## 2 Structural Principle, Detection Mechanism, and Performance Indicators
### Refractive Index Sensing Mechanism
#### 2.2.2 MZI Sensor Detection Mechanism  
To derive the transfer functions of the input and output light intensity, the light beam splitter and coupler consisting of the two Y-shaped branch structures in the MZI refractive index sensor are divided into different sections, as shown in Figure 5.  
Figure 4: (**a**) Resonant wavelength shift method; (**b**) intensity variation method.  
Figure 3: Sensing flowchart of resonance wavelength drift method.  
Let \(E_{m}\) be the normalized mode field distribution (\(I_{in}=|E_{in}|^{2}=1\)) of the waveguide; after the incident light passes through the first Y-branch structure, the light field is divided into \(E_{1}\) and \(E_{2}\) components. In addition, the field distribution in section B can be expressed as follows:  
\[\begin{pmatrix}E_{1}\\ E_{2}\end{pmatrix}=\begin{pmatrix}a\\ b\end{pmatrix}\cdot E_{in}, \tag{8}\]  
where \(a\) and \(b\) represent the amplitude distribution coefficients of the Y-branch beam splitter and satisfy the condition \(0\leq a^{2}+b^{2}\leq 1\). When \(a^{2}+b^{2}=1\), this is considered an ideal situation without transmission loss. When \(E_{1}\) and \(E_{2}\) pass through the sensing and reference arms of the MZI refractive index sensor, respectively, the field distribution in section C can be expressed as follows:  
\[\begin{pmatrix}E_{1}^{\prime}\\ E_{2}^{\prime}\end{pmatrix}=\begin{pmatrix}exp[i(\Delta\phi_{m})]&0\\ 0&1\end{pmatrix}\cdot\begin{pmatrix}E_{1}\\ E_{2}\end{pmatrix}; \tag{9}\]  
\[\Delta\phi_{m}=\frac{2\pi}{\lambda}\cdot\Delta N\cdot L, \tag{10}\]  
where \(\Delta\phi_{m}\) is the phase difference introduced by the object under test, \(\lambda\) is the wavelength of the incident light, \(L\) is the window length of the sensing arm of the sensor, and \(\Delta N\) is the difference in the refractive index between the sensing and reference arms. When the light components pass through the sensing arm and recombine at the Y-shaped branch, the field distribution in section D can be expressed as follows:  
\[E_{out}=\begin{pmatrix}a&b\end{pmatrix}\begin{pmatrix}E_{1}^{\prime}\\ E_{2}^{\prime}\end{pmatrix}=\begin{pmatrix}a&b\end{pmatrix}\begin{pmatrix}exp[i( \Delta\phi_{m})]&0\\ 0&1\end{pmatrix}\begin{pmatrix}a\\ b\end{pmatrix}E_{in}. \tag{11}\]  
The relationship between the output and input light intensities of the MZI refractive index sensor can then be expressed based on Equation (8) as follows:  
\[\frac{I_{out}}{I_{in}}=\bigg{|}\frac{E_{out}}{E_{in}}\bigg{|}^{2}=a^{4}+b^{4}+ 2a^{2}b^{2}cos(\Delta\phi_{m})=(a^{4}+b^{4})\bigg{[}1+\frac{2a^{2}b^{2}}{a^{4} +b^{4}}cos(\Delta\phi_{m})\bigg{]}. \tag{12}\]  
Therefore, when \(a^{2}+b^{2}=1\), there is no loss in the Y branch, and the energy transfer coefficient is 1 in this case. Ideally, there is no loss in the Y branch; that is, (\(a^{2}+b^{2}=1\)) is satisfied, and the transfer function between the output and input light intensities can be expressed as follows:  
\[\frac{I_{out}}{I_{in}}=\frac{1}{2}[1+cos(\Delta\phi_{m})]. \tag{13}\]  
Generally, only the most ideal situation is considered; that is, the relationship between the output and input light intensities satisfies Equation (13).  
Optical waveguide sensors mainly function based on the evanescent field principle. In an MZI sensor, the evanescent field changes via interaction with the external object under  
Figure 5: MZI refractive index sensing structural division.  
test, resulting in a change in the refractive index. The refractive index change is converted into a measurable change in the output power or transmission spectrum. Therefore, to improve the sensitivity of the entire sensing device, the interaction strength between the evanescent field and external environment must be enhanced. This can be achieved mainly by changing the structure or material of the device.