# Switchable Fiber Ring Laser Sensor for Air Pressure Based on Mach-Zehnder Interferometer
## 2 Principle of Operation and Its TCFMI Pressure Analysis  
The TCFMI was fabricated using a segment of TCF with a small core diameter spliced between two SMFs. The SMF sections of Nufern 1060-XP (East Granby, CT, USA), with a core diameter of 5.8 \(\upmu\)m for wavelengths ranging from 980-1600 nm, were employed. Meanwhile, the model of a segment of TCF is Nufern 460-HP; it has a core diameter of 2.5 \(\upmu\)m and operates from 450 nm to 600 nm. It was manufactured via fusion splicing using an FSM-100M (Singapore) Fujikura splicer. Table 1 shows the main parameters of the fibers used to produce the TCFMI. Figure 1 shows the segment of TCF(460-HP) in the middle section of the interferometer has a small length of \(L=8.5\) cm, which was used to obtain minimal insertion losses in the interference pattern, approximately \(-11\) dB, and reaching a depth in the fringes of \(-20\) to \(-25\) dB. Figure 2 shows the spectrum signal generated by the proposed TCFMI. The main principle of the proposed TCFMI can be explained by considering the general theory of the operation of the modal interferometer principle. Equation (1) determines the spectrum response:  
\[I=I_{1}+I_{2}+2\sqrt{I_{1}I_{2}}cos\left[\frac{2\pi\left(n_{eff}^{co}-n_{eff}^{ cl}\right)}{\lambda}L\right] \tag{1}\]  
where \(I_{1}\) and \(I_{2}\) are the intensities of the core and cladding, respectively, of the Nufern 460-HP fiber, \(\lambda\) is the free space wavelength of the light fiber material, \(L\) is the length of the segment of TCF, and finally, \(n_{eff}^{co}\) and \(n_{eff}^{cl}\) represent the effective refractive index involved. When the fundamental mode propagates through the first section of the Nufern 1060-XP fiber until it reaches the first interface of the TCMFI, high-order cladding modes are generated in the Nufern 460-HP fiber due to the misalignment of the core of the fibers involved in this. Upon reaching the second interface, part of these modes are recoupled to the Nufern 1060-XP fiber core, and the excited fiber cladding modes interfere with the core mode, forming some local maximum or minimum interference points, respectively. By \begin{table}
\begin{tabular}{l c c} \hline
**Items** & **SMF** & **TCF** \\ \hline Core diameter (\(\upmu\)m) & 5.8 & 2.5 \\ \hline Cladding diameter (\(\upmu\)m) & 125 & 125 \\ \hline Refractive index of core & 1.4512 & 1.4505 \\ \hline Refractive index of cladding & 1.447 & 1.447 \\ \hline \end{tabular}
\end{table} 
Table 1: 1060-XP and 460-HP fiber parameters. considering a local minimum at the endpoint, i.e., a transmit dip in the transmit spectrum, the relative phase shift of the two interfering modes can be described as:  
\[2\pi\Big{(}n_{eff}^{co}-n_{eff}^{cl}\Big{)}\frac{L}{\lambda_{D}}=(2k+1)\pi \tag{2}\]  
Here, \(k\) is an integer; then, the intensity of interference light reaches a minimum, corresponding to the wavelength as follows [41]:  
\[\lambda_{D}=\frac{2\Big{(}n_{eff}^{co}-n_{eff}^{cl}\Big{)}L}{(2k+1)} \tag{3}\]  
Equation (3) shows that either the effective refractive index or length differences can affect the fringe pattern or both changes. In our case, these parameters were controlled when the TCFMI interacted with the pressure inside the gas cell. The scheme in Figure 3 was implemented to study the pressure response of the TCFMI. This setup employed a QPHOTONICS QDFBLD-980-350 (Ann Arbor, MI, USA) semiconductor laser diode inside a Thorlabs CLD1015 controller as our pump source, followed by a WDM, which is a wave division multiplexer; as an active medium, we used ytterbium-doped fiber (Thorlabs YB1200, Newton, NJ, USA), with a length of 2.8 m, a core diameter close to 10 um, and a high concentration of nearly 9 \(\times\) 10\({}^{19}\) ions/cm\({}^{-3}\), which worked at the same wavelength as the pumping source. Afterward, a polarization controller was set to manipulate the polarization stage; moreover, an optical fiber isolator  
Figure 1: TCFMI schematic diagram. Figure 2: Optical spectrum of the TCFMI.