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Fano resonance-induced high-purity circularly polarized spectra for high-precision refractive index sensing from hybrid resonator-graphene meta-surfaces

Yazheng Hao and Rui Yang

Open Access Open Access

Abstract

We demonstrate the perfect synthesis of terahertz circularly polarized Fano resonant reflecting spectra from hybrid resonator-graphene meta-surfaces for highly sensitive refractive index sensing of the biochemical analyte. Such a hybrid resonator-graphene meta-surface, consisting of periodic multi-node split ring resonators on the top of the grounded polyimide substrate inserted with a monolayer graphene sheet, can perfectly transform the linearly polarized electromagnetic fields into circularly polarized waves. Especially, the greatest polarization purity of the reflecting spectra can readily be obtained at the Fano resonance by tuning the Fermi level of the graphene, thus offering an alternative way to identify the difference between the given test specimens and other analytes with a very close refractive index on the basis of the polarization extinction ratio. The proposed methodology, capable of distinguishing the samples with a difference in the refractive index of ten thousandths, should pave the way for tangible applications of precision detections in biochemical assays with high accuracy.

© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Graphene meta-surfaces have demonstrated the great efficiency in generating Fano resonances for high precision detections of dielectrics, possessing the merits of being accurate, fast and harmless to the human body in the biological sensing of proteins, and simultaneously offering the high-quality and cost-effective detection solutions [1–3]. Generally, the sensitivity of the meta-surface based sensors can be determined by observing the frequency shifts of the resonances varied with different refractive index of the analyte. Su et al. proposed a high-performance Fano resonant sensor using two silicon elliptical cylinders with asymmetric minor axis on silica substrate coated graphene [4]. Zhang et al. designed a high-Q, high-sensitivity, all-dielectric four-resonance sensor using periodical asymmetric paired bars in the near-infrared regime [5]. Liu et al. demonstrated a Fano resonance sensor with high modulation depth and polarization insensitive characteristics [6]. Sun et al. presented a Fano resonant sensor that can independently adjust the absorption mode by examining the time domain signal [7]. These researches, using Fano resonant meta-surface sensors as the first choice for biochemical detections, have paved the way for the quest of tangible applications of more advanced sensing technologies.

However, the present proposals often lack the polarization manipulations of the electromagnetic fields at Fano resonances, and testing the polarization purity of the reflected wave should offer an alternative way to identify the difference between the given test specimens and other analytes with very close refractive index on the basis of polarization extinction ratio (PER) [8]. Different from determining the refractive index of the analyte on the basis of the frequency offset, the PER method compares the greatest polarization purity of the given test specimens. As a result, the PER method often depends on the polarization conversion, where the interactions between the incidence and the proposed testing devices will offer multidimensional analysis with much sharper spectra possessing less aliasing effects and huge distinguishability of refractive index, other than solely the shifted frequency. On the other hand, the reconfigurable characteristics of graphene meta-surface also enable the modulations of the polarization states of the reflected waves to achieve the high-purity circularly polarized (CP) fields with efficient linear-to-circular polarization conversions. One can thus use the modulation effect of the graphene layer by tuning the Fermi levels to achieve the maximum polarization purity of the reflections from the given test specimens. Based on these consideration, we demonstrate the high-precision refractive index sensing from dynamically modulated Fano resonances of CP waves using hybrid resonator-graphene meta-surfaces. We will show that the proposed strategy can distinguish the samples with a difference in refractive index of ten thousandths by comparing the greatest polarization purity of the reflecting spectra at Fano resonances.

2. Modeling and simulation results

Figure 1 schematically demonstrates interactions between the electromagnetic fields and the proposed hybrid resonator-graphene meta-surfaces, where the biochemical analyte can be loaded over the meta-surface. Such a hybrid resonator-graphene meta-surface consists of periodic multi-node split ring resonators (SRRs) on the top of the grounded polyimide ($\varepsilon _{r}$ = 3+0.05j) substrate inserted with a monolayer graphene sheet. The detailed geometric parameters are h = 5 $\mu$m, P = 36 $\mu$m, w = 1.25 $\mu$m, g = 1.25 $\mu$m, L = 20 $\mu$m, ${d}_{1}$ = 5 $\mu$m, ${d}_{2}$ = 4 $\mu$m. When the x-polarized wave is casting over the meta-surface, Fano resonant reflecting spectrum will be created by the multi-node-SRR array. In the meanwhile, the chirality of the resonators also enables the polarization conversion of the reflected waves [9,10], transforming x-polarized incidence into the CP reflections. The greatest polarization purity of the reflecting spectra can readily be obtained at the Fano resonance though tuning the Fermi level of the graphene. The permittivity ($\varepsilon _g$) of graphene is depending on the surface conductivity function ($\sigma _g$):

