2.2.1. Structural Design

In order to enhance the sensitivity of the fiber MZI, the SMBMS MZI was developed on the basis of the SMS MZI, as shown in Figure 2; a waist-enlarged bitaper was fused to the middle multimode segment. The function of the fusion point in front of the waistenlarged bitaper was to distribute the light transmitted through the single mode fiber to the multimode fiber core and fiber cladding, while the function of the waist-enlarged bitaper was to redistribute the light transmitted in the core and the cladding, which excited higher-order modes and entered the multimode cladding transmission section behind the waist-enlarged bitaper. The function of the fusion point behind the waist-enlarged bitaper was to couple the light transmitted in the core and the cladding to the output fiber, and through the spectrometer for storage and analysis. In order to verify whether the position of the waist-enlarged bitaper had an effect on the sensor performance, we made a control sensor, and set the position of the waist-enlarged bitaper at one-third of the sensing arm, as shown in Figure 3. *Micromachines* **2022**, *13*, x FOR PEER REVIEW 2 of 19

**Figure 2.** Schematic diagram of the Single mode–Multimode–Bitaper–Multimode–Single mode Mach–Zehnder Interferometer (SMBMS MZI). **Figure 2.** Schematic diagram of the Single mode–Multimode–Bitaper–Multimode–Single mode Mach–Zehnder Interferometer (SMBMS MZI).

**Figure 3.** Schematic diagram of the SMBMS MZI with the position of the bitaper set to one third of the sensing arm. **Figure 3.** Schematic diagram of the SMBMS MZI with the position of the bitaper set to one third of the sensing arm.

### 2.2.2. Analysis of Coupling Response Characteristics of Waist-Enlarged Bitaper

The following section focused on analyzing the response characteristics of the optical fiber waist-enlarged bitaper. The inverted taper was composed of a multimode fiber and an inverted taper fiber with a gradually enlarged diameter. The waist-enlarged bitaper part played the role of expanding the beam of the fundamental mode spot. The waistenlarged bitaper directly affected the propagation response characteristics of the beam in it, as shown in Figure 4. Therefore, we used the waist-enlarged bitaper to make more light waves leak into the cladding and excite higher-order modes. In the following, by explaining the propagation response characteristics of Gaussian beams in tapered fibers, the reasons for the improvement of the coupling efficiency by the waist-enlarged bitaper are theoretically given. *Micromachines* **2022**, *13*, x FOR PEER REVIEW 4 of 19

**Figure 4.** Optical path analysis of waist-enlarged bitaper in SMBMS MZI. **Figure 4.** Optical path analysis of waist-enlarged bitaper in SMBMS MZI.

The propagation response characteristics of Gaussian beams in tapered fibers are given by the spot size *ω(z)* and the wavefront curvature radius *R(z),* used to describe the The propagation response characteristics of Gaussian beams in tapered fibers are given by the spot size *ω(z)* and the wavefront curvature radius *R(z),* used to describe the propagation response characteristics of Gaussian beams [17].

$$
\omega^2(z) = \omega^2(0) \cdot \left[A^2 + \left(\mathcal{B}/k\omega^2(0)\right)^2\right]^{\frac{1}{2}}\tag{7}
$$

ω

$$\frac{1}{R(z)} = \frac{\omega'(z)}{\omega(z)}\tag{8}$$
 
$$\text{fations can change to the following form:}$$

sin

2

θ

First, the above two equations can change to the following form:

propagation response characteristics of Gaussian beams [17].

ω

$$
\omega^2(z) = \omega^2(0)[(g(0)/g(z))\cos^2\theta + \frac{1}{g(0)g(z)} \cdot \frac{\sin\theta^2}{k\_1^2\omega(0)}]\tag{9}
$$

2 2 2 This can be obtained by derivation:

ω

med need by derivation:

$$
\omega^2(z) = a\_0 f(z) / (\frac{2\pi m\_1}{\lambda})\sqrt{2\Delta} \tag{10}
$$

This can be obtained by derivation: The spot magnification factor M can be expressed as:

ω

$$M = \omega(z) / \omega(0) = f(z) = 1 + c \cdot \exp \gamma(z) \tag{11}$$

λ The spot magnification factor M can be expressed as: *M* =ω(*z*)/ω(0) = *f* (*z*) =1+ *c* ⋅expγ (*z*) (5) Therefore, at the position of the waist-enlarged bitaper, the light spot was enlarged. In a coupled system, the coupling tolerance of the system was only related to the spot size. The larger the spot size, the larger the vertical and horizontal tolerances. In a word, the waist-enlarged bitaper played the role of spot amplification in the optical fiber coupling system, improving the coupling tolerance of the system, and redistributing the spectrum.

0

### Therefore, at the position of the waist-enlarged bitaper, the light spot was enlarged. 2.2.3. Manufacture

2.2.3. Manufacture

In a coupled system, the coupling tolerance of the system was only related to the spot size. The larger the spot size, the larger the vertical and horizontal tolerances. In a word, the Fiber MZI waist-enlarged bitaper is manufactured using a common fusion splicer (FITEL, S178A). In the sensing system, a fiber cleaver (Furukawa, S325) was used to obtain

waist-enlarged bitaper played the role of spot amplification in the optical fiber coupling system, improving the coupling tolerance of the system, and redistributing the spectrum.

Fiber MZI waist-enlarged bitaper is manufactured using a common fusion splicer

(FITEL, S178A). In the sensing system, a fiber cleaver (Furukawa, S325) was used to obtain a flat fiber end, and make a waist-enlarged bitaper based on commercial fusion splicers. Multimode optical fiber has a core diameter of 40 µm and a cladding diameter of 125 µm. By increasing the splicing time to 500 ms, and increasing the splicing strength to 155 unit, the waist-enlarged bitaper is obtained. Figure 5 is the photo of the waist-enlarged bitaper observed through a microscope. The diameter of the waist-enlarged bitaper is expanded

to 152 µm, and the length of the cone region is 350 µm.

a flat fiber end, and make a waist-enlarged bitaper based on commercial fusion splicers. Multimode optical fiber has a core diameter of 40 µm and a cladding diameter of 125 µm. By increasing the splicing time to 500 ms, and increasing the splicing strength to 155 unit, the waist-enlarged bitaper is obtained. Figure 5 is the photo of the waist-enlarged bitaper observed through a microscope. The diameter of the waist-enlarged bitaper is expanded to 152 µm, and the length of the cone region is 350 µm. *Micromachines* **2022**, *13*, x FOR PEER REVIEW 5 of 19

**Figure 5.** The waist-enlarged bitaper in SMBMS MZI. **Figure 5.** The waist-enlarged bitaper in SMBMS MZI.

*2.3. Etched Single mode–Multimode–Bitaper–Multimode–Single mode Mach–Zehnder Interferometer (ESMBMS MZI)*
