1. Introduction

The terahertz (THz) sensing has been a hot topic in recent years due to their significant scientific and technological potential application in various fields, such as non-invasive sensing, biomedicine, atmospheric environment monitoring and so on [1–5]. Among all the sensing technologies, the emerging metamaterials [6–11] represent a revolutionary technology to produce sensitive sensors in the THz region by utilizing periodically-arranged subwavelength planar structures on a dielectric substrate.

As one of the simplest planar mesometrial structures, split-ring-resonator (SRR) is widely used to demonstrate the exotic characteristics of metamaterials in controlling the electromagnetic fields. For the ultrasensitive THz sensing applications, the achievement of high quality factor (Q) resonance of SRR is of particular importance. When the quality factor Q of SRR increases, the confinement of electromagnetic field becomes stronger, leading to an enhanced interaction between electromagnetic field and the specific analyte. However, the quality factor of a conventional SRR resonance is quite low due to its high radiative and Ohmic losses at THz frequency, which limits its sensing performance to detect small frequency shifts [12]. In order to improve the quality factor of SRR resonance, the Fano-like resonance with sharp and asymmetric lineshape [13] has been introduced by employing asymmetrical split-ring resonator (ASR) structure [14–19] or by inducing the coherent interference between a broad resonance and a narrow resonance [20–23]. In addition, high-Q resonances can also be excited by constructing different coupling configurations among the meta-molecules that consists of a group of closely spaced planar SRRs in a supercell [24].

In this paper, we put forward a new type of high-Q Fano-like resonance with a SRR-dimer structure consisting of two successive layer of identical SRR arrays separated by a thin dielectric spacer shown in Fig. 1. At normal incidence, when the electric-dipolar even-eigen mode (n = 2) of SRRs is excited [27], two apparent resonances ω⁻ and ω⁺ are observed. Around the lower-frequency resonant point ω⁻, a narrow Fano-like lineshape is present with more than twenty times larger Q factor. This narrow linewidth resonance is counterintuitive because electric-dipole resonance mode (n = 2) of SRR has low Q factor due to its highly radiative nature [27]. Full wave numerical simulations were performed to obtain the surface current and electromagnetic field distributions at the relevant resonances in order to gain insight into the coupling mechanisms and explain the narrow linewidth resonance features. We also utilize the quantum optics formula to qualitatively understand the origin of this high-Q resonance, which could benefit future works on high-Q Fano-like resonance generation. To investigate the sensing performance of the SRR-dimmer structure, we calculated the sensitivities of ω⁻ and ω⁺ resonances by varying the thickness as well as the refractive index of the analyte layer at THz frequency. We found the figure of merit (FOM) of the Fano-like resonance is about ten times larger than that of the ordinary electric dipole resonance. This high quality Fano-like resonance of the SRR-dimer structure is an alternative effort to enhance the sensitivity of SRR devices and offer another important sensing platform at THz frequency.

 

Fig. 1 Geometry of bilayer SRR-dimer structure with dimensions l = 60 μm, w = 6μm, g = 3μm. The periodicity for the array is fixed at 75μm and the spacing is denoted as t.

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2. Structure and resonance properties

Figure 1 illustrates the designed bilayer metallic SRR-dimer structure together with their geometric parameters. Each square SRR element in the dimer has a side length of l = 60 μm, a width of w = 6 μm, a capacitive gap of g = 3 μm, and the periodicity of SRR array is fixed at p = 75 μm. The planar SRR arrays are patterned as 200 nm aluminium layers on both sides of a polyimide spacer layer (refractive index of 1.6 + 0.02i from [25, 26]) with thickness of t. The unit cell consists of two vertically spatially separated SRRs with identical geometry (same constitution) as well as the same orientation (same configuration).

For the numerical simulation (performed using COMSOL Multiphysics), a electromagnetic field excitation propagating normally to the xy plane and polarized along the x direction is assumed. Floquet periodic boundary conditions along the x and y directions are employed to the unit cell and perfect matched layers (PMLs) are set at the ends of the truncated computational domain in the z direction. Considering experimental demonstration, we chose metallic material aluminum due to its high conductivity at THz regime as well as the stability of the sample. Electromagnetic properties of aluminum in the THz region is described by the Drude model ${∊}_m=1-{\omega }_p^2/\left({\omega }^2+i\omega \mathrm{\Gamma }\right)$ with plasma frequency ω_p/2π = 3606 THz and collision frequency Γ/2π = 19.6 THz, respectively [28].

