1. Introduction

Metamaterials (MMs) are manmade engineered material to exhibit fascinating optical properties not occurring in nature, such as negative refractive index [1–4], perfect lens [5,6], electromagnetic cloaking [7,8],extraordinary transmission [9] and asymmetric transmission [10].Recently, MMs have become a flourishing research in the terahertz domain demonstrating application potential in sensing [11]. In order to make MMs perform efficiently on biological sensing, a resonance with a high quality factor (defined by the resonant frequency over width of the resonance) is desired [12,13].However, conventional planar MMs normally have a modest Q factor due to the rather small volume confinement of electromagnetic fields and strong coupling to vacuum [14–16].Fano resonance (FR) is attributed to the interference of bright (radiating) and dark (non radiating) modes [17–21] and possessing a sharp feature in the transmission spectrum [22].Therefore, the problem of the low sensitivity in MMs sensor can be tackled if MM structural design is engineered in such a way that it could support FRs. FRs were originally considered in the photonics structures such as dielectric photonic crystal slabs [23–27] and perforated metal films (in the context of extraordinary optical transmission) [28]. Another way to obtain FRs is to break the symmetry of MMs [29]. By introducing an asymmetry in the shape of the MM structural element, dark modes (trapped modes) which are not accessible in the symmetric MMs can be evoked [21]. These dark mode resonances weakly couple to free space and strongly interact with the bright modes thus giving rise to a FR with a steep dispersion.This FR is able to concentrate the electromagnetic field in a small region to interact with the matter strongly, making it suitable for sensitively sensing [30]. Alternatively, FR can also be achieved by the coupling between symmetric and anti-symmetric resonance in symmetric MMs [31]. However, most of the works demonstrate FRs in planar MMs based on a single metallic layer. It has been suggested that multiple layers of perforated metal–dielectric-metal(MDM) stacks (for 100 and 200 layers) along the propagation direction established a promising approach for a three dimensional (3D) optical MM [32–34].Such a 3D MM can possess a thickness much larger than the free space wavelength in the near infrared(NIR) region and excite a strong magneto-inductive coupling between neighboring functional layers under a normally incident light [33].The tight coupling between adjacent LC resonators through mutual inductance results in a low loss [35].Therefore, achievement of 3D multilayer MMs is an inevitable requirement for nearly all possible applications, like cloaking and invisible materials [33–36], the observation of FRs [37] as well as tunable metamaterials [38].

As discussed above, multilayer MMs have strong magnetic responses which can be described by effective magnetic dipoles. In detail, giant magnetic dipolar moments are emerged inside the dielectric interlayer with the formation of closed displacement current (J_D) loops [39,40].This phenomenon especially shows the possibility of attaining the double negative index material [41]. Most notably, several recent works show that magnetic activity in multilayer MMs is also capable of exciting FR. Seo et al. have shown that a FR can be observed in double layer symmetric electric ring resonators at GHz regime [37]. Soltani et al. have investigated a MDM mulilayer MM with half ellipse patterns, exhibiting a sharp FR around 1THz [11]. It is realistic and well known that broken symmetry of the planar MM structures can result in a high order mode thus enhancing the FRs [42].However, to the best of our knowledge, limited research work looks at the influence of the symmetry breaking in MDM multilayer MMs on the FRs, specifically in the NIR regime considering the potential of the NIR spectrum for sensing.

