## Formation mechanism of Fano peaks

In order to investigate the optical properties of the structure, the transmission spectra of the structure without baffle, the MIM waveguide with baffle, and the whole structure are shown in Fig. 2. The geometric parameters of the structure are set as *r* = 180 nm, *R* = 180 nm, *d* = 20 nm, and *g* = 10 nm. For the structure without baffle, as shown in the red line, there are three transmission dips at 1390 nm, 1800 nm, and 2235 nm in the transmission spectrum, which can be treated as narrow discrete states. Meanwhile, the MIM waveguide with baffle can formed a broad continuum state as shown in the black line. For the whole structure, as a result of the interference of the discrete states and the continuum state, there are three Fano peaks (FR1, FR2, and FR3) at the wavelengths of 1364 nm, 1790 nm, and 2222 nm, as shown in the green line.

To further understand the mechanism of Fano resonances of the designed structure, the magnetic field distributions (Hz) at the resonant wavelengths are simulated as shown in Fig. 3. It can be seen that the energy is almost distributed in the SSC at the wavelengths of 1364 nm and 2222 nm. On the contrary, most of the energy is distributed in the RC at the wavelength of 1790 nm. This phenomenon indicates that both FR1 and FR3 may be more effected by SSC and the RC may have more powerful affect on FR2. The two magnetic field distributions of FR1 and FR3 show that the SPPs coupled into SSC meet the resonance conditions, and the resonance mode orders at FR1 and FR3 are 2 and 1, respectively. The Fig. 3(b) shows that the SPPs in RC produce a phase shift of 2π and the resonance mode order is 1.

## Tunable Fano resonance and its applications for sensing

First, we investigate the influences of the SSC and RC sizes on Fano resonances of the structure. In Fig. 4(a), the inner radius *R* of SSC increases from 180 nm to 200 nm in step of 5 nm, and the structural parameter *d* is fixed at 20 nm. The approximately linear relationships between the resonance wavelengths and the inner radius of the SSC are shown in Fig. 4(b). When the structure parameter *R* increases from 180 nm to 200 nm, the wavelengths of FR1 and FR3 are increased by 100 nm and 230 nm, respectively, but FR2 keeps unchanged. Analyzing of these results, FR1 and FR3 meet the resonance conditions in SSC, while FR2 meets the resonance conditions in RC. According to Eq. (3), only changing the size of SSC will change the wavelengths of FR1 and FR3, while that of FR2 remains unchanged. Figure 5(a) shows the transmission spectra for different effective radii of the RC from 205 nm to 235 nm with step of 10 nm, and the linear fitting curves between the resonance wavelengths and the effective radius of the RC are shown in Fig. 5(b). As the effective radius *r* increases, the wavelength of FR2 increases from 1790 nm to 2048 nm, and those of FR1 and FR3 remain unchanged. This phenomenon shows that the size of RC only affects FR2. Therefore, FR1 and FR3 can be tuned by changing the structural parameter *R* of the SSC, while FR2 can be tuned by changing the effective radius of the RC. The triple Fano resonances can be well tuned independently.

Successively, we investigate the effect of the refractive index of the insulator in the resonators on the position of Fano resonance peaks. The refractive indexes of the insulator in the SSC and RC increases from 1.0 to 1.12 with a step of 0.04, and the other area refractivity remains unchanged. The corresponding transmission spectra is exhibited in Fig. 6(a). It can be seen that the resonance wavelengths of the triple Fano resonances all appear obvious red-shifts and the Fano resonances are sensitive to the insulator refractive index of the resonators. The phenomenon can be explained by Eq. (1)-(4). With the increase of the refractive index of the insulator in the SSC and RC, the effective refractive index increases, and then the resonant wavelength red shift. Figure 6 (b) shows the linear fitting relationship between the resonant wavelengths of the three Fano peaks and the refractive index of the insulator in the SSC and RC. It can be seen that FR1, FR2, and FR3 all have good performances in linearity. The linear correlation coefficients of FR1, FR2, and FR3 are 0.99992, 0.9993, and 0.99995, respectively.

Based on the linear adjustment of Fano resonance by refractive index of insulator, the designed SPPs waveguide can be used for the refractive index sensing. As an important index for evaluating sensing performance, sensitivity is defined as [32–34]\(S=\Delta \lambda /\Delta n\), where \(\Delta n\) is the change of the refractive index of the insulator in the resonant cavities, and \(\Delta \lambda\) is the variation of the corresponding resonant wavelength. Therefore, the refractive index sensitivities of FR1, FR2 and FR3 can be expressed by the slope of the fitting lines in Fig. 6(b). Those are 1330 nm/RIU, 1785 nm/RIU and 2235 nm/RIU, respectively. It can be seen that FR3 has the largest sensitivity compared with FR1 and FR2. Based on the same dielectric properties of material for RC and SSC, FR3 has the greater sensitivity due to its higher resonant wavelength [35]. In addition, another key factor to sensing performance is FOM which is defined as [36] \({\text{FOM}}=\Delta T/T\Delta n.\) Where represents the transmittance and \(\Delta T\) represents the transmittance change. The structure parameters are the same as the initial values. The FOM values at different wavelengths are shown in Fig. 7. It can be seen that the highest FOM reaches 3387 at 2395 nm. This result shows that the Fano peak is very sharp, which is conducive to the accurate measurement of the peak position [37].

Based on the refractive index sensing performance, the SPPs waveguide structure is suitable for detecting biological parameters that have definite correlation with refractive index, such as the glucose solution concentration. The refractive index of glucose solution can be expressed as [38]:

$${n_{{\text{glucose}}}}=1.1889 \times {10^{ - 4}}C+1.33230545,$$

5

where *n*glucose is the refractive index of solution, *C* represents the concentration of glucose solution. In Fig. 8(a), the concentration of glucose solution in the SSC and RC increases from 0 g/L to 400 g/L with a step of 100 g/L. According to Eq. (5), the refractive index of the resonators can be calculated. The simulation results show that the resonance wavelengths of the triple Fano resonances exhibit red-shifts with the increase of solution concentration. The linear fitting relationship between resonant wavelengths and solution concentration is shown in Fig. 7(b). The slope of fitting straight lines of FR1, FR2 and FR3 are 0.164, 0.214 and 0.266, respectively. The linear correlation coefficients of FR1, FR2 and FR3 are 0.99921, 0.99965, and 0.99977, respectively. Exploiting the equation \({S_{{\text{glucose}}}}=\Delta \lambda /\Delta C\), the calculated sensitivity of glucose solution sensing for FR1, FR2 and FR3 are 0.164 nm/(g/L), 0.214 nm/(g/L) and 0.266 nm/(g/L), respectively. Therefore, the concentration of glucose solution in the resonant cavity can be detected according to the resonant wavelengths.