*2.1. Single mode–Multimode–Single mode (SMS) Mach–Zehnder Interferometer (MZI)* 2.1.1. Structural Design

The basic structure of the fiber MZI involved in this paper is SMS MZI, as shown in Figure 1. This MZI included the following parts: the left side was the single mode fiber of the input light, the middle section was a 20 mm long multimode fiber, and the right side was the single mode fiber of the output light. The working principle of MZI is as follows: When light is transmitted from a single mode fiber into a multimode fiber, a part of the cladding mode is excited. Due to the different refractive indices of the core and cladding, there is a certain optical path difference between the core mode and the cladding mode. When light re-entered the single mode fiber from the multimode fiber, the core mode and the cladding mode were coupled and interfered to form interference fringes.

**Figure 1.** Schematic diagram of the Single mode–Multimode–Single mode Mach–Zehnder Interferometer (SMS MZI). **Figure 1.** Schematic diagram of the Single mode–Multimode–Single mode Mach–Zehnder Interferometer (SMS MZI).

### 2.1.2. Principles of Refractive Index Sensing

The occurrence of the interference phenomenon depends on the optical path difference between the two beams. It can be seen from the structure that the core path and cladding path of the MZI sensing arm are of equal length. There is a direct relationship to the frequency spectrum of wavelengths, which is the effective refractive index difference between the different light wave modes. Assuming two different guided modes *LP*0*<sup>m</sup>* and *LP*0*<sup>n</sup>* (*m*, *n* are positive integers) are in the multimode fiber, the phase difference between them is:

$$
\Delta\varphi = \frac{2\pi (n\_{\varepsilon ff}^m - n\_{\varepsilon ff}^n)L}{\lambda} = \frac{2\pi \Delta n\_{\varepsilon ff}^{m\_{\varepsilon}n}L}{\lambda} \tag{1}
$$

where *L* is the length of the fiber and *λ* is the input wavelength of the light source. ∆*n m*,*n e f f* is the effective refractive index difference between *LP*0m and *LP*0n. The equation for the effective refractive index is as follows:

$$m\_{eff} = \frac{\pi \cdot s}{L} \tag{2}$$

where *n*eff is the effective refractive index, *n* is the refractive index in the medium, *s* is the distance traveled by the light, and *L* is the length of the interference arm. Equation (2) shows that when the refractive index of the medium and *L* are constant, the mode with the higher order has a larger diffusion angle. That is, the effective refractive index of the high-order mode is greater than that of the low-order mode.

The intensity of the output light can be expressed as:

$$I = I\_1 + I\_2 + 2\sqrt{I\_1 I\_2} \cos \Delta \varphi \tag{3}$$

where *I* is the output light intensity of the MZI, *I<sup>1</sup>* and *I<sup>2</sup>* are the light intensity of the guided mode *LP*0*m*, and the light intensity of *LP*0*<sup>n</sup>* in the interference core. According to Equation (3), the phase difference equation corresponding to the valley is as follows:

$$
\Delta \varphi = (2m+1)\pi \tag{4}
$$

*Micromachines* **2022**, *13*, x. https://doi.org/10.3390/xxxxx www.mdpi.com/journal/micromachines

Substituting Equation (1) into Equation (4), we obtain:

$$\frac{2\pi\Delta n\_{eff}^{m,n}L}{\lambda\_m} = (2m+1)\pi\tag{5}$$

where *m* is an integer, representing the interference order and *λ<sup>m</sup>* is the center wavelength of m-order interference.

According to Equations (4) and (5), Equation (6) is obtained:

$$
\delta\lambda\_m \approx 2\pi \text{L}\delta n\_{eff} \tag{6}
$$

where *δλ<sup>m</sup>* is the center wavelength shift of the m-th order interference fringes and *δne f f* is the change caused by the refractive index of the sucrose solution. Equation (6) demonstrates that when the interference length *L* is constant, the shift amount of the interference valley wavelength changes with the change of the refractive index of the external liquid. Therefore, the refractive index of the sucrose solution is measured by monitoring the shift in the wavelength of the *m*-th valley of the MZI.
