## 2 Theoretical Description  
### Multimode Interference (MMI) Sensor Principle  
A common geometry of multimode interference-based fiber detection device is known as an SMS configuration. This configuration contains a segment of multimode fiber (MMF) linked between two single-mode fibers (SMFs). If the MMF coating is removed or absent, as in the case of an NC-MMF, then the structure can be used as a sensor of the surrounding media [35], as shown in Figure 1.  
As the signal transitions from the SMF to the central portion of the fiber, various modes are stimulated and interact while traversing the NC-MMF. Along the optical axis of the fiber, specific locations generate self-images, which essentially replicate the initial input field [36]. Due to the circular symmetry of the input field, only the linearly polarized mode \(LP_{0m}\) is excited within the central fiber [37; 38; 39]. The electric field at the entrance of the MMF can be described as [36; 39]:  
\[E(r,0)=\sum_{0}^{M}b_{m}\psi_{m}(r) \tag{1}\]  
where \(m\) represents the mode number that is guided, \(\psi_{m}(r)\) stands for the electric field within the NC-MMF for the \(m^{th}\) mode, \(r\) denotes the radial position within the fiber's cross-section, \(M\) signifies the overall count of radial modes stimulated in the NC-MMF segment, and \(b_{m}\) indicates the mode expansion coefficient. This coefficient is determined by the overlap integral between \(E(r,0)\) and \(\psi_{m}(r)\), and the field at distance \(L\) of the NC-MMF expressed by [40; 41]:  
\[E(r,L)=\sum_{m=1}^{M}b_{m}\psi_{m}(r)e^{-i\beta_{m}L} \tag{2}\]  
In this context, \(\beta_{m}\) represents the longitudinal propagation constant specific to the \(LP_{0m}\) mode within the NC-MMF. According to Equation (2), whenever the phase factor \(\beta_{m}L\) equals \(p\pi\), where \(p\) takes on integer values (1, 2, 3, etc. ), the input field is replicated. This replication point is essentially the result of constructive interference among the modes as they propagate along the NC-MMF. The assessment of the transmission output power is determined by [38; 42]:  
\[T(\lambda,L)=\left|\sum_{m=1}^{M}b_{m}^{2}\psi_{m}(r)e^{-i\beta_{m}L}\right|^ {2} \tag{3}\]  
By Equation (3), the wavelength transmitted through the MMI filter is affected by both the diameter and length of the NC-MMF. The NC-MMF length is meticulously selected to correspond to a specific wavelength by the expression [39; 43]:  
\[L_{MMF}=p\frac{n_{eff}D_{eff}^{2}}{\lambda_{peak}} \tag{4}\]  
Figure 1: Schematic representation of the SMS fiber optic sensor configuration (sketched by the authors). where \(n_{eff}\) denotes the effective refractive index of the core, \(D_{eff}\) represents the diameter of the NC-MMF, \(L_{MMF}\) is the length of the NC-MMF, and \(p\) is an integer indicating the self-image order. The reader is referred to Refs. [39; 44] for a detailed derivation of Equation (4) within the context of the integrated MMI devices and to Ref. [11] for a simplified extension to fiber-optic MMI devices. The adjustment of the wavelength occurs through modifications in dimensions or refractive index. The self-image position is influenced by the MMF dimension, and the wavelength is propagated. When the NC-MMF length is precisely aligned with the self-image point for a specific wavelength and spliced with the output SMF, maximum transmission is achieved at that wavelength. However, for other wavelengths, self-image points may fall either before or after the NC-MMF-SMF interface, resulting in reduced light coupling to the SMF. Consequently, when a broadband spectrum passes through such a cascade structure, it gives an MMI passband filter [41]. To reach genuine self-images and minimize loss, most studies opt for the fourth self-image point when selecting the MMF length [45; 46]. A detailed description of the multimodal interference effect can be consulted in the Reference [39].