, tyx, and tyy of the bilayer metasurface should be first obtained. Then the parameters of the transmission matrix can be retrieved by theoretically fitting the four transmission spectra.
The scheme of the proposed metasurface structure is shown in Figs. 1(a) and 1(b). The period of the unit cell is p = 140 µm, the thickness of the BCB interlayer is t = 4.5 µm with dielectric constant ε = 2.67 and a loss tangent of δ = 0.012. Two Ag (the conductivity σAg = 7 × 106 S ∕ m) SRRs have the same size. The other parameters are the side length L = 50 µm, the width w = 10 µm, the gap of SRRs g = 9 µm. The parameter s denotes the horizontal distance between the two structures as one SRR is projected in the plane with the other SRR, here s = 10 µm. VO2 is positioned within the gap of one SRR. The permittivity of VO2 in the THz region is described by the Drude model [27]
The co- and cross-polarized transmission coefficients of the bilayer metasurface with different VO2 conductivities are simulated by use of CST Microwave Studio, as shown in Fig. 2(a). The amplitudes of resonance in tyy gradually increases while that in txx, txy and tyx decreases due to the increased dissipation loss rate Γx as σ increases. The resonance frequencies of the resonator are ωx = 0.833 THz and ωy = 0.834 THz. This difference is caused by the incorporation of VO2 into SRR oriented in the x-direction. The fitted transmission spectra are illustrated in Fig. 2(b), the main difference is reflected in the profiles of txx and tyy, which is caused by the phase delay of the bilayer structure and the dissipative loss of the intermediate layer [22]. The corresponding fitting parameters are shown in Fig. 3. It is shown that γx, γy, Γy and κ do not change much with the increase of σ since the conductivity change of VO2 only affects one of the SRRs. The dissipation loss rate Γx of the ${\tilde{a}_x}$ increases as σ increases owing to the shorting effect of the capacitance gap. The radiative loss rates γx and γy are also found to be different from each other.
Fig. 2. (a) Simulated and (b) fitted transmission spectra of the metasurface for different conductivity(σ) of VO2. The resonance dips in co-polarized txx and the cross-polarized output txy (tyx) gradually decrease while the resonance amplitude tyy increases with increasingσ.
Fig. 3. Fitted parameters of the transmission as a function of σ. The model parameters derived from the fitting are γx = 0.016 THz, γy = 0.018 THz, Γy = 0.009 THz, κ = 0.013 THz. Γx changes from 0.03 to 0.038 THz with the increasing σ while other parameters are nearly constant.
In order to locate the EP and analyze the PT symmetric phase transition, the eigenvalues of transmission spectra and their matching eigenpolarization states at ω0 = 0.83 THz are shown in Fig. 4. The transition from PT symmetry to PT symmetry broken phase can be observed according to the transmission spectra and eigenpolarization states. The transmission spectra of eigenpolarization state 1 and eigenpolarization state 2 have different resonance frequencies, as shown in Fig. 4(a). When σ increases, the two resonance frequencies gradually move towards each other, but the resonances have slightly different linewidths in the two transmission spectra as shown in Fig. 4(c). Especially, when the conductivity of VO2 is σ = 582 S ∕ m, the amplitude and phase spectra of two eigenpolarization states nearly coincide with each other, indicating that the EP appears, as shown in Fig. 4(b). Although the transmission spectra are not completely degenerate due to the difference in resonant frequency and loss, the transition from PT symmetry to PT symmetry breaking can be obviously observed. Correspondingly, in the PT symmetry regime where σ is less than 582 S ∕ m, the eigenpolarization states are two polarization ellipses with their major axes oriented along ±45°. At the EP, two eigenpolarization states nearly coalesce into one circularly polarized state. In the PT symmetry broken regime where σ is large than 582 S ∕ m, the eigenpolarization states are two polarization ellipses with their major axes oriented along 0° and 90°, respectively. The aforementioned results can be clearly seen in Figs. 4(d)–4(f).
Fig. 4. (a)-(c) Transmission spectra for the eigenpolarization states in the metasurfaces with different conductivities σ = 500 S/m, σ = 582 S/m, σ = 700 S/m. (d)-(f) Eigenpolarization states in the metasurfaces with different conductivities σ = 500 S/m, σ = 582 S/m, σ = 700 S/m.
As mentioned previously, one of the important differences between a non-Hermitian system and a Hermitian system is the degeneracy of eigenvalues. In a Hermitian system, conventional degeneracies known as diabolic points (DP) have a double-cone topology around them. While in a non-Hermitian system two intersecting Riemann sheets centered around an EP are formed [6]. This type of topology is important in the non-Hermitian systems. Therefore, self-intersecting Riemann surface structure will serve as an important signature of an EP. Figures 5(a) and 5(b) show the calculated surfaces of the magnitude and phase of the complex eigenvalues in (ω, σ) parameter space. Depending on the variation of σ, the crossing behavior is opposite as a function of the frequency between the magnitude and phase of the eigenvalues. Such behavior is a result of the unique topology of the self-intersecting Riemann surface formed near an EP [18].
Fig. 5. Theoretically calculated surfaces of the magnitude (a) and phase (b) of the eigenvalues in (ω, σ) parameter space.