$$\varepsilon _{g} (\omega )=1+\frac{i\sigma_{g}(\omega ) }{\varepsilon _{0}\omega \Delta }$$
where $\varepsilon _0$ is the vacuum permittivity and $\Delta$ = 1 nm is the thickness of the mono-layer graphene sheet. Graphene conductivity can be obtained from the Kubo formula [11–13]
$$\sigma _{g} (\omega )=\frac{ie_{0}^{2}k_{B}T }{\pi \hbar (\hbar \omega +i\Gamma )} (\frac{\mu _{c}}{k_{B}T }+2\mathrm{ln}[\mathrm{exp}(\frac{-\mu _{c}}{k_{B}T }) +1 ])$$
where $\mu _c$ is the chemical potential of graphene, $\Gamma =-(e_{0}\hbar v_{f}^{2} )/(\mu \mu _{c} )$ is the damping coefficient, $v_{f} = 10^6 \mathrm {m/s}$ is the Fermi velocity, $\mu$ is the electron mobility, $e_{0}$ is the electronic unit-charge, $k_B$ is the Boltzmann constant, and $\hbar$ is the reduced Plank’s constant. The temperature T = 300 K is kept constantly throughout the whole investigation.

 figure: Fig. 1.

Fig. 1. The hybrid resonator-graphene meta-surfaces interacting with an x-polarized electromagnetic field and creating the highest-purity CP reflection. (a) Schematic view and the geometric dimensions of the meta-atom as well as the detailed layered structures. (b) The flow chart of the high-precision sensing.

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The relationship between the incidence and reflected fields of hybrid resonator-graphene meta-surfaces can thus be represented using the Jones matrix with

$$\begin{pmatrix} \textbf{E} _{xr} \\ \textbf{E}_{yr} \end{pmatrix}=\begin{pmatrix} \textbf{R} _{xx} & \textbf{R} _{xy}\\ \textbf{R} _{yx} & \textbf{R} _{yy} \end{pmatrix}\begin{pmatrix} \textbf{E} _{xi}\\ \textbf{E} _{yi} \end{pmatrix}$$
where the incident $(\textbf {E} _{xi} , \textbf {E} _{yi})$ and reflected $(\textbf {E} _{xr} , \textbf {E} _{yr})$ fields are related by the reflection coefficient $(\textbf {R} _{xx} , \textbf {R} _{yx} , \textbf {R} _{xy} , \textbf {R} _{yy})$ and the coefficient $\textbf {R} _{yy}$ and $\textbf {R} _{xy}$ can be ignored when the incidence is an x-polarized wave. We can use PER to quantitatively define the polarization purity of the CP wave converted from the x-polarized incidence in the form of
$$\mathrm{PER(dB)=20log_{10} } \frac{|\textbf{R} _{xx}+i\textbf{R} _{yx}|}{|\textbf{R} _{xx}-i\textbf{R} _{yx}|}$$

Full-wave simulations (CST Studio Suite) are carried out to verify the proposed hybrid resonator-graphene meta-surfaces with an x-polarized incidence along the -z direction. Figure 2 demonstrates the reflections and the PER of the proposed hybrid resonator-graphene meta-surfaces, where we mimic the interactions between the electromagnetic fields and the proposed hybrid resonator-graphene meta-surface using the Floquet mode analysis as shown in Fig. 2(a) with boundary conditions virtually repeating the modeled structure periodically in x and y directions. The reflection spectra of the meta-surface and phase difference ($\Delta \varphi =\Phi _{yx} -\Phi _{xx}$) of the co- and cross- components are shown in Fig. 2(b) with 0.49 eV Fermi energy imposed over the graphene sheet. It can be found that the $\Delta \varphi$ is around $\pm 90^{\circ }$ between 4 THz and 6 THz, performing the linear-to-circular polarization conversion. We use the ellipticity $\chi$ derived from Stokes Parameters to qualify the polarization purity of the reflected wave [14,15].