Here, we numerically present that a FR in the NIR region can be observed from the symmetric elliptical nanoholes array (ENA) embedding through metal-dielectric-metal layers. This FR response is caused by the interaction between the bright modes and the dark modes. Particularly, the elliptical nanoholes resonator array shows an electric resonance (localized surface plasmon) to excite the bright modes whereas the MDM multilayers support a magnetic resonance (inductive-capacitive resonance) to induce the dark modes [43,44].As soon as the elliptical holes are displaced from their central positions, the single magnetic resonance would be divided into two magnetic resonances at different wavelengths, and a double FR appears owing to the destructive interference between the electric resonance and two magnetic resonances at different wavelengths. This interference leads to a sharp transparency peak for the asymmetric MDM-ENA. Moreover, the degree of the asymmetry allows for the tuning of the amplitude and bandwidth of the double FR window. The sensing capabilities of both the symmetric MDM-ENA and asymmetric MDM-ENA are evaluated through the sensing sensitivity and figure-of-merit (FOM). Our proposed structure possesses a simple geometry which remains compatible with standard photolithography patterning and can be easily experimentally realized at the optical region.

2. Metamaterials design

The proposed MM structures are developed from our previously studied double negative index MMs at optical regime [39] consisting of a two metallic films (30nm thick Au) separated by a dielectric interlayer (60nm thick Al₂O₃). Here, we simulate two sets of multilayer MMs starting from a symmetric element with both the top and bottom elliptical holes exactly at the centre [see Fig. 1(a)-(b)] and then gradually displaced the upper elliptical hole from the centre by a distance “δ” while keeping the lower elliptical hole fixed at the centre in order to break the symmetry [see Fig. 1(c)-(d)]. For both of the structures, the elliptical nanoholes array (ENA) (lattice constant, L = 400nm; hole diameters, d₁ = 260nm, d₂ = 160nm) has been perforated through the entire Au/Al₂O₃/Au structure, β is a cross-section plane of the structure, and the elliptical holes are periodically arranged in both the x and y directions. The Au bottom layer interacts with the upper Au layer to give rise to a closed loop of displacement current (J_D) and localize an electromagnetic (EM) field within the dielectric interlayer. The two structures both reside on a 200μm thick BK7 glass. Au is selected as the metal due to its stability and low ohmic loss. The geometry of the unit cell pattern and the thickness of the sandwich layers have been chosen to allow for the impedance matching between the metamaterials and impinging plane wave [45]. In order to justify the influence of magnetic activity on the FR, we have simulated the identical lateral structure to the symmetrical MDM-MMs, however, with a single metal layer (30nm Au) as shown in Fig. 1(e)-(f). The optical properties of the structures are calculated using 3D EM Explorer Studio [46], a commercial Finite Difference Time Domain (FDTD) code. In the simulation, the dielectric properties of Au are described by a drude-type dielectric function, where ω_p = 1.37x10¹⁶ Hz is the plasma frequency and ω_c = 4.08x10¹³ Hz is the collision frequency for bulk Au [47]. A plane wave source is simulated at normal incidence to the structure with the electric field direction along the small axis of the elliptical hole. The computational domain (400 nm × 800 nm × 1000 nm) has perfectly match layer (PML), absorbing boundaries, in the z direction and periodic boundaries in the x-y plane [48]. A uniform FDTD mesh size is adopted, the mesh size is the same along all Cartesian axes: ∆x = ∆y = ∆z = 2nm, which is sufficient to minimize the numerical errors arising from the FDTD method.

 

Fig. 1 (a) Schematic of the symmetric MDM structure consisting of a 60nm thick Al₂O₃ dielectric layer between two 30nm thick Au films perforated with a square array of elliptical holes residing on a 200μm thick BK7 glass. (b) Illustration of the symmetric element of ENA, the lattice constant is L = 400nm and hole diameters are d₁ = 260nm, d₂ = 160nm. (c) Schematic of the asymmetric MDM structure consisting of a 60nm thick Al₂O₃ dielectric layer between two 30nm thick Au films perforated with a square array of elliptical holes residing on a 200μm thick BK7 glass. (d) Illustration of the asymmetric element of ENA, the lattice constant is L = 400nm, hole diameters are d₁ = 260nm, d₂ = 160nm, δ is the distance of the upper elliptical hole from the centre. (e) Schematic of the symmetric structure consisting of a 30nm thick Au film perforated with a square array of elliptical holes residing on a 200μm thick BK7 glass. (f) Illustration of the symmetric element of ENA, the lattice constant is L = 400nm and hole diameters are d₁ = 260nm, d₂ = 160nm.

