A metasurface is a two-dimensional single-layer version of artificial electro-magnetic material^[1-2] that has aroused widespread interest in recent years because it can flexibly manipulate the phase, propagation mode and polarization of electro-magnetic waves^[3-5]. A series of novel physical effects can be achieved through metasurfaces such as the negative refractive index^[6], the dielectric constant of infinity^[7] and strong chirality^[8]. Due to these abnormal physical properties, metasurface structures can be applied to sensing^[9-10], absorbers^[11], slow-light generation^[12-14] and light manipulation^{[4, 15]}.

The periodic and neighboring interactions of metasurface structures are used to control combined radiative and non-radiative loss in resonance^[2]. The classic approach to reduce radiative losses is to employ the Fano resonance which is caused by destructive interference between the broad bonding mode and narrow anti-bonding mode in a plasmonic nanostructure^[16-19]. Fano resonance has shown an extremely-high sensitivity to the surrounding dielectric environment of the nanostructure^[20-23], which allows great potential applications in fields including chemical bio-sensors^[24], optical switches^[25], etc. Therefore, the designing of a Fano resonance nanostructure is a fundamental issue. Broken symmetry is an effective method for the generation of Fano resonance in nanostructures^[26-27]. Researchers have used many methods to break the symmetry of nanostructures to achieve Fano resonance: Removing a wedge from a nanodisk^[27]; the complex nanostructure composed of differently shaped nanoparticles, such as rod-ring nanostructure, rod-sphere nanostructure, ring-disk nanostructure et al^[28-29]; ruptured symmetric nanostructures with a consistent geometric shape but different arrangement, material composition and size^[30], such as nanoshells^[29]; metal-dielectric-metal nanoplates, etc.^[31-33].

In this paper, a metasurface array consisting of parallel-arranged nanorods with different lengths on a SiN_x substrate is designed and used to study double Fano resonance and the sensitivity of its refractive index. The transmission spectrum, the steady electric field distribution and the charge density distribution are calculated by using the finite element method (FEM). The double Fano resonances are observed in the projective spectrum. The proposed structure can be easily combined with the nanorod-like structure and a substrate with a high dielectric constant to realize double Fano resonance, which greatly reduces the dependence of Fano resonance on the nanostructure and the requirements for the preparation process. The perfectly matched layers are employed at the top and the bottom boundaries, while the periodic boundary conditions are employed at other boundaries of the cuboid’s unit. The quasi-static approximation model is employed to explain the coupling mechanism of double nanorods. Concurrently, the influence of structural parameters on the transmission characteristics is also studied.

Fig. 1(a) (color online) illustrates the metasurface array of double parallel nanorods with different lengths, which is constructed on a SiN_x substrate with a dielectric constant of ε = 8.5 and the thickness of 100 nm. The structure is arranged with a period of 2000 nm and the center of the parallel nanorod dimer coincides with the center of the unit cell. The distance between the incident port and the exit port is 1800 nm. Fig. 1(b) (color online) is the planar graph of this structure. The blue and yellow areas represent the substrate and Au, respectively. The length of the long nanorod and the short nanorod are L₁ = 400 nm and L₂ = 800 nm, and their widths are w = 100 nm. The thickness of the nanorods is fixed at t = 50 nm. The s is the horizontal separation between the short nanorod center and the long nanorod and the g is the vertical separation between them. The polarization is employed in the direction of the electric field along the -x axis.

The optical responses of the metasurface are calculated numerically by the FEM. In order to improve computational efficiency, the computational domain only contains one cell of the array. Bloch-periodic boundary conditions (BPBC) are employed at the lateral boundaries of the elementary cell (in the x and y directions) for simulating the periodic array, and perfectly matched layers (PML) are applied to the propagation direction (the -z direction) to eliminate nonphysical reflections. The incident light transmission is in the -z direction and polarizes in the -x direction. The transmittance is defined as T = |S₂₁|², where S₂₁ is the transmittance of the cell. The permittivity of Au is characterized by the Drude mode^[34] as:

$ \varepsilon \left( \omega \right) = 1 - \frac{{\omega _{\rm{p}}^2}}{{{\omega ^2} + i\gamma \omega }} \quad,$ (1)

here $ \gamma = 4.065 \times {10^{13}} $ Hz is the collision frequency, $ {\omega _{\rm{p}}} = 1.37 \times {10^{16}} $ rad/s is the plasma oscillation frequency and ω is the incidence wave frequency.