$$\chi=\frac{2|\textbf{R} _{xx}||\textbf{R} _{yx}|\mathrm{sin}(\Delta \varphi) }{|\textbf{R} _{xx}|^{2}+|\textbf{R} _{yx} |^{2} }$$
$\chi =\pm 1$ represents the prefect conversion of right-handed CP (RHCP) (−1) and left-handed CP (LHCP) (+1) under the condition of $|\textbf {R} _{xx}|=|\textbf {R} _{yx}|$ and $\Delta \varphi =90^{\circ }$. Figure 2(c) indicates the frequency range that the meta-surface transforms the linearly polarized (LP) incidence into the CP fields. It can be found that the LHCP reflection spectrum appears in the range of 4.52-4.81 THz, and the RHCP spectrum arises in the range of 5.15-5.28 THz. We can also observe that pretty high polarization purity of LHCP reflections can be achieved from 4.67 THz to 4.72 THz possessing $\chi =1$ with the corresponding PER value of −72.24 dB at 4.698 THz. Figure 2(d) plots the five normalized scattering multipoles close to the PER peak, where the multipole analysis is performed to evaluate the role of Fano resonance in the generation of such a high polarization purity LHCP wave. The electric dipole P, magnetic dipole M, electric quadrupole $\textbf {Q}_{e}$, magnetic quadrupole $\textbf {Q}_{m}$ and toroidal T can be calculated as [16]
$$\textbf{P}=\frac{1}{i\omega } \int \textbf{j}d^{3} r$$
$$\textbf{M}=\frac{1}{2c } \int (\textbf{r}\times \textbf{j })d^{3} r$$
$$\textbf{Q}_{e}=\frac{1}{i\omega } \int [r_{\alpha } j_{\beta } +r_{\beta } j_{\alpha } -\frac{2}{3}( \textbf{r} \cdot \textbf{j})] d^{3}r$$
$$\textbf{Q}_{m}=\frac{1}{3c } \int [(r\times j)_{\alpha }r_{\beta } +(r\times j)_{\beta }r_{\beta } ] d^{3}r$$
$$\textbf{T}=\frac{1}{10c } \int [(\textbf{r}\cdot \textbf{j})\textbf{r}-2r^{2}\textbf{j} ] d^{3}r$$
where j is the current density and r is the displacement vector from any point to the origin. The total scattered power of each multipole in the far field from all unit cells of the graphene meta-surface can be expressed as [17]
$$I= \frac{2\omega ^{4} }{3c^{3} } |\textbf{P}|^{2} +\frac{2\omega ^{4} }{3c^{3} } |\textbf{M}|^{2} + \frac{2\omega ^{6} }{3c^{5} } |\textbf{T}|^{2}+ \frac{\omega ^{6} }{5c^{5} } \textbf{Q}_{e}\cdot \textbf{Q}_{e} +\frac{\omega ^{6} }{20c^{5} } \textbf{Q}_{m}\cdot \textbf{Q}_{m}$$
We can observe that the electric dipole P (the bright mode) with a large value in the full frequency range is suppressed, and the strong toroidal T (the dark mode) is excited. Such a strong toroidal resonance interferes destructively with the electric dipole, possessing the characteristic of the Fano resonance. In other words, the PER peak representing high polarization purity is the result of the Fano resonance generated by the destructive coupling of the poles [18,19].

 figure: Fig. 2.

Fig. 2. The electromagnetic responses of the proposed hybrid resonator-graphene meta-surface. (a) The Floquet mode analysis. (b) Reflection spectrum and the phase difference at the Fermi level of 0.49 eV. (c) Linear-to-circular conversion polarization purity in 4-6THz. (d) The expansion of multipoles around PER peak. The color maps of normalized electric field (e) and the vector distributions of surface current (f) and magnetic field (g) for the corresponding frequency points of the electric dipole at $P_{1}$, PER peak at $P_{2}$, toroidal modes at $P_{3}$.

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Figures 2(e) to (g) continue to demonstrate the multipole analysis of the interaction between the hybrid resonator-graphene meta-surface and the x-polarized incidence by demonstrating the distribution of the normalized electric field $E_{z}$ on the x-y plane and also the current over the multi-node SRR array, as well as the magnetic field on the x-z plane. For the comparison purpose, we choose three points of $P_{1}$ = 4.2 THz dominated by electric dipole, $P_{2}$ = 4.698 THz at the PER peak, and $P_{3}$ = 4.96 THz dominated by toroidal mode for the demonstration. It can be seen from the x-z plane that the magnetic field directions at the intermediate gap of the multi-node SRR at $P_{1}$ are all in the -y direction, and the current over the multi-node SRR can thus be equivalent to an electric dipole when other modes are weaker. However, the partial ring current on the multi-node SRR at $P_{2}$ will lead to the generation of magnetic fields in $\pm z$ directions at the gap. The magnetic field is confined inside the multi-node SRR gap, with the magnetic ring near the center in the x-z plane creating the toroidal mode. On the other hand, the opposite current on the multi-node SRR is more pronounced at $P_{3}$, while the magnetic fields in the x-z plane distribute almost linearly oriented along the z axis with opposite directions as the distinct feature of the toroidal mode [20–22].

