## 3 Numerical Results and Discussion
### Sensor-Performance Investigations  
It was clearly found that the proposed D-shaped sensor's performance relied on the resonance wavelength moving, induced by changing the detected analyte RI, which might have been caused by environment variations surrounding the sensing layer. According to Section 3.1, due to the uniform distribution of the loss spectra and the single peak in the investigated band, it was very suitable for sensing applications.  
Figure 5a describes loss spectra for different analyte RIs between 1.33 and 1.39 with \(d_{1}=2.8\)\(\upmu\)m, \(d_{2}=d_{3}=7.0\)\(\upmu\)m, \(\Lambda_{1}=6.0\)\(\upmu\)m, \(\Lambda_{2}=6.2\)\(\upmu\)m, and \(T_{Au}=40\) nm. By the WIM, when the RI varied from 1.33 to 1.39, the corresponding resonance wavelength could be tuned from 2.9 to 3.6 \(\upmu\)m. Results showed that the average \(\Delta\lambda_{peak}\) was about 88.5 nm; hence, average WS was 8850 nm/RIU and RI resolution was \(1.13\times 10^{-5}\) RIU in this sensor. Maximal WS emerged when the analyte RI varied from 1.36 to 1.37. The maximal value could reach 11,500 nm/RIU, and the corresponding RI resolution was \(8.7\times 10^{-6}\) RIU. Figure 5b describes amplitude sensitivity with different analyte RIs by the AIM. We found that the AS peak decreased as RI changed from 1.33 to 1.39, and maximal AS was 230 RIU\({}^{-1}\) when n\({}_{\text{a}}\) was 1.33.  
Figure 5: **(a)** Loss spectra for different analyte RI \(n_{a}\) between 1.33 and 1.39; **(b)** amplitude sensitivity (AS) with different analyte RIs when \(d_{1}=2.8\)\(\upmu\)m, \(d_{2}=d_{3}=7.0\)\(\upmu\)m, \(\Lambda_{1}=6.0\)\(\upmu\)m, \(\Lambda_{2}=6.2\)\(\upmu\)m, and \(T_{Au}=40\) nm; **(c)** linear correlation degree for resonance wavelength versus different analyte RIs; **(d)** figure of merit (FOM) of this sensor versus different analyte RIs.  
Next, the linearity of the sensor was analyzed. Figure 5c shows that resonance points were distributed on both sides of the red solid line. This indicated an approximately linear relationship between analyte RIs and resonance wavelengths. By calculation, the fitting equation for this sensor was expressed as follows:  
\[\lambda_{o} = 8.67143n_{a}-8.57857,\ (1.33\leq n_{a}\leq 1.39) \tag{8}\]  
where \(n_{a}\) and \(\lambda_{o}\) denote analyte RI and operating wavelength, respectively. The adjusted \(R^{2}\) was a powerful tool to help us judge linearity [43]. According to these resonance points, an adjusted \(R^{2}\) of 0.99519 was obtained that verified a highly linear correlation degree between \(n_{a}\) and \(\lambda_{o}\).  
Figure 5d shows the FOM of our sensor versus different analyte RIs. As the analyte RI varied between 1.33 and 1.39, FOM first increased and then decreased; it was obvious that, when \(n_{a}=1.37\), an adequate FOM of 142 could be achieved.  
For comparison, Table 1 displays the major performance parameters of our sensor and some existing D-shaped sensors. We found that the maximal WS, RI resolution, linearity, and FOM of our D-shaped sensor could then reach the highest order of magnitude, but sensors simultaneously with these characteristics are uncommon. Among existing midinfrared sensors, for example, [PERSON] et al. proposed a dual open-ring structured sensor with average WS of 5500 nm/RIU, AS of 333.8 RIU\({}^{-1}\), and maximal resolution of \(7.69\times 10^{-6}\) RIU [44]. [PERSON] et al. obtained a D-shaped sensor with maximal WS of 11055 nm/RIU and RI resolution of \(9.05\times 10^{-6}\)[21], but they did not consider linearity and FOM. In comparison, our sensor combined four excellent performance advantages and possesses broad prospects for various midinfrared applications.