## 2 Structural Principle, Detection Mechanism, and Performance Indicators
### Structural Principle of Optical Waveguide Refractive Index Sensing
#### 2.1.1. Refractive Index Sensing Principle of a Microring Structure  
Optical waveguide MRRs have good wavelength selection characteristics, high quality factor (Q), and compact structure, and their ring cavity can provide optical feedback. Therefore, they can be easily integrated with other optoelectronic devices and are widely used in integrated optoelectronics, optical communications, and other fields. An MRR is generally composed of a microring waveguide and a strip waveguide. Light first enters the strip waveguide from the input end and then passes through the coupling area of the ring waveguide. The evanescent waves are excited by the light in the waveguide, some of them couple to the ring waveguide, while some couple to the strip waveguide in the near-field. The light coupled to the ring waveguide interferes with the light in the strip waveguide after traveling around the ring. In addition, the light waves that satisfy the resonant condition of the ring waveguide form a relatively strong optical field; that is, a resonance enhancement effect is generated, and the ring mode propagates circularly in the ring waveguide. Therefore, the guided wave optics theory and coupled mode theory form the theoretical basis of the MRR transmission principle [15]. The shapes of the MRR can be circular (Figure 1a), disk-shaped (Figure 1b), or racetrack-like. The theory of guided wave optics involves studying the transmission and electromagnetic field distribution of light in a waveguide. The function of the optical waveguide is to confine the light beam to the core layer and guide it to propagate along a certain direction. Because light is also an electromagnetic wave, the theory of electromagnetism based on [PERSON]'s equations applies to optical research. Furthermore, the Helmholtz equations were obtained using [PERSON]'s equations. Therefore, when analyzing the optical field in the optical waveguide, the specific electromagnetic field distribution can be obtained by solving the Helmholtz equations, as follows:  
\[ \begin{split}\
abla^{2}\overset{\rightarrow}{E}+k^{2}\overset{ \rightarrow}{E}=0\ ;\\ \
abla^{2}\overset{\rightarrow}{H}+k^{2}\overset{\rightarrow}{H}=0,\end{split}  \tag{1}\]  
where \(\
abla\) is Hamiltonian, \(k\) is wave number, \(\overset{\rightarrow}{E}\) is electromagnetic vector, and \(\overset{\rightarrow}{H}\) is magnetic field vector. The coupled-mode theory is derived from the transmission matrix theory,  
Figure 1: (a) Annular MRR; (b) disk MRR. which is a simplification of the guided-wave optics theory in specific cases and can rapidly solve the problems of certain types of waveguides [16]. The MRR structure can be considered as a coupler, and the general form of the coupled-mode equation is derived. Directional coupling can be used to exchange the optical power between the strip waveguides, and bending coupling can cause the light in the microring waveguide to resonate. An MRR can be designed using the resonance and filtering characteristics of the microring. The principle of resonance in the microring is based on the interference of light. Light is input into the waveguide from one end and transmitted to the coupling area. Next, the light is coupled to the ring from the straight waveguide. When the wavelength of the light is an integer multiple of the length of the ring, resonances occur, yielding an output of light of the resonance wavelength at the other end. Incident light refers to a certain range of broad-spectrum light sources rather than specific wavelengths. In related research on MRRs, the microring resonance equation must be satisfied, as shown in Equation (2). \[2\pi Rn_{eff}=m\lambda, \tag{2}\]  
where \(R\) is the resonance radius of the microring, \(n_{\mathit{eff}}\) is the effective refractive index, \(m\) is the resonance base and a positive integer, and \(\lambda\) is the resonance wavelength. The resonance base \(m\) is an important parameter because it is the mode order.