In order to understand the electromagnetic response of the SRR-dimer structure, at first we study the resonance behavior of a single layer SRR with the same geometrical parameters. Figure 2(a) shows the simulated transmission spectra of a single SRR on the polyimide substrate with different spacer thicknesses t = 5, 10, 15, 20, 25 μm. The gradual red shifting of the resonance frequency is observed while increasing the thickness of polyimide substrate, which is mainly caused by the increase in the capacitance of the SRR split gap. The electric dipole resonance frequency for substrate thickness 5, 10, 15, 20 and 25 μm is 1.415, 1.345, 1.310, 1.295 and 1.290 THz, respectively. Saturation of this red-shifting behavior emerges when thickness t approaches 15 μm indicating the penetration depth of electromagnetic field into the substrate at the resonance frequency is around 15 μm.

 

Fig. 2 (a) The transmission spectra of a single SRR under even-mode (n = 2) excitation with different spacer thickness. (b) Electric field strength and surface current density distributions (red arrows) at resonant frequency (skyblue dot) with t = 10 μm.

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We performed analytic fittings of the transmission spectra using Lorentzian lineshape formula $\left|S_{21}\right|=\left|1-\text{Im}\left(K/\left({\omega }^2-{\omega }_0^2+i\omega \gamma \right)\right)\right|$ to obtain the quality factors of the resonances [29], where K is the fitting parameter, ω₀ is the resonance frequency and γ represents the damping rate of the resonance. In contrast to the sharpening effect of the resonance dip for fundamental RLC resonance (n = 1) [12], we find that the quality factors of electric dipole resonances stay around Q = ω₀/2γ ≈ 0.8 for different substrate thicknesses. The electric field strength as well as the surface current density distributions at the resonance frequency for t = 10 μm are illustrated in Fig. 2(b) which reveals the features of electric dipole resonance. When the SRR is excited in the electric dipole resonance mode, two parallel surface currents are induced by the incident electromagnetic field. As a result, negative and positive charges are accumulated at the end of split gap and its opposite point respectively, denoted as the − and + signs in Fig. 2(b).

The transmission spectra of the bilayer SRR-dimer structure with substrate thickness t = 5, 10, 15, 20 and 25 μm are shown in Fig. 3(a). For different ts, there are apparently two observable resonances with frequencies ω⁻ and ω⁺, respectively. When the separation t becomes larger, it results weaker electromagnetic interaction between the top (SRR₁) and bottom (SRR₂) layers of SRR arrays leading to a smaller energy splitting (Δω = ω⁺ − ω⁻) between higher- and lower-frequency resonances.

 

Fig. 3 (a) The transmission spectra of a bilayer SRR under different spacer thickness. (b) The Fano and Lorentzian lineshape fitting for ω⁻ and ω⁺ at t = 10μm, respectively. (c) Electric field strength and surface current density distributions of the top SRR (SRR₁) and the bottom SRR (SRR₂) at the lower (ω⁻), higher (ω⁺) resonant frequencies as well as the off-resonance frequency ω_off with t = 10μm. (d) The electric field strength distributions on the x − z cut-plane of the SRR-dimer structure at ω⁻ and ω⁺.

