## 2 Model and Theory  
The midinfrared D-shaped PCF-SPR sensor's structure was obtained by side-polishing a three-layer hexagonal-lattice PCF to form a plane that was coated with a gold layer by the sputter-deposition method (SDM) [24]. When the sensor was operating, the gold layer was directly in contact with the analyte to be detected. Figure 1a displays the cross-section illustration of our sensor. The lattice pitch was \(\Lambda_{\rm I}=6.0\upmu\)m, the cladding holes' diameters and two central holes are represented by \(d_{1}\) and \(d_{2}\), respectively, and we first set \(d_{1}=2.8\upmu\)m. Other structural parameters, namely, channel length, gold-layer thickness, and analyte RI are characterized by \(\Lambda_{2}\), \(T_{Au}\), and \(n_{a}\), respectively. The background material of the PCF was Si. Resonance wavelength between Si and the excitation gold layer was located exactly in the midinfrared band. The RI of Si can be characterized by a Sellmeier equation [25] that is described as  
\[n_{Si}(\lambda)\ =\ \sqrt{\varepsilon+\frac{A}{\lambda^{2}}+\frac{B\lambda_{1}^{2} }{\lambda^{2}-\lambda_{1}^{2}}} \tag{1}\]  
where \(\lambda\) represents a free space wavelength, and detailed parameters were \(\varepsilon\) = 11.6858, \(\lambda_{1}\) = 1.1071 \(\upmu\)m, \(A\) = 0.939816 \(\upmu\)m\({}^{2}\), and \(B\) = 8.10461\(\times\)10\({}^{-3}\), respectively. The dielectric constant of the gold sensing layer in the wavelength region from 0.248 to 6.2 \(\upmu\)m can be characterized by the Drude-Lorentz (DL) model [26; 27; 28]:  
\[\varepsilon(\omega)\ =\ \varepsilon_{1}+i\varepsilon_{2}\ =\ \varepsilon_{\infty}- \frac{\omega_{p}^{2}}{\omega(\omega+i\omega_{c})} \tag{2}\]  
where \(\omega\ =\ 2\pi/\lambda\), and detailed parameters were \(\varepsilon_{\infty}\ =\ 9.75\), \(\omega_{p}\ =\ 1.36\ \times\ 10^{16}\), and \(\omega_{c}\ =\ 1.45\times\ 10^{14}\), respectively. In the sensor's performance analysis, we adopted confinement loss (CL) [29; 30] to characterize the detection signal. Usually, CL can be expressed by  
\[\alpha\ =\ 8.686\times\frac{2\pi}{\lambda}\times Im\Big{(}n_{eff}\Big{)}\times 1 0^{4}(dB/cm) \tag{3}\]  
where \(\lambda\) represents the incident-light wavelength and \(Im\Big{(}n_{eff}\Big{)}\) represents the effective RI imaginary part for the core fundamental mode (FM). In order to facilitate understanding the following formulas, \(\alpha\) in dB/cm here represents CL. In our works, sensor performance was examined on the basis of wavelength sensitivity (WS) [31], amplitude sensitivity (AS) [32], RI resolution [33], linearity [34], and FOM [35]. Common wavelength-interrogation method (WIM) and amplitude-interrogation method (AIM) [36] were employed for analyzing these performance parameters. Figure 1: (**a**) Schematic illustration of designed sensor and (**b**) finite-element-method (FEM) discretized mesh. Usually, WS, AS, and RI resolution are the three most common parameters for measuring sensor performance. WS is described by  
\[S(\lambda)\ =\ \Delta\lambda_{peak}/\Delta n_{a}\ (nm/RIU) \tag{4}\]  
where \(\Delta\lambda_{peak}\) represents the shift of the resonance wavelength and \(\Delta n_{a}\) represents RI variations; here, \(\Delta n_{a}\) is 0.01. AS is described by  
\[S_{A}(\lambda)\ =\ \frac{1}{\alpha(\lambda,n_{a})}\frac{\partial\alpha(\lambda,n _{a})}{\partial n_{a}}\Big{(}RIU^{-1}\Big{)}, \tag{5}\]  
where \(\alpha(\lambda,n_{a})\) represents CL when the detected RI was \(n_{a}\), and \(\partial\alpha(\lambda,n_{a})\) represents the CL difference when RI variation was \(\partial n_{a}\). RI resolution is described by  
\[R\ =\ \Delta n_{a}.\frac{\Delta\lambda_{min}}{\Delta\lambda_{peak}}(RIU), \tag{6}\]  
where \(\Delta\lambda_{min}\) is wavelength resolution assuming that \(\Delta\lambda_{min}\) is 0.1 nm for the purposes of calculation [37; 38]. The FOM was adopted for evaluating accuracy of detection in sensors.