The meta-surface can be regarded as a multi-layer cascaded structure as shown in Fig. 3(a), where each layer is equivalent to a generalized two-port network. The layer-1 is composed of multi-node SRR array and a half-layer-thick polyimide substrate. The layer-2 refers to a mono-layer graphene sheet embedded in the middle of the polyimide to control the electromagnetic fields between the multi-node SRR array and the ground. The layer-3 is another half-layer substrate and the ground. Using generalized signal flow analysis, the reflection matrix of the proposed hybrid graphene-meta-surface can be expressed as follows

$$\begin{aligned} \textbf{R}_{in}&=\begin{bmatrix} \widetilde{R}_{xx} & \widetilde{R}_{xy}\\ \widetilde{R}_{yx} & \widetilde{R}_{yy} \end{bmatrix}\\ &=\textbf{S}_{11}^{1}+\textbf{S}_{12}^{1}(\textbf{S}_{11}^{2} +\textbf{S}_{12}^{2} \Gamma _{3}(\textbf{I }-\textbf{S}_{22}^{2}\Gamma _{3})^{{-}1} \textbf{S}_{21}^{2})\\ &\cdot (\textbf{I }- \textbf{S}_{22}^{1}(\textbf{S}_{11}^{2}+\textbf{S}_{12}^{2} \Gamma_{3}(\textbf{I }-\textbf{S}_{22}^{2}\Gamma_{3})^{{-}1} \textbf{S}_{21}^{2}))^{{-}1} \textbf{S}_{21}^{1} \end{aligned}$$
where $\textbf {S}_{ij}^{1/2}$ is the scattering matrix associated with the outgoing wave vector $\textbf {a}_{j}^{1/2}$ and the incoming wave vector $\textbf {b}_{i}^{1/2}$, satisfying $\textbf {b}_{i}^{1/2}=\textbf {S}_{ij}^{1/2}$ $\textbf {a}_{j}^{1/2}$. The scattering matrix of the layer-1 can be expressed as
$$\begin{aligned}&\textbf{S}_{11}^{1}=\begin{bmatrix} \widetilde{R}_{cox}^{11} & \widetilde{R}_{cr}^{11}\\ \widetilde{R}_{cr}^{11} & \widetilde{R}_{coy}^{11} \end{bmatrix}\quad \textbf{S}_{12}^{1}=\begin{bmatrix} \widetilde{T}_{cox}^{12} & \widetilde{T}_{cr}^{12}\\ \widetilde{T}_{cr}^{12} & \widetilde{T}_{coy}^{12} \end{bmatrix}\\ &\textbf{S}_{21}^{1}=\begin{bmatrix} \widetilde{T}_{cox}^{11} & \widetilde{T}_{cr}^{11}\\ \widetilde{T}_{cr}^{11} & \widetilde{T}_{coy}^{11} \end{bmatrix}\quad\textbf{S}_{22}^{1}=\begin{bmatrix} \widetilde{R}_{cox}^{12} & \widetilde{R}_{cr}^{12}\\ \widetilde{R}_{cr}^{12} & \widetilde{R}_{coy}^{12} \end{bmatrix} \end{aligned}$$
where the co- and cross-reflection/transmission coefficients excited by port-1 are denoted as $\widetilde {R}_{co}^{11}/\widetilde {T}_{co}^{11}$ and $\widetilde {R}_{cr}^{11}/\widetilde {T}_{cr}^{11}$, the co- and cross-reflection/transmission coefficients excited by port-2 are denoted as $\widetilde {R}_{co}^{12}/\widetilde {T}_{co}^{12}$ and $\widetilde {R}_{cr}^{12}/\widetilde {T}_{cr}^{12}$. Based on the feature of multi-node SRR array being non-axisymmetric, $\widetilde {R}_{cox}^{11/12}\ne \widetilde {R}_{coy}^{11/12}$, $\widetilde {T}_{cox}^{11/12}\ne \widetilde {T}_{coy}^{11/12}$. We can observe that layer-1 decomposes the incidence into reflected/transmitting co- and cross-LP components from Figs. 3(b) to (e). The reflection/transmission spectra of the two ports are different due to the asymmetry property of the air-SRRs-polyimide of layer-1 in the -z direction.

 figure: Fig. 3.