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To qualitatively understand the spectral characteristics of these resonances, the surface currents and electric field distributions at relevant resonances with t = 10 μm are calculated as shown in Fig. 3(c). For the lower-frequency resonance dip ω⁻, the excited parallel surface currents at the two arms of SRR₁ and SRR₂ are in the opposite direction, resulting antiparallel electric dipoles for the top and bottom SRRs in the unit cell. The negative electric interaction between two antiparallel electric dipoles (out-of-phase state) lower the total energy of the bilayer SRR-dimer structure leading to a smaller resonance frequency ω⁻ than that of a single-layer SRR array. While for the higher-frequency resonance dip ω⁺, the induced electric dipoles of SRR₁ and SRR₂ are in the same direction (in-phase state) resulting in a larger resonance frequency. For the transmission peak at 1.1THz (denoted as ω_off in Fig. 3(a)), the surface currents mainly concentrate on SRR₁, while there are only very weak surface currents on SRR₂. Such current density distributions reveal the transition process from the out-phase state into the in-phase state. Figure 3(d) illustrates the electric field distributions on the central xz cut-plane of SRR-dimer structure at ω⁻ and ω⁺, respectively. We found that the electric field was more concentrated in the substrate at ω⁻ than that at ω⁺ as a result of the attractive interaction between the antiparallel electric dipoles of SRR₁ and SRR₂ at the lower resonance frequency. On the other hand, at the higher resonance frequency, the electric field extended much farther into the air (see Fig. 3(d)), making it more sensitive to the environment changes. It is also noteworthy to mention that the lower-frequency resonance ω⁻ has the feature of Fano-like resonance with a sharp and asymmetric lineshape.

Following, we use quantum optics formula to qualitatively explain the interference mechanism of this high Q resonance, which could give insights to future researches to achieve high-Q resonances. In terms of quantum optics, we use |1〉 (|2〉) to stand for the state when SRR₁ (SRR₂) is individually excited. The coupling strengths of SRR₁ and SRR₂ to the external electromagnetic field are proportional to their electric dipoles, which can be written as 〈1|E⃗ · r⃗₁|1〉 = γ₁ E⃗ and 〈2|E⃗ · r⃗₂|2〉 = γ₂ E⃗. Here, we set γ₁ = γ₂ = γ considering SRR₁ and SRR₂ has the same constitution as well as the same configuration. The Hamiltonian of the SRR-dimer structure are in the form of H₀ = ω₀|1〉〈1| + ω₀|2〉〈2| + g(|1〉〈2| + |2〉〈1|), where ω₀ is the resonance frequency of SRR₁ and SRR₂, g is the interaction strength. By diagonalizing the above Hamiltonian, we obtained the eigen-states $|-〉=\frac{1}{\sqrt{2}}\left(|1〉-|2〉\right)$(out-of-phase state) and $|+〉=\frac{1}{\sqrt{2}}\left(|1〉+|2〉\right)$ (in-phase state) as well as the corresponding eigen energies ω⁻ = ω₀ − g and ω⁺ = ω₀ + g. For the out of-phase state, the total electric dipole of SRR dimer is 〈−|(E⃗ · r⃗₁ − αE⃗ · r⃗₂)|−〉 = 1/2(1 − α)γE⃗, while for the in-phase state, the total electric dipole is 〈+|(E⃗ · r⃗₁ + αE⃗ · r⃗₂)|+〉 = 1/2(1 + α)γE⃗. Here, the parameter α is slightly smaller than 1, representing the attenuation of electromagnetic field after passing SRR₁. From the above analysis, we can see that the total electric dipole of the out-of-phase state is much smaller than that of the in-phase state. Thus, the coupling between the out-of-phase state to the external electromagnetic field is much weaker, resulting a larger Q -factor for the out-of-phase state. The quantum optics formula explain the interference nature for Fano-like resonance, which agrees well with our simulation results.

Detailed analysis of Q factors of the Fano-like resonance and broader higher-frequency resonance were performed by fitting the transmission spectra using Fano-formula and Lorentzian lineshape respectively, as shown in Fig. 3(b). The Fano formula is given by $\left|S_{21}\right|=\left|a_1+i{a}_2+\frac{b}{\omega -{\omega }_0+i\omega \gamma }\right|$, where a₁, a₂, and b are constant real numbers, ω₀ is the Fano resonance frequency, and γ is the overall damping rate of the resonance [30, 31]. The concerned fitting parameters and Q factors of ω⁻ and ω⁺ are listed in Table 1. As shown in Table 1, the quality factor of ω⁻ resonance is more than 20 times larger than that of ω⁺, which agrees well with our quantum optics explanation.

Table 1. The concerned fitting parameters and corresponding Q factors for ω⁻ and ω⁺ at different substrate thicknesses.