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3. Results and discussions

The ENA has a lower transmission for s polarized light due to the electric field’s orientation with respect to the metallic stripe width hence the polarization of the incident wave was set to be p polarized [39]. Here s polarization means the incident electric field vector is parallel to the long axis of the ENA and p polarization means that the incident electric field vector is parallel to the short axis of the ENA. Figure 2 shows the transmissions for both the symmetric MDM-ENA [see Fig. 1(a)] and its reference structure consisting of the identical ENA embedding through a single Au layer [see Fig. 1(e)].In the transmission of the multilayer structure, it is seen that a FR appears at the wavelength of 1284nm [marked as P₁ in Fig. 2] possessing a resonance linewidth of 77nm and Q factor of 21,the detailed calculation of which is presented in Section 4. Whereas the single layer ENA structure doesn't have a FR response under the normally incident light owing to the absence of the magnetic dipole resonance. Therefore, the origin of the FR in the symmetric MDM-ENA can be traced to the interaction between the electric resonance and the magnetic resonance, formed by closed loops of displacement current(J_D) [40]. Such a magnetic resonance is normally inaccessible but can be excited if, for example, the MMs are based on metal/dielectric/metal films.

 

Fig. 2 3D- FDTD simulation of the transmission spectrum of the symmetric multilayer ENA [see Fig. 1(a)] and its reference structure consisting of the identical ENA embedded through single Au layer [see Fig. 1(e)] for p polarization at normal incidence.

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To gain further understanding on the nature of the resonance, we look at total electric field intensity distributions$E=\sqrt{{\left|E_x\right|}^2+{\left|E_y\right|}^2+{\left|E_z\right|}^2}$, total magnetic field intensity distributions $H=\sqrt{{\left|H_x\right|}^2+{\left|H_y\right|}^2+{\left|H_z\right|}^2}$ and J_D along the β plane at the resonant wavelength of 1284nm for the symmetric MDM-ENA in Fig. 3(a)-(b), and for the single Au layer ENA in Fig. 3(c)-(d). In the field maps of Fig. 3, the arrows show J_D, whereas the colour shows the magnitude of the electric field and magnetic field. In Fig. 3(b), the formation of the closed J_D loops is observed to support a magnetic resonance at which light is trapped and strongly absorbed thus the magnetic field can be efficiently confined between two Au layers. However, Fig. 3(a) shows a relatively low concentration of the electric field intensity in the dielectric interlayer and elliptical apertures, indicating a weak electric resonance. The FR (P₁) in the symmetric MDM-ENA is caused by the interaction between the bright modes and the dark modes. Particularly, at the wavelength of 1284nm, the ENA resonator shows an electric resonance namely localized surface plasmon resonance whereas the MDM multilayers support a magnetic resonance stemming from the retardation of the wave propagating in the dielectric spacer namely inductive-capacitive resonance [49].Thus, the ENA resonator and the MDM multilayers serve as the bright modes and dark modes, respectively. Once the configuration changes to the single Au layer in Fig. 3 (c)-(d), it shows that H cannot be trapped in the metal layer since the loops of J_D no longer exist hence showing no FR response in the transmission spectrum. To excite the magnetic resonance of such a planar ENA, one may use an oblique incident light which possesses the required magnetic field component [50–52]. This restriction will lower the coupling efficiency between the magnetic field of the incident light and the resonant element.

