## 2 Structural Principle, Detection Mechanism, and Performance Indicators
### Performance Indicators for Refractive Index Sensing  
Optical waveguide refractive index sensing is evaluated based on performance indicators, such as the free spectral range (\(FSR\)), quality factor (\(Q\)), wavelength full width at half maximum (FWHM), distinguish ratio (\(DR\)), sensitivity (\(S\)), and limit of detection (\(LOD\)) [21]. The sensitivity and detection limit are the two main criteria used for evaluating the performance of the optical biosensors, whereas the performance of waveguide optical biochemical sensors depends on the strength of the interaction between the solution or substance on the surface and light. The key performance indicators are as follows. 1. \(FSR\): This parameter refers to the wavelength difference between two adjacent resonance peaks in the spectrum and can be expressed as follows: \[\lambda_{m}=\frac{2\pi n_{eff}R}{m};\] (16) \[FSR=\lambda_{m}-\lambda_{m-1}.\] (17)  
Let the group effective refractive index be \(n_{g}=n_{eff}-\lambda\frac{dn_{eff}}{d\lambda}\), then Equation (17) becomes:  
\[FSR=\frac{\lambda^{2}}{2\pi n_{g}R}. \tag{18}\]  
It can be seen from Equation (10) that factors affecting the size of the \(FSR\) include the device radius \(R\), light wavelength \(\lambda\), and effective refractive index \(n_{eff}\) of the waveguide. In refractive index sensing applications, the \(FSR\) is an important performance indicator, which typically represents the detection range of the refractive index sensor. 2. Q: This parameter describes the sharpness of the resonant peak position in the spectrum of the MRR. The higher the \(Q\) value, the more sensitive the microring is to wavelength changes, and the better the performance of the sensor. It is expressed as follows: \[Q=\frac{\lambda}{\Delta\lambda_{3 dB}}.\] (19)
3. \(F\)_Fwhm_: In the transmission spectrum of the microring, the full width at half maximum is the distance between two adjacent wavelengths when the power is reduced to half of the maximum power, which is also referred to as 3 dB bandwidth. This parameter determines the maximum data rate that can be processed and affects the detection lower limit of the sensor. FWHM can be determined using the following expression: \[\Delta\lambda_{3 dB}=\frac{\lambda^{2}}{2\pi Rn_{eff}}\cdot\frac{1-\alpha t^{2 }}{t\sqrt{\alpha}}.\] (20)
4. \(DR\): This parameter is the resolution representing the minimum change that can be detected by the refractive index sensor. The output changes only if the change in the substance under test (such as the refractive index) is greater than the resolution. The resolution varies at each level within the sensing range of the sensor. This resolution can be expressed as three times the ratio between the system standard deviation \(\sigma\) and the sensitivity \(S\): \[DR=3\frac{\sigma}{S}.\] (21)
5. \(S\): Sensitivity and quality factors have an inverse relationship. In optical waveguide refractive index sensing, sensitivity can be expressed as follows: \[S=\frac{\partial I_{out}}{\partial(n_{m})}=\frac{\partial I_{out}}{\partial( \Delta\phi_{m})}\frac{\partial(\Delta\phi_{m})}{\partial(\Delta N)}\frac{ \partial(\Delta N)}{\partial(n_{m})}.\] (22) Each component in the formula can be further defined as follows:\(\frac{\partial I_{out}}{\partial(\partial\mu_{m})}=-\frac{\alpha I_{in}}{2}Vsin( \Delta\varphi_{m}+\Delta\varphi_{0})\), the sensitivity function is a periodic function, and sensitivity changes periodically during testing; \(\bullet\)\(\frac{\partial(\Delta\varphi_{m})}{\partial(\Delta N)}=2\pi\frac{L}{A}\), where \(L\) is the size of the sensing window; the larger the value of \(L\), the more contact occurs between the sample under test and the sensing arm, the bigger the effective refractive index change of the guided mode, and the higher the sensitivity.