The quasi-static approximation model is employed to interpret the coupling mechanism of the two nanorods (as shown in Fig. 2). The total interactive energy V_t of the double resonators of the Au nanorod dimer can be expressed as^[35]:

$ {V_{\rm{t}}} \propto \frac{{{{\vec p}_{\rm{a}}} \cdot {{\vec p}_{\rm{b}}}}}{{{r^3}}} - \frac{{3\left( {{{\vec p}_{\rm{a}}} \cdot \vec R} \right)\left( {{{\vec p}_{\rm{b}}} \cdot \vec R} \right)}}{{{r^5}}}\quad, $ (2)

here, $ \vec R $ is the displacement vector connecting the point dipoles $ \left( {r = \left| {\vec R} \right|} \right) $ and ${\vec P_{\rm{a}}}$, ${\vec P_{\rm{b}}}$ denotes the electric dipole’s moment.

The Lorentzian polarizabilities of the individual dipoles is represented as α_a and α_b, the incident plane wave is defined as $\vec E{\text{ = }} x{E_0}{e^{{\rm{i}}(\omega t + kz)}}$. Then the system follows the equations^[33]:

$ {\vec P_{\rm{a}}} = {\alpha _{\rm{a}}}({\vec E_{\rm{a}}} + {V_{\rm{ab}}}{\vec P_{\rm{b}}}) \quad,$ (3)

$ {\vec P_{\rm{b}}} = {\alpha _{\rm{b}}}({\vec E_{\rm{b}}} + {V_{\rm{ba}}}{\vec P_{\rm{a}}})\quad, $ (4)

here, V represents the dipolar interaction, $ \vec P $ is the polarizations of the dipoles, and the incident E-fields are evaluated by the E_a and E_b at the corresponding dipole locations. By performing algebraic operations on equations (3) and (4), the effective polarizability of the constituent elements can be described as^[30]:

$ {\tilde \alpha _{\rm{a}}} = \frac{{{\alpha _{\rm{a}}}\left( {1 + {\alpha _{\rm{b}}}{V_{\rm{ab}}}} \right)}}{{1 - {\alpha _{\rm{a}}}{\alpha _{\rm{b}}}{V_{\rm{ab}}}{V_{\rm{ba}}}}}\quad, $ (5)

$ {\tilde \alpha _{\rm{b}}} = \frac{{{\alpha _{\rm{b}}}\left( {1 + {\alpha _{\rm{a}}}{V_{\rm{ba}}}} \right)}}{{1 - {\alpha _{\rm{b}}}{\alpha _{\rm{a}}}{V_{\rm{ba}}}{V_{\rm{ab}}}}} \quad,$ (6)

Here, V_ij is the electro-static propagator which can be affected by the locations of the dipoles and their orientation, and the interaction term between them is caused by the hybridization of eigenstates.

The transmission spectra of the short nanorod, the large nanorod and the double parallel nanorods (s is fixed at 0 nm and 80 nm) are shown in Fig. 3. The nanorod as dipole antennas can cause a strongly localized electric field when the nanorod is directly excited by incident light. Two and one Fano resonance peaks are observed in the transmission spectrum of the short and large nanorods, respectively. For the double parallel nanorods, the resonance peaks are different for s = 0 nm and s = 80 nm. We found a new transmission peak appearing (as shown in Fig. 3, color online) at the short wavelength in the spectrum due to the destroyed symmetry of the double parallel nanorods. Two asymmetric Fano resonances appeared in the transmission spectrum due to the single short nanorod coupled with a high dielectric substrate while a single Fano resonance appeared for the long nanorods because the resonance peak at the large wavelength redshifted out of the given wavelength range. For the nanorod dimer structure, the electrons’ vibration path changed for both the single short and the long nanorods due to the coupling effect between the two nanorod dimers. The Fano resonance at the short and long wavelengths is blue- and red-shifted with respect to the single short nanorods when s = 0 nm, respectively. For s = 80 nm, the coupling between the two nanorods causes an asymmetric change in the electronic vibration path due to the shift of the center of the short nanorods relative to the long nanorods, which results in a new resonance valley appearing at the short wavelength.