 

Fig. 3 3D- FDTD simulation of (a) total electric field intensity distribution, (b) total magnetic field intensity distribution and J_D distribution along β plane for the symmetric MDM-ENA, at normal incident angle where λ = 1284nm; Simulation of (c) total electric field intensity distribution, (d) total magnetic field intensity distribution and J_D distribution for the symmetric single layer ENA at normal incident angle where λ = 1284nm

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Figure 4 presents the transmissions of the symmetric MDM-ENA and the asymmetric MDM-ENA. Starting from the symmetric MDM-ENA (δ = 0), where only a single FR (P₁) is observed at λ = 1284nm, we move to an asymmetric MDM-ENA with δ = 60nm [see Fig. 1(c)]. As can be seen, the single FR (P₁) is split in two distinct resonant peaks at 1300nm (marked as P₂) and 1228nm (marked as P₄) by an asymmetric resonant dip (marked as P₃) at 1252nm. Namely, the single magnetic resonance at λ = 1284nm would be divided into two magnetic resonances at different wavelengths of 1228nm and 1300nm by breaking the structural symmetry. The appearance of the double FR is due to the destructive interference between the electric resonance and two magnetic resonances at different wavelengths. In particular, the transmission dip is due to the excitation of the dark mode with a weak coupling to free space, created by a certain weak structural asymmetry of the elements of MMs [19]. As can be seen in Section 4, a narrow transparency sub-band resonance window with a spectral width of 42nm is achieved at P₂ mode resulting in a high Q factor of 34.

 

Fig. 4 3D-FDTD simulation of the transmission spectrums of symmetric MDM-ENA [see Fig. 1(a)] and asymmetric MDM- ENA [see Fig. 1(c)] for p polarization at normal incidence.

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Figure 5(a)-(f) describe the distribution of E field intensity, H field intensity and J_D along the β plane at the resonant wavelengths of 1284nm (P₁) for the symmetric MDM-ENA and at 1300nm (P₂) and 1228nm (P₄) for the asymmetric MDM-ENA, respectively. It is obvious that the P₂ and P₄ modes correspond to different magnetic dipole resonances. Figure 5(c) shows that E field intensity at the P₂ mode is efficiently concentrated in both the dielectric interlayer and the elliptical apertures thus giving rise to the strongest electric resonance. As can be seen, the H field intensity distributions are confined well within the dielectric layer for both P₁ [see Fig. 5(b)] and P₂ modes [see Fig. 5(d)].However, H field intensity at P₂ mode is higher than the P₁ mode indicating the larger magnetic dipole resonance. Thus, at the P₂ mode of the asymmetric MDM-ENA, the interaction of the enhanced magnetic resonances and electric resonances leads to a much narrower FR than the P₁ mode of the symmetric MDM-ENA. In Fig. 5(e)-(f), we present the distributions of E field, H field and J_D associated with the P₄ mode, it demonstrates that the closed J_D loops are still maintained to provide the magnetic dipolar moment hence leading to a resonant peak. However, the resonance is broadened owing to the weak magnetic and electric resonances resulted from the low strength of H and E fields in both the dielectric layer and elliptical apertures.

 

Fig. 5 3D-FDTD simulation of (a) total electric field intensity distribution, (b) total magnetic field intensity distribution and J_D distribution along β plane for the symmetric multilayer ENA, where λ = 1284nm; Simulation of (c) total electric field intensity distribution, (d) total magnetic field intensity distribution and J_D distribution for the asymmetric multilayer ENA where λ = 1300nm. (e) total electric field intensity distribution, (f) total magnetic field intensity distribution and J_D distribution for the asymmetric multilayer ENA where λ = 1228nm.

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Figure 6 depicts the calculated transmission for different values of from 10nm to 60nm where the splitting of the single resonant peak at λ = 1284nm is observed when the upper elliptical hole's center is shifted by a distance of δ = 20nm. In this graph, the vertical axis describes the center shift of the top elliptical hole as a function of the parameter δ. As the distance δ is increased, the splitting of the resonant peak leading to a transmission dip in the spectrum around 1252nm becomes more obvious since the resonant dip window broadens and its amplitude nearly reaches 0. It can be explained by the fact that the asymmetry increases the free-space coupling and consequently the mode energy losses. Moreover, the asymmetry parameter “δ” leads to the broadening and enhancement on the transmission dip window centered around the same wavelength of 1252nm hence playing a control role in the structure [42].