Fig. 3. Multi-level cascaded generalized two-port network representation of the hybrid resonator-graphene meta-surfaces. (a) The generalized signal flow analysis. The amplitude (b) and phase (c) of co-reflection/transmission coefficient $\widetilde {R}_{cox}^{11}/\widetilde {T}_{cox}^{11}$ and cross-reflection/ transmission coefficient $\widetilde {R}_{cr}^{11}/\widetilde {T}_{cr}^{11}$. The amplitude (d) and phase (e) of co-reflection/transmission coefficient $\widetilde {R}_{cox}^{12}/\widetilde {T}_{cox}^{12}$ and cross-reflection/ transmission coefficient $\widetilde {R}_{cr}^{12}/\widetilde {T}_{cr}^{12}$. (f) Reflection/transmission coefficient $\widetilde {R}^{2}/\widetilde {T}^{2}$ of the graphene layer when imposed with 0.49 eV Fermi level. (g) Theoretical calculation result of the total network reflection coefficients $R_{xx/yx}$ and phase difference $\Delta \varphi$ when imposed with 0.49 eV Fermi level.

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The scattering matrix of Layer-2 for a mono-layer graphene sheet sandwiched in a polyimide substrate can be expressed as

$$\begin{aligned}\textbf{S}_{11}^{2}=\textbf{S}_{22}^{2}=\begin{bmatrix} \widetilde{R}^{G} & \\ & \widetilde{R}^{G} \end{bmatrix}\\ \textbf{S}_{21}^{2}=\textbf{S}_{12}^{2}=\begin{bmatrix} \widetilde{T}^{G} & \\ & \widetilde{T}^{G} \end{bmatrix} \end{aligned}$$
where the reflection coefficient $\widetilde {R}^{G}$ and transmission coefficient $\widetilde {T}^{G}$ of graphene sheet follow the expressions [23]
$$\widetilde{R}^{G}={-}\frac{\sigma _{g}\sqrt{\mu _{0}/(\varepsilon _{r}\varepsilon _{0} ) }/2 }{1+\sigma _{g}\sqrt{\mu _{0}/(\varepsilon _{r}\varepsilon _{0} ) }/2}$$
$$\widetilde{T}^{G}={-}\frac{1 }{1+\sigma _{g}\sqrt{\mu _{0}/(\varepsilon _{r}\varepsilon _{0} ) }/2}$$
We can find in Fig. 3(f) that the graphene layer can be equivalent to a partially transmissive sheet when imposed with 0.49 eV Fermi level, and changing the Fermi level will influence the interactions between the multi-node SRR array and the ground, offering the opportunities to obtain high-purity CP spectra through Fermi level modulation.

The layer-3 of the ground is equivalent to a short circuit terminal in the transmission line with the reflection coefficient of

$$\Gamma_{3}=[1-\frac{2}{1+i\mathrm{tan}(\sqrt{\varepsilon _{r}}k_{0} h/2 ) } ]\cdot \begin{bmatrix} 1 & \\ & 1 \end{bmatrix}$$

The theoretical values from Eq. (12) in Fig. 3(g) of the amplitude and phase of the scattering spectrum are basically agree with the full-wave simulation, and we can conclude that the layer-1 and layer-3 only depend on their structural composition, and the response of the entire system can be manipulated through the dynamic modulation of the Fermi level so as to achieve the high polarization purity with efficient linear-to-circular conversion.

Figures 4(a) and (b) demonstrates the variations of PER curve with different Fermi levels imposed over the graphene sheet. We can clearly observe that Fermi level is not only manifested in the blue shift of the PER spectrum, the polarization purity of the linear-to-circular polarization conversion is also significantly changed. The sharpest PER curve at 0.49 eV with the highest polarization purity paves the way for high-precision refractive index sensing. The equivalent circuit model of the SRR is inserted in Fig. 4(b) [24]. $C_{analyte}$ represents the equivalent analyte capacitance and will become larger with the increase of the refractive index. $L_{SRR1}$, $L_{SRR2}$ are controlled by the branches of SRR and their values will gradually increase with the width and the length of the branch. $C_{SRR}$ is determined by the two gaps of the SRR, and the narrower gap will lead to more charge accumulation, resulting in an increase in $C_{SRR}$. The total impact of $L_{SRR1}$, $L_{SRR2}$ can be described by $L_{SRR}$ and as long as any branch becomes wider or longer, $L_{SRR}$ will definitely increase. Figures 4(c) to (f) show the influences of different structural parameters L, ${d}_{1}$, g, w of the multi-node SRR on the PER curve, respectively. As we can observed, the PER peak will have red-shifts with the increase of L and ${d}_{1}$ because $L_{SRR}$ increases due to the elongation of the SRR main arms and broadening of the wide branches along the x direction. On the other hand, it will experience blue-shifts as g becomes bigger because the gaps become wider and the strong electric field in the gap weakens, leading to a decrease in $C_{SRR}$. In addition, the variation of w will not significantly influence the operating frequency of the PER peak, the reason is that the widening of the main branches of SRR lead to the increase of ${L_{SRR}}$, the charge aggregation at the gap is simultaneously weakened because the gap is also widened, ${C_{SRR}}$ is almost inversely proportional to ${L_{SRR}}$. As a result, we selected L = 20 $\mu$m, ${d}_{1}$ = 5 $\mu$m, g = 1.25 $\mu$m and w = 1.25 $\mu$m from all geometric parameters to ensure the maximum polarization purity for the precise sensing purpose.

