## 3 Simulations and Results  
As shown in Figure 2, the red curve indicates the transmission spectrum of the structure without two rectangular stubs, the blue one describes the transmission spectrum of the structure without the elliptical ring resonator, the green one shows the transmission spectrum of the structure with only one rectangular stub, while the black one shows the whole structure's transmission spectrum. The parameters of the structure are \(R_{1}~{}=~{}300\) nm, \(R_{2}~{}=~{}140\) nm, \(r_{1}~{}=~{}150\) nm, \(r_{2}~{}=~{}70\) nm, \(d~{}=~{}10\) nm, \(l~{}=~{}480\) nm, \(h=60\) nm.. When the wavelengths are 1105 and 1545.5 nm, the transmission spectrum of the black curve decreases sharply, and two narrow resonance lines are formed. Among them, the left dip is a symmetric Lorentzian resonance, while the right one is a typically asymmetric Fano resonance. In general, when the continuous broadband state and the discrete narrowband state interact with each other, a Fano resonance will be generated [37; 38]. When there are only two rectangular stubs and no elliptical ring resonator in the structure, a broad continuous spectrum with relatively high transmittance is formed. The transmission spectrum of the single elliptical ring resonator has shapes of the Lorentzian resonance and Fano resonance, which are considered as discrete narrowband states. When both two rectangular stubs and the elliptical ring resonator exist in the meantime, the coupling of two discrete narrowband states and a broad continuous state will generate a symmetric Lorentzian resonance, as well as an ultrasharp and asymmetrical Fano resonance in the system, as is shown in Figure 2. In this article, we mainly study the characteristics of the Fano resonance. Figure 1: The 2D schematic of the structure composed of a metal–insulator–metal (MIM) with two symmetric rectangular stubs coupled with the elliptical ring resonant cavity. The expression of the incident electric and magnetic field is shown in the upper right corner of the picture. To deeply understand the principle of the Fano resonance, we study normalized Hz field distributions of the MIM waveguide coupled with the single elliptical ring resonator, the MIM waveguide with single two rectangular stubs, as well as the entire structure at the resonance dip. When \(\lambda=1105\) nm, a strong Hz field appeared in the left rectangular stub and elliptical ring resonator (as shown in Figure 3a). The field distribution of the elliptical ring resonator is symmetric on the reference line. SPPs are confined in the left stubs and elliptical ring resonator because of the destructive interference of the two excitation pathways. The two pathways include a broad resonance spectrum from the rectangular stub resonator and a narrow one from the elliptical ring resonator. As a result, the SPPs do not transmit to the other side of the MIM waveguide. When \(\lambda=1545.5\) nm, the SPPs scarcely resonate in the rectangular stub, which shows that the incident SPPs and the SPPs which escape from the stub resonator into the MIM waveguide have an interaction. That is to say, at this time a coherence enhancement occurs in the MIM waveguide. Herein, the rectangular stub resonator can be thought as an FP cavity. The FP cavity is composed of an air dielectric layer sandwiched between two silver dielectric layers. The FP cavity forms a grating structure. The structure makes sure that the incident light diffracts in order to excite the propagation mode in the waveguide. As a result, the interference between the diffracted light and the directly transmitted light is more likely to happen. It accounts for the phenomenon that the SPPs scarcely resonate in the rectangular stub. Compared with Figure 3d, it is obvious that when two rectangular stubs are added into the structure, a stronger resonance generates in the elliptical ring resonator, which can facilitate the Fano resonance. Meanwhile, when a strong resonance is generated, the SPPs are nearly limited in the elliptical ring resonator.