 

Fig. 6 3D-FDTD simulation of spectrum of transmission of asymmetric multilayer ENA for different values of δ from 10nm to 60nm at normal incidence.

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4. Fano formula fitting to the transmission spectrums simulated through FDTD method

In Fig. 7, we fit the transmission curves of both the symmetric MDM-ENA and asymmetric MDM-ENA using the single FR lineshape in Eq. (1) [53,54] and double FR lineshape in Eq. (2) [12], respectively:

 

Fig. 7 (a) 3D-FDTD simulation of the transmission spectrums (red dot) of the symmetric MDM-ENA and best-fits to the single FR lineshape Eq. (1) (blue solid); (b) 3D-FDTD simulation of the transmission spectrums (red dot) of the asymmetric MDM- ENA and best-fits to the double FR lineshape Eq. (2) (blue solid).

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where ${\epsilon }_1=\frac{2\left(\omega -{\omega }_1\right)}{{\Gamma }_1}$, ${\epsilon }_2=\frac{2\left(\omega -{\omega }_2\right)}{{\Gamma }_2}$, ${\epsilon }_4=\frac{2\left(\omega -{\omega }_4\right)}{{\Gamma }_4}$. Particularly, q₁, q₂, q₄ are the phenomenological shape parameters (so-called asymmetry parameter), ω₁, ω₂, ω₄ are the resonance frequencies and Γ₁, Γ₂, Γ₄ are the width of the autoionized state (the resonance linewidth at half maximum, FWHM) of the modes P₁, P₂, and P₄ respectively [54,55]. A₀, A₁, F₁, F₂, F₄ are constant factors. With the asymmetric parameters shown in Table 1, the calculated transmissions based on Eq. (1) and Eq. (2) approximately reproduce our FDTD simulation data with the different surrounding dielectric refractive index of n = 1.0, 1.2 and 1.4.

Table 1. Aymmetric Parameters used in the FR Lineshape to Fit the Transmissions of the Symmetric MDM-ENA and Asymmetric MDM-ENA

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For example, in Fig. 7 we use the Fano formula to fit the transmission curves of both the symmetric MDM-ENA and asymmetric MDM-ENA at the surrounding dielectric refractive index of 1.0 as shown by the solid blue curves, respectively. It is found that the Fano formula mostly matches the FDTD simulation data around the resonant modes. Q factor is defined by the resonant frequency over the width of the autoionized state [12,13], where the resonant frequency and the width of the autoionized state are obtained from the Fano formula fitting transmission curves shown in Fig. 7.The resonant frequencies, the widths of the autoionized state and Q factors are 1455.6THz and1443.3THz, 67.4THz and 42.3THz, 21 and 34 for both the symmetric MDM-ENA (P₁ mode) and asymmetric MDM-ENA (P₂ mode), respectively.