In order to research the double Fano resonance effect, the steady state electric field distribution and the charge density distribution are shown in Fig. 4 and Fig. 5 (color online), respectively. For s = 0 nm, the normalized square of electric field (|E|²) and the normalized charge density distribution of the nanorod dimer peaks at A (λ_A = 2.32 μm) and C (λ_C = 2.92 μm), and dips at B (λ_B = 2.36 μm), D (λ_D = 3.00 μm) were shown in Fig. 4. For the λ_A = 2.32 μm (Fig. 4(a)) and the λ_B = 2.36 μm (Fig. 4(b)), the incident light interacts with the electrons of the nanorods to form electric dipoles, which generates an enhanced electric field near the nanorod’s surface. As a result of charge induction, different charges are accumulated on the surface of adjacent long nanorods. For λ_C = 2.92 μm with λ_D = 3.00 μm, the normalized square of the electric field and the normalized charge density distribution are shown in Fig. 4(c) and 4(d), respectively. The obvious electric dipole vibration mainly occurs on the short nanorods. However, we can find that the enhanced electric field of the short nanorods surface is strongly coupled with the free electrons on the long nanorods, which results in a large amount of heterogenic charges being concentrated in the surface region adjacent to the short nanorods. Therefore, the Fano resonance of the long wavelength of the double nanorods system is derived from the long wavelength Fano resonance of the single short nanorods.

The electric field (|E|²) and the charge density distribution of the double parallel nanorods at the peaks E, G, and I and dips F, H, J with s of 80 nm are shown in Fig. 5. It can be seen that the nanorods are excited by incident light at λ_E = 2.26 mm (a), λ_F = 2.30 mm (b), λ_G = 2.36 mm (c), λ_H = 2.90 mm (d), λ_I = 2.98 mm (e), and λ_J = 3.00 mm (f). Fig. 5(a) and Fig. 5(b) illustrate that the electric field is mainly distributed at the top of the dimer. Many positive charges are concentrated in the inner side of the short nanorod, and the positive and negative charges are distributed in the higher and lower parts of the short nanorod at λ_E = 2.26 μm. The charge distribution of the long nanorod as a multipolar mode is opposite to that of the short nanorod. The electric field will be distributed in the middle of the dimer and the inside of the short nanorod has only negative charges at λ_F = 2.30 μm. However, there are many positive charges in the middle of the long nanorod and just a few negative charges distributed at both ends. When the nanorod dimer is excited by the incident light field at λ_G = 2.36 μm, the electric field is distributed at the bottom of the dimer which will induce many negative charges and few positive charges on the inside of the short nanorod. The charge distribution on the long nanorod is approximately the same as that shown in Fig. 5(d). The electric field distribution at λ_H = 2.90 μm is similar to that at λ_G = 2.36 μm and the charge distribution is just opposite to that at λ_G = 2.36 μm. The electric field at λ_H = 2.90 μm is concentrated in the interior of the double parallel nanorods and distributed at both ends of the dimer, and the positive and negative charges of the short nanorod are distributed in the top and bottom of the dimer. The charge distribution of the long nanorod is opposite to that of the short nanorod and presents multiple dipoles. The electric field distribution at λ_I = 2.98 μm is similar to that at λ_J = 3.00 μm and the charge distribution is opposite to that at λ_J = 3.00 μm. The electric field strength and charge density at λ_I = 2.98 μm is significantly enhanced compared with that at λ_H = 2.90 μm.