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3. Thickness and refractive index sensing

It is well known that the sharp Fano-like resonance can be applied as a ultrasensitive sensor for both thickness and refractive index sensing. In the following, we compare the sensitivities of the narrow resonance ω⁻ and the broad resonance ω⁺ under different thicknesses and refractive index of coated analyte. Firstly, different thicknesses of analyte (refractive index 1.6) were coated on the top of SRR₁, due to the fact that the electromagnetic field strength is stronger on the top SRR array (see Figs. 3(c)(d)). Figure 4(a) shows the transmission spectra of the bilayer SRR-dimer where gradual red shifts are observed for both ω⁻ and ω⁺ resonances while increasing the thickness of analyte. As summarized in Fig. 4(b), for ω⁻, the red shift for analyte thickness h_ana = 1, 4, 8 and 16 μm is 0.02, 0.04, 0.053, and 0.063 THz, respectively, when compared to that without any analyte layer on top of the SRR-dimer i.e. h_ana = 0 μm. While for the higher-frequency resonance ω⁺, the corresponding red shift for the same analyte thickness is 0.047, 0.110, 0.153, and 0.187 THz respectively. The largest red shift occurs for the first 1 μm thickness of analyte and a saturation behavior is observed for both cases.

 

Fig. 4 (a)The transmission spectra of a bilayer SRR for different analyte thickness. (b) The relationship between the red shift of the resonant frequencies ω⁻ and ω⁺ and analyte thickness. (c) The transmission spectra of a bilayer SRR for different refractive index of analyte. (d) The frequency shifts of ω⁻ and ω⁺ versus refractive index of analyte.

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Since the resonance frequency shifts of ω⁻ and ω⁺ are different for the same thickness of coated analyte on top of the SRR-dimer structure, we analyze the sensitivities of both resonances by varying the refractive index of the analyte with a fixed thickness. Figure 4(c) shows the transmission spectra of SRR-dimer structure with a constant analyte thickness of h_ana = 4 μm under several different refractive indices of the analyte. The total red shift of the lower resonance frequency ω⁻ by increasing the refractive index of analyte from n = 1.0 to n = 2.0 is found to be 0.083 THz. For the higher resonance frequency ω⁺, the total shift in this case is observed to be 0.163 THz. We plotted resonance frequency shift versus the refractive index of analyte in Fig. 4(d) and then estimated the sensitivities of both resonances. The sensitivities for ω⁻ and ω⁺ turned out to be 83 GHz/refractive index unit (RIU) and 163 GHz/RIU, respectively. The standard sensitivities of both resonances are obtained using the formula $S=\left|d\lambda /dn\right|=\frac{c}{f{0}^2}\times \frac{df}{dn}$, here c is the speed of light in vacuum, f₀ is the resonance frequency, and n represents the refractive index of the analyte. In terms of standard sensitivity, the corresponding sensitivities for 4 μm thick analyte of the lower and higher resonance frequencies are 2.49 × 10⁴ nm/RIU and 3.26 × 10⁴ nm/RIU, respectively.

For the sensing application, a dimensionless parameter, figure of merits (FOM = S/Δλ) is usually applied to evaluate the performance of sensor. We plot the figure of merits of ω⁻ and ω⁺ under different analyte thicknesses, as shown in Fig. 5. Although the sensitivity of ω⁻ is slightly smaller than that of ω⁺, the FOM of ω⁻ about ten times larger. The superior performance in terms of FOM for ω⁻ resonance lies in the fact that the linewidth (Δλ) of this resonance is much narrower.

 

Fig. 5 Figure of merits of ω⁻ and ω⁺ under different analyte thicknesses.

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4. Conclusion

In summary, we have demonstrated Fano-like high-Q resonance using symmetric SRR-dimer structure at THz frequency with more than twenty times enhanced quality factor. After analyzing the resonance properties of this high-Q resonance, we show that ultrasensitive sensing can be performed using this SRR-dimer structure with about ten times enhanced FOM for the Fano-like resonance. In this model, the freestanding SRR-dimer structure could be possibly fabricated [32,33]. We believe this new type of high-Q resonance revealed in this paper would afford intriguing opportunity for future design and realization of real time on-chip chemical and bio-molecular detection in the THz regime.