 figure: Fig. 4.

Fig. 4. The variations of PER curve with different Fermi levels imposed over the graphene sheet and different geometric parameters of the multi-node SRR. The varied Fermi levels in the range of 0-1 eV (a) and 0.46-0.52 eV (b). The embedded picture refers to the equivalent circuit model of the multi-node SRR. The varied geometric parameters of L (c), ${d}_{1}$ (d), g (e) and w (f).

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3. Sensing applications

When loaded a superstrate of the analyte with thickness of ${h}_{1}$ as shown in Fig. 5(a), the proposed hybrid resonator-graphene meta-surface can readily start the applications. The temperature and other conventional variables, such as Polyimide substrate thickness and SRR geometry parameters, are kept constantly when performing the test. Clearly, different ${h}_{1}$ will lead to the variations of sensitivity ($S=\Delta f/\Delta n$) from the reflection spectrum as illustrated in Fig. 5(b). We can observe the sensitivity increases continuously as the ${h}_{1}$ increases, but the uptrend of the curve is gradually slowing down. As a result, we take ${h}_{1}$ = 3 $\mu$m in the following two specific examples of biochemical detections to verify the great detection accuracy of our proposed hybrid resonator-graphene meta-surface, where a micropipette is required to control the total volume of the analyte and keeps ${h}_{1}$ at 3 $\mu$m with the assistance of an optical 3D surface profiler.

 figure: Fig. 5.

Fig. 5. The sensing performance of the graphene meta-surfaces. (a) Numerical simulation of analyte solutions. (b) Relationship between analyte thickness and sensitivity. Detecting anhydrous ethanol from high-concentration ethanol using graphene-modulated PER method (c) and frequency shift method (d). DNA amplification detection from graphene-modulated PER method (e) and frequency shift method (f).

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Case-1: Anhydrous alcohol has important applications in solvents, cosmetics, and fuels nowadays. However, distinguishing anhydrous ethanol from ultra-high-concentration ethanol often needs very precision sensing with a 1/10,000 refractive index discrimination capacity. The alcohol’s refractive index is 1.3604 with 97$\%$ concentration, 1.3599 with 98$\%$ concentration, and 1.3593 for anhydrous ethanol with more than 99.5$\%$ concentration [25]. In this way, we can adjust the graphene Fermi level imposed over the graphene meta-surface to 0.635 eV so that the PER curve of anhydrous ethanol reached the sharpest peak as shown in Fig. 5(c). On the contrary, if we still depend on the method of observing the frequency shift of the resonance peak, we will fail to tell the difference as the reflection spectra overlap in a wide range as shown in Fig. 5(d).

Case-2: DNA amplification method has been recognized as an alternative to polymerase chain reaction for virus detection nowadays [26–28]. Microcavity inline Mach-Zehnder interferometer ($\mathrm {\mu }$IMZI) offers a direct, real-time and label-free isothermal DNA amplification monitoring method with the advantages of high sensitivity and high linearity in a short time [29]. However, the detection principle of the DNA amplification method is actually the change of the refractive index of the solution induced by the synthesis of new DNA strands. Figure 5(e) demonstrates the process of the DNA amplification, where the refractive index of the solution changes from 1.3333 to 1.3343 with the synthesis of new DNA strands. Given 0.640 eV Fermi energy imposed over the graphene, the polarization purity reaches the peak with the refractive index of 1.3333. It can be seen that the peak value of PER decreases continuously as the refractive index increases, and the corresponding frequency of the peak is red-shifted compared with the peak frequency corresponding to n =1.3333. Compared with $\mathrm {\mu }$IMZI sensing in Fig. 5(f), graphene-modulated PER sensing is more sensitive and accurate when the refractive indices are very close.