To investigate the effect of the offsets on the transmission spectra of the double parallel nanorods, we increase s from 0 nm to 80 nm every 20 nm with fixed L₁ = 400 nm, w₁ = 100 nm, L₂ = 800 nm, w₂ = 100 nm, t = 50 nm and g = 20 nm. The transmission spectra of the double parallel nanorods for s = 0, 20, 40, 60, 80 nm are presented in Fig. 6(a) (color online). For s = 20 nm, the left shoulder of the double parallel nanorods mode peak has a dip emerge with a profile that is observed more clearly with an increase in s. The dip is regarded as a Fano resonance dip, which is caused by the symmetrical breaking of the dimer. We notice that the right shoulder of the double parallel nanorods mode peak red-shifted slightly and the intensity of the Fano resonance effect will be weakened with an increase in s. There is no significant change in the left shoulder. Figure 6(b) (color online) plots the transmission spectra for L₁ = 400, 420, 440, 460, 480 nm with fixed s = 0 nm, w = 100 nm, L₂ = 800 nm, t = 50 nm and g = 20 nm. With an increase in L₁, the Fano resonance becomes more pronounced and then shows redshift. Because both of these Fano resonances are derived from the dipole vibrations of L₁, there is an increase in the mean free path of the electrons on the nanorods.

Figure. 7(a) (color online) depicts the transmission spectra for g = 20, 40, 60, 80, 100 nm with fixed s = 0 nm, L₁ = 400 nm, w = 100 nm, L₂ = 800 nm, t = 50 nm. We notice that with increasing g, the intensity of the double Fano resonance effect does not change. However, the peak and dip of the Fano resonance on the right blue shifts and the transparency windows of the right Fano resonance narrows with no change in the Fano resonance on the left. According to the above description, g is not an important parameter for double Fano resonance. In order to study the influence of the polarization direction of an electric field on the double Fano resonance effect, the transmission spectra for θ = 0°, 30°, 60°, and 90° are calculated with fixed s = 0 nm, L₁ = 400 nm, w = 100 nm, L₂ = 800 nm, t = 50 nm and g = 20 nm in Fig. 7(b) (color online) and the polarization of θ = 90° is along the x direction. With an increase in θ, the position of the double Fano resonance does not change but its intensity will gradually decrease. The double Fano resonance will disappear with a polarization of θ = 90° because the nanorods are perpendicular to the incident light field to present a dark mode, which can't be directly excited by incident light due to the lack of coupling between the nanorods.

To investigate the sensitivity of Fano resonance to the refractive index, the air around the double parallel nanorods (s = 80 nm, L₁ = 400 nm, w = 100 nm, L₂ = 800 nm and t = 50 nm) is replaced by water, turpentine, glycerin, and olive oil. Their refractive indices are n = 1.0, 1.1, 1.2, 1.333, 1.354, 1.4, 1.472, 1.473, and 1.476, respectively. The transmission spectra are displayed in Fig. 8 (a)(color online). As n increases, the transmission peak λ₁ and λ₂ red-shift because the increase in the refractive index of the medium around the double parallel nanorods results in an increase in the effective refractive index of the surface plasmons, which leads to an increase in the mean free path of electrons in the nanorod. A linear relationship can be observed from two resonance dips with a change in the refractive index from Fig. 8(b) (color online). In addition, the sensitivity curves at λ₁ and λ₂ of the double parallel nanorods are shown in Fig. 8(c) (color online), whose sensitivities are calculated as S_λ1 = δλ/δn = 1.137 μm/RIU and S_λ2 = δλ/δn = 0.899 μm/RIU, respectively.

In addition, as shown in Table 1, we have made a comparison with previous similar works and found that the proposed design scheme is relatively simple in structure and easy to process. Its sensor performance also has certain advantages over that of the refractive index sensor designed in previous works.

In this study, the double parallel nanorods composed of two nanorods with different lengths is designed, and the FEM and quasi-static approximation models are employed to theoretically study the nanorods’ coupling mechanism. The simulation results show that the double Fano resonance effect can be observed, and the transparency window peak can be easily controlled by the structural parameters. We note that the offset between the geometric centers of the double nanorods is an important coupling parameter to induce Fano resonance while other parameters will have an influence on its position. The sensitivity of Fano resonance was studied for different refractive index by changing the double parallel nanorods’ surrounding medium and we found that the sensitivity of the double Fano resonance is S_λ1 = δλ/δn = 1.137 μm/RIU and S_λ2 = δλ/δn = 0.899 μm/RIU. This will provide a theoretical basis for designing plasmonic switches and sensors.

Data Availability Statement: The data that support the findings of this study are available upon reasonable request to the authors.

Conflicts of Interest: The authors declare no conflict of interest.