## 2 MZI Fabrication and Filter Principle  
We fabricate an MZI by using a core-offset fiber array as the filtering mechanism. The MZI was constructed using a standard single-mode SMF28 fiber with a length of 2 m. We utilized a 2 cm piece of the TCEDF (RightWave EDF R37003X; Furikawa, OFS Fitel, Norcross, GA, USA), with a core diameter of 2.90 \(\upmu\)m and cladding of 80 \(\upmu\)m, which was spliced in half using a U-shape [17; 23] core-offset splice. The U-shape was achieved by holding the SMF28 to a steel ruler with masking tape (Figure 1a), applying small marks on the fiber, ruler, and tape to prevent any unintentional movement and keeping it as stable as possible during splicing to avoid any twisting. We made the splice with a 6 cm piece of the TCEDF (shown in Figure 1b), which we manually cut to a length of 2 cm using a ceramic razor (CSW12-5 Thorlabs ceramic fiber scribe; Thorlabs, Newton, NJ, USA), enabling us to control the position of the MZI segments (Figure 2b). We performed the splicing using a Fitel S178 fusion splicer and modified the SMF program parameters to a pre-fusion time of 100 ms, a 1 st first-arc duration time of 400 ms, a \(Z\) pressure distance of 4 \(\upmu\)m, a first-arc power start of +45, an arc offset of +5 \(\upmu\)m, a cleaning arc power offset of \(-\)20, and a cleaning time duration of 100 ms, always placing the TCEDF on the right holder side and the SMF on the left holder side, with the remaining fiber attached to the steel rule. We made the first splice (Figure 2a) by aligning the cores of the SMF (left holder) and the TCEDF (right holder) and then moving the left holder of the fusion splicer 4 \(\upmu\)m upwards on the X-axis. Figure 2b illustrates how the SMF/TCEDF tip was cut to leave a measurement of 2 cm. InFigure 2c, we show how the second splice was made by aligning the TCEDF (left holder) and SMF (right holder) cores and then moved the motor on the X-axis 4 \(\upmu\)m down, resulting in the U-shape sensor depicted in Figure 2d and viewed from the z-side. We tested the MZI by powering it with the ASE source, using the LDC operating at 135 mW and setting the temperature controller at 25 \({}^{\circ}\)C.  
A U-shaped Mach-Zehnder interferometer was fabricated using a segment of thin-core erbium-doped fiber (TCEDF) with a length \"\(L\)\". The operation principle of the fabricated MZI is based on the light input into the SMF. At the left junction of the core offset, the device acts as a splitter, with one part of the light coupled to the core, while the other part  
Figure 1: Fiber clamping for SMF/TCEDF/SMF splices. (**a**) This demonstrates how to adhere the fiber to the metric rod with the masking tape and (**b**) illustrates how splicing is accomplished with the fusion splicer. Figure 2: Core-offset splice SMF/TCEDF/SMF design. (**a**) Illustrates the splicing between fibers, (**b**) shows the TCEDF cut, (**c**) U-shape of the MZI filter, (**d**) exhibits the z-side view of the MZI. is coupled to the cladding of the TCEDF. At the right junction of the core offset, the core and the cladding modes are recombined into the SMF to generate interference. The output intensity of the interference can be expressed by [24,25]:  
\[I=I_{core}+I_{clad}+2\sqrt{I_{core}I_{clad}}\text{cos}\,\emptyset \tag{1}\]  
where \(I\), \(I_{core}\) and \(I_{clad}\) are the intensity of the output in the core mode and cladding modes, respectively, and \(\emptyset\) is the phase difference between the core and cladding mode and is approximated as:  
\[\varnothing=\frac{2\pi\Delta n_{eff}L}{\lambda} \tag{2}\]  
where \(\lambda\) is the wavelength, \(L\) is the length of the TCEDF, and \(\Delta n_{eff}\) is the effective index difference between the core and the cladding modes of TCF. If \(\varnothing\) equals (\(2m+1\)) in Equation (2), with \(m=1,2,3,\dots\), The dip of the interference spectrum can be expressed as:  
\[\lambda_{m}=\frac{2\Delta n_{eff}L}{2m+1} \tag{3}\]  
where \(\lambda_{m}\) is the monitoring wavelength.