In fact, the proposed hybrid graphene meta-surface has excellent performance not only in the refractive index range of 1.3-1.4 given in the above two examples, but can also satisfy the detection accuracy over the refractive index range of 1-1.6 by setting the proper Fermi level of graphene. A refractive index distinguishability of 1/10,000 can be achieved when PER < −50 dB. Table 1 demonstrates when the refractive index is changed from 1 to 1.6 in 0.1 intervals, the maximum polarization purity can always be found to meet the 1/10,000 refractive index distinguishability requirement through the change of graphene Fermi level. This refractive index range includes many common biochemical solutions, which makes our proposal universal.

Tables Icon

Table 1. The optimized Fermi levels imposed over the graphene to achieve the maximum peak of PER curve when testing different refractive index.

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Table 2 presents a detailed comparison between the previous studies on meta-surface sensors and our proposed design. Reference [8] proposed highly sensitive graphene plasmonic meta-surfaces to characterize refractive index of viruses by detecting the polarization state of the reflected electric fields spectrum at 1-2 THz in the refractive index range of 1-1.5 with a 1/1,000 distinguishability. Reference [30] proposed a graphene meta-surface-based cross polarization converter operating for detecting biomolecules such as SARS-CoV-2 virus at 1-3.5 THz in the refractive index range of 1.29-1.37 with a 1/100 distinguishability. Reference [31] investigated a chiral meta-surface sensor filled with ferromagnetic nanofluids for LHCP and RHCP sensing of nanoparticle concentration at 0.5-1 THz in the refractive index range of 1-1.9 with a 1/100 distinguishability. Based on coupled mode theory, Ref. [32] proposed a graphene meta-surface composed of a continuous graphene strip and a truncated graphene strip for sensing at 3-7 THz in the refractive index range of 1-1.62 with a 1/100 distinguishability. Reference [33] proposed a metallic toroidal dipole meta-surface for sensing applications at 2.5-5 THz in the refractive index range of 1-2 with a 1/100 distinguishability. Reference [34] demonstrated high-Q Fano terahertz resonance in all-dielectric meta-surface for refractive-index sensing at 1.65-1.95 THz in the refractive index range of 1-2 with a 1/10 distinguishability. Reference [35] proposed a plasmon induced tunable meta-surface for multiband superabsorption and terahertz sensing at 1.25-1.75 THz in the refractive index range of 1-2 with a 1/10 distinguishability. Different from the present sensors above achieving the highly sensing characteristics on the basis of observing the frequency shifts of the resonances varied with different refractive index of the analyte, our proposal employs the Fano resonances induced high-purity CP spectra for high-precision refractive index sensing on the basis of PER analysis. Compared with these literatures, our design should achieve much more accurate sensing applications with the great capacity of refractive index distinguishability of 1/10,000 in refractive index range of 1-1.6.

Tables Icon

Table 2. Comparisons of the meta-surfaces used as precision sensors.

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4. Conclusions

In conclusion, we have demonstrated the high-precision refractive index sensing through Fano resonances induced high-purity CP spectra using hybrid resonator-graphene meta-surfaces. The greatest polarization purity of the reflecting spectra with linear-to-circular polarization conversion can readily be obtained at the Fano resonance though tuning the Fermi level of the graphene, and two cases of biochemical detections have validated the advantages our proposal of using the PER peak variations compared with the observations of the frequency shifts of the resonances, thus should offer promising prospects in the high-precision detection of biochemical solutions.

Funding

National Natural Science Foundation of China (61301072, 61671344).

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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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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Figures (5)
Fig. 1.
Fig. 1. The hybrid resonator-graphene meta-surfaces interacting with an x-polarized electromagnetic field and creating the highest-purity CP reflection. (a) Schematic view and the geometric dimensions of the meta-atom as well as the detailed layered structures. (b) The flow chart of the high-precision sensing.

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Fig. 2.
Fig. 2. The electromagnetic responses of the proposed hybrid resonator-graphene meta-surface. (a) The Floquet mode analysis. (b) Reflection spectrum and the phase difference at the Fermi level of 0.49 eV. (c) Linear-to-circular conversion polarization purity in 4-6THz. (d) The expansion of multipoles around PER peak. The color maps of normalized electric field (e) and the vector distributions of surface current (f) and magnetic field (g) for the corresponding frequency points of the electric dipole at $P_{1}$, PER peak at $P_{2}$, toroidal modes at $P_{3}$.

