## 2 Fiber Filter Fabrication and Its Principle Operation  
The modal fiber interferometer (MFI) was fabricated by using the arc splice technique, where the catastrophic splicing of an optical microfiber (<60 \(\upmu\)m diameter) at the tip of conventional single-mode fiber (125 \(\upmu\)m diameter) is performed. The microfiber used in this work was fabricated using the drawing flame technique [39]. During the fabrication process, only an arc splicer (Fitel S175, Furukawa FITEL Optic Products Co., Ltd., Peachtree, GA, USA) was used in manual mode, setting the single-mode fiber default program to the following parameters: power 94 A.U and Z push distance 11\(\upmu\)m. Both fibers were cleaved, set, and aligned into the splicer machine; then, ten discharges were applied until a first catastrophic splice zone (Z1) was achieved (Figure 1). Zone Z1 was translated at a specific distance from the electrodes, and the microfiber was exposed at the electrode's area; then, several discharges are applied to cut the microfiber (\(\leq\)18 discharges). As a result, the second catastrophic zone, Z2, was obtained; the final structure can be observed in Figure 1. The Z1 and Z2 collapse regions had a size of 50\(\upmu\)m (this value was estimated using the arc splicer). The distance, D, between Z1 and Z2 oscillated between 400 \(\upmu\)m and 500 \(\upmu\)m. The fabrication process was monitored in real-time to minimize the insertion loss. Hence, the number of discharges varied, considering the minimal insertion loses; consequently, the distance, D, was also affected.  
Figure 1: Physical characteristics of the MFI structure: (**a**) structure diagram; (**b**) Scanning Electron Microscope Image (SEM) of the structure.  
The MFI was operated in reflection mode; its interference spectrum is shown in Figure 2a. The spectrum showed visibility oscillating between 8 dB and 10 dB; its free spectral range was 1.6 nm. The Fourier transform was applied to the reflection interference spectrum to identify the modal contribution (see Figure 2b). The fundamental reflected core mode is considered the peak centered at 0 nm\({}^{-1}\). Two spatial frequency regions with some intensity peaks can be seen; these modal energies correspond to higher-order reflection modes. All the modes described above are excited by several refractive index interfaces located at zones Z1 and Z2. The first refractive index interface can be found at the Z1 zone, where the bulk of the collapsed material and the core and cladding of the fibers involved produces a refractive index change. At this point, part of the light arriving at the Z1 zone is reflected, and the rest travels through the microfiber. The light then travels to the Z2 zone, where the tip of the bulk collapsed material, and the surrounding medium produces a refractive index interface, resulting in the generation of some reflections.  
The amplitude of the total reflected electric vector can be described by [40]:  
\[E_{R}=\left[\sqrt{R_{1}}+\sum_{k=1}^{n}\left(e^{-2j(\sum_{k=1}^{k}\varnothing_ {i})}\sqrt{R_{k+1}}\prod_{l=1}^{k}(1-R_{i})\right)\right]E_{0} \tag{1}\]  
each reflectivity can be determined by the relation \(R=\left|\left(n_{x}-n_{y}\right)\left(n_{x}+n_{y}\right)\right|^{2}\); here, \(n_{x}\) and \(n_{y}\) represent the effective refractive index involved. \(R_{1}\) is the first reflection and \(R_{k+1}\) represents the other reflections gene rated by the structure. The phase between the reflected modes generated is \(\varnothing_{i}=\left(4\pi dn/\lambda\right)\), where \(\lambda\) is the operation's wavelength, \(d\) is the distance of the optical path and \(n\) is the effective refractive indexes of the different zones. More detailed information is presented in reference [41].