5. Sensitivity to the dielectric environment

The dependence of the spectral position of surface plasmon resonance (SPR) on the permittivity of the surrounding medium environment is important for the refractive index sensor [56]. Developing from the SPR technologies, the multilayer MMs proposed here have the sharp FR making them interesting for sensor applications. We evaluate the sensing performance of the symmetric MDM-ENA and asymmetric MDM-ENA by testing the index of the medium surrounding the structures varied from n = 1.0 to n = 1.4. Here, the test samples with the different refractive index is placed on the top of the MM sensor [28,57]. The sensing capability of our multilayer fishnet MMs can be understood by the fact that the resonance wavelength is proportional to the square root of capacitance between the top and bottom Au films according to LC circuit resonance model, which is in turn proportional to the surrounding dielectric constant [43]. As can be seen in Fig. 8(a) and (b), both the single FR and the double FR exhibit a resonance red shift with respect to the increment of the surrounding material index. Figure 8(c) presents the refractive index sensitivity$S=\frac{\Delta \lambda }{\Delta n}$ of the MM sensors, where ∆λ is the resonant wavelength shift obtained for the refractive index variation(∆n) of the media surrounding the MMs sensors. It shows that the resonant wavelength shift dependences are linear.The double FR reveals a higher response to change in the dielectric environment compared to the single FR.The calculated sensitivity is 120nm/ RIU for the P₂ mode and 60nm/RIU for the P₁ mode.The resonance linewidth directly dictates the MM sensor’s minimum resolution [58].Hence, a higher Q factor or narrower resonance linewidth are desired to yield a higher sensitivity. Only then small analyte induced shifts of a resonance wavelength can be accurately measured. As we presented in this work, upon the symmetry breaking between the neighboring holes, the single FR(P₁) splits into two plasmon modes (P₂ and P₄) resulting in an increase in the strength of the resonance at the P₂ mode shown in Fig. 5. The interaction between the enhanced magnetic resonance and electric resonance leads to a FR with a narrower linewidth and higher Q factor thus leading to a higher sensitivity.

 

Fig. 8 3D-FDTD simulation of (a)transmission spectra of symmetric MDM-ENA for different values of the refractive index of the surrounding medium; (b)transmission spectra of asymmetric MDM-ENA for different values of the refractive index of the surrounding medium; (c)wavelength shift of symmetric MDM-ENA and asymmetric MDM-ENA as a function of the surrounding dielectric; (d)change of the phenomenological shape parameter as a function of the surrounding dielectric.

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In order to provide better quantification of the spectroscopic sensors operating at the different regime, the sensing performance in terms of the figure of merit (FOM) is defined as follows [58]:

where Γ is the width of the autoionized state (FWHM) of the resonant feature [54,56]. Γ is 77nm at the P₁ mode leading to FOM of 0.8 RIU⁻¹. However, at the P₂ mode of the asymmetric MDM-ENA, it possesses a higher FOM of 3.0 RIU⁻¹ with Γ = 42nm, which is close to the measurement results of FOM in the NIR region in [56,59]. Therefore a weakly asymmetric MDM-ENA appears more suitable for the development of a simple spectroscopic sensor, given the higher values of both sensitivity and FOM. After extracting the asymmetric parameters q₁, q₂, q₄ fitted by the Fano functions in Table 1, we present the variation of the asymmetric parameters to the change of the surrounding dielectric in Fig. 8(d). It is found that the absolute values of the asymmetric parameters of q₁ and q₂ decrease when altering the surrounding dielectric refractive index from 1.0 to 1.4. Specifically, for the case of q₄ at the refractive index of 1.4, the resonance linewidth increases to a large value thus leading to the degeneration of the P₄ resonant mode [shown by the red dot line in Fig. 8(b)] as well as a worse fitting through the Fano formula.

6. Conclusion

In conclusion, we show that a FR can be obtained using the symmetric multilayer MMs. This fact is due to the interplay between the bright modes and the dark modes. The bright modes are induced by the elliptical nanoholes resonator array which shows a typical localized surface plasmon resonance, whereas the dark modes correspond to the inductive-capacitive resonance excited by the closed loops of J_D from the MDM multilayers. Furthermore, by setting a low degree of the structural asymmetry in the ENA element, the single FR from the symmetric MDM-ENA splits in two resonant peaks, resulting in a double FR. It is concluded that the strong interaction between the electric resonance and two magnetic resonances at different wavelengths plays the key role to produce the enhanced double FR.This interference leads to a sharp transparency peak with ultrahigh Q factor in the NIR regime for the asymmetric MDM-ENA. Moreover, the degree of asymmetry allows for the tuning of the amplitude and bandwidth of the double FR window. Finally, the sensitivity and FOM of the MDM-ENA to the external dielectric environment are calculated to show the possibilities of sensing application in the NIR region.