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Fig. 3.
Fig. 3. Multi-level cascaded generalized two-port network representation of the hybrid resonator-graphene meta-surfaces. (a) The generalized signal flow analysis. The amplitude (b) and phase (c) of co-reflection/transmission coefficient $\widetilde {R}_{cox}^{11}/\widetilde {T}_{cox}^{11}$ and cross-reflection/ transmission coefficient $\widetilde {R}_{cr}^{11}/\widetilde {T}_{cr}^{11}$. The amplitude (d) and phase (e) of co-reflection/transmission coefficient $\widetilde {R}_{cox}^{12}/\widetilde {T}_{cox}^{12}$ and cross-reflection/ transmission coefficient $\widetilde {R}_{cr}^{12}/\widetilde {T}_{cr}^{12}$. (f) Reflection/transmission coefficient $\widetilde {R}^{2}/\widetilde {T}^{2}$ of the graphene layer when imposed with 0.49 eV Fermi level. (g) Theoretical calculation result of the total network reflection coefficients $R_{xx/yx}$ and phase difference $\Delta \varphi$ when imposed with 0.49 eV Fermi level.

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Fig. 4.
Fig. 4. The variations of PER curve with different Fermi levels imposed over the graphene sheet and different geometric parameters of the multi-node SRR. The varied Fermi levels in the range of 0-1 eV (a) and 0.46-0.52 eV (b). The embedded picture refers to the equivalent circuit model of the multi-node SRR. The varied geometric parameters of L (c), ${d}_{1}$ (d), g (e) and w (f).

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Fig. 5.
Fig. 5. The sensing performance of the graphene meta-surfaces. (a) Numerical simulation of analyte solutions. (b) Relationship between analyte thickness and sensitivity. Detecting anhydrous ethanol from high-concentration ethanol using graphene-modulated PER method (c) and frequency shift method (d). DNA amplification detection from graphene-modulated PER method (e) and frequency shift method (f).

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Tables (2)

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Table 1. The optimized Fermi levels imposed over the graphene to achieve the maximum peak of PER curve when testing different refractive index.

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Table 2. Comparisons of the meta-surfaces used as precision sensors.

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Equations (17)

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ε g ( ω ) = 1 + i σ g ( ω ) ε 0 ω Δ
σ g ( ω ) = i e 0 2 k B T π ℏ ( ℏ ω + i Γ ) ( μ c k B T + 2 l n [ e x p ( − μ c k B T ) + 1 ] )
( E x r E y r ) = ( R x x R x y R y x R y y ) ( E x i E y i )
P E R ( d B ) = 20 l o g 10 | R x x + i R y x | | R x x − i R y x |
χ = 2 | R x x | | R y x | s i n ( Δ φ ) | R x x | 2 + | R y x | 2
P = 1 i ω ∫ j d 3 r
M = 1 2 c ∫ ( r × j  ) d 3 r
Q e = 1 i ω ∫ [ r α j β + r β j α − 2 3 ( r ⋅ j ) ] d 3 r
Q m = 1 3 c ∫ [ ( r × j ) α r β + ( r × j ) β r β ] d 3 r
T = 1 10 c ∫ [ ( r ⋅ j ) r − 2 r 2 j ] d 3 r
I = 2 ω 4 3 c 3 | P | 2 + 2 ω 4 3 c 3 | M | 2 + 2 ω 6 3 c 5 | T | 2 + ω 6 5 c 5 Q e ⋅ Q e + ω 6 20 c 5 Q m ⋅ Q m
R i n = [ R ~ x x R ~ x y R ~ y x R ~ y y ] = S 11 1 + S 12 1 ( S 11 2 + S 12 2 Γ 3 ( I  − S 22 2 Γ 3 ) − 1 S 21 2 ) ⋅ ( I  − S 22 1 ( S 11 2 + S 12 2 Γ 3 ( I  − S 22 2 Γ 3 ) − 1 S 21 2 ) ) − 1 S 21 1
S 11 1 = [ R ~ c o x 11 R ~ c r 11 R ~ c r 11 R ~ c o y 11 ] S 12 1 = [ T ~ c o x 12 T ~ c r 12 T ~ c r 12 T ~ c o y 12 ] S 21 1 = [ T ~ c o x 11 T ~ c r 11 T ~ c r 11 T ~ c o y 11 ] S 22 1 = [ R ~ c o x 12 R ~ c r 12 R ~ c r 12 R ~ c o y 12 ]
S 11 2 = S 22 2 = [ R ~ G R ~ G ] S 21 2 = S 12 2 = [ T ~ G T ~ G ]
R ~ G = − σ g μ 0 / ( ε r ε 0 ) / 2 1 + σ g μ 0 / ( ε r ε 0 ) / 2
T ~ G = − 1 1 + σ g μ 0 / ( ε r ε 0 ) / 2
Γ 3 = [ 1 − 2 1 + i t a n ( ε r k 0 h / 2 ) ] ⋅ [ 1 1 ]
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