In order to verify the coalescence of the eigenpolarization states in the system into right-handed circular polarization at the EP, the transmission spectra of the VO2-integrated metasurface with σ = 582 S ∕ m are simulated for incident circularly polarized waves, as shown in Fig. 6(a). Here, the subscripts r and l represent right-handed and left-handed circularly polarized wave, respectively. It can be found that two co-polarized transmission curves tll and trr are almost identical to each other while the cross-polarized transmission curves tlr and trl are very different, i.e. The asymmetric transmission phenomenon of circularly polarized light occurs. The response of the VO2-integrated metasurface to circularly polarized light is directionally asymmetric. The cross-polarized transmissions tlr (red curve) at 0.83 THz almost decreases to zero. The corresponding phase profile and derivative of the phase change at the resonant frequency are presented in Figs. 6(b) and 6(c), respectively. A significant phase jump can be observed near the EP, which makes the metasurface suitable for sensing applications. As VO2 is an insulator-to-metal phase change material, the VO2 conductivity can be calculated as a function of temperature [37,38]. Maxwell-Garnett EMT model and Boltzmann function can be used to described the relation between conductivity of VO2 and temperature. The giant phase jump due to the change of VO2 conductivity can be correspondingly used for measurement of the temperature. The dependence of the phase variation on temperature near the EP is illustrated in Fig. 6(c). To demonstrate this phenomenon more clearly, the absolute gradient of the phase change was calculated according to the formula $\Delta \phi = |d\phi /df|$ [39]. Remarkably, a slight perturbation in temperature leads to a significant change in $\Delta \phi $. Such nonlinear changes are very important for highly sensitive temperature sensing, especially in biology [11,18,25].
Fig. 6. (a) Transmission spectra of the VO2-integrated metasurface in circular polarization basis with σ = 582 S ∕ m for incident circularly polarized waves. (b) Phase spectra of tlr for metasurface with different σ. (c) Derivative of the phase change ($\Delta \phi $) spectra when the temperature varies.
In the following, the ability of bilayer metasurface to perform refractive index sensing will be investigated. We consider the case that a thin analyte layer with refractive index n = 1.01 and a thickness of 2.7 µm covers on the top side of the bilayer metasurface and show numerically how it affects the transmission spectra and the eigenpolarization states. Figure 7 shows the comparison between the resonance frequency shift of co-polarization transmission, Δωy and the eigenfrequency splitting, ΔωE, in presence of a sensing layer, when the incident wave is y-polarized. For the refractive index difference of 0.01, the resonant frequency shift is Δωy = 1.5 GHz in Fig. 7(a) and an eigenfrequency splitting is ΔωE = 4.4 GHz in Fig. 7(b), indicating a significant enhancement in sensitivity. In Fig. 7(c), resonance frequency ωy and the eigenfrequencies ωEig.1 and ωEig.2 are presented as a function of the refractive index of the analyte layer. When the increase of refractive index occurs, the eigenfrequency 2 (red dot) decreases significantly while the eigenfrequency 1 (blue dot) remains almost constant. The variation of the resonant frequency ωy nearly exhibits a linear dependence upon the refractive index as shown in Fig. 7(c). The Hamiltonian of a non-Hermitian system with 2 × 2 matrix can be written as follows [10,40,41],
When adding a non-Hermitian perturbation εH1, the eigenvalues of the perturbed Hamiltonian H = $H_0^{EP}$ + εH1 will split by that is nonlinear in the perturbation strength ε, but scales with the square root of ε. Figure 7(d) plots the eigenfrequency splitting of the bilayer structure as a function of the detuning of the resonant frequency. A linear relationship is observed from Fig. 7(d) in the log-log axis (the slope of the curve is 0.618), indicating the presence of a square root behavior with the induced perturbation. For larger perturbations, the slope for the EP sensor is slightly larger than 1/2. [10,22]. The results show that the sensitivity in the proposed reaches a maximum value of 163 GHz RIU−1 µm−1.Fig. 7. (a) Transmission spectra of the metasurface with (black dotted line) and without (cyan solid line) an analyte layer. (b) Transmission spectra of two eigenpolarization states in the bilayer PT symmetry metasurface with (dotted line) and without (solid line) an analyte layer. (c) Resonance frequency ωy and the eigenfrequencies ωEig.1 and ωEig.2 as a function of the refractive index of the analyte layer. (d) Eigenfrequency splitting ΔωE of the bilayer EP structure as a function of the resonance shift Δωy of the transmission spectra. The inset is a log-log plot with a slope of 0.618.
3. Conclusion
In conclusion, we propose a bilayer PT symmetry metasurface intergrated with VO2, and two EP sensing schemes to achieve high sensitivity sensing of temperature and refractive index. By changing the conductivity of VO2, the metasurface undergoes a variation from PT symmetry to PT symmetry breaking and the generation of EP is clearly observed. At the EP, the two eigenpolarization states coalesce. A slight perturbation of temperature leads to a significant change of phase, leading to high-sensitivity sensing of both temperature and refractive index sensors.
Funding
National Natural Science Foundation of China (62275061, 62175049); Natural Science Foundation of Heilongjiang Province (ZD2020F002); Harbin Engineering University (B13015); Fundamental Research Funds for the Central Universities (3072021CFT2501, 3072022CF2505, 3072022TS2509).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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