# High-Sensitivity Displacement Sensor Using Few-Mode Optical Fibers and the Optical Vernier Effect
## 2 Principle of the OVE and Harmonics in Cascading SFS Sections  
The OVE occurs by superimposing two independent interferometers, with a slight phase shift in their Free Spectral Range (FSR). This is achieved by splicing FMF segments of nearly identical lengths between two SMFs. When the sensor length \(L_{Sen}\) closely matches the reference length \(L_{Ref}\), the characteristic OVE is observed, as illustrated in Figure 1.  
Building upon the fundamental equation for modeling the OVE from [6] and adapting it to an SFS structure specifically supporting two modes, as demonstrated in [15; 16], the transmission of the sensor (IS), reference (IR), and combined interferometer (IC) can be described by the following expressions:  
\[I_{S}=a+bcos(\varphi_{S});\ I_{R}=a+bcos(\varphi_{R}) \tag{1}\]  
\[I_{C}=a^{2}+ab[cos(\varphi_{S})+cos(\varphi_{R})]+\frac{b^{2}}{2}[cos(\varphi _{S}+\varphi_{R})+cos(\varphi_{S}-\varphi_{R})] \tag{2}\]where \(a=t_{1}^{2}+t_{2}^{2}\); \(b=2t_{1}t_{2}\), and \(t_{1}\) and \(t_{2}\) represent the optical power transfer ratios from the SMF to the fundamental mode and the first higher-order mode within the FMFs. The phase differences developed between these modes in the sensing and reference FMF segments are denoted as \(\varphi_{S}=\Delta\beta_{S}L_{S}\) and \(\varphi_{R}=\Delta\beta_{R}L_{R}\), respectively. The propagation constants for the first and higher order mode in the sensing and reference FMF segments are \(\Delta\beta_{S}\) and \(\Delta\beta_{R}\), with corresponding lengths \(L_{S}\) and \(L_{R}\). Factor \"\(a\)\" represents the total optical intensity of the two modes when no interaction occurs between them, while factor \"\(b\)\" reflects the effect of constructive and destructive interference, which is influenced by the cosine of the phase difference between the two waves. This phase difference changes with variations in the bending of the optical fiber. In an SFS structure, with \(\varphi_{S}\) as a function of both wavelength \(\lambda\) and curvature \(c\), the phase difference can be expressed as follows:  
\[\Delta\varphi_{S}=\bigg{[}\frac{\partial\big{(}\Delta\beta_{S}\big{)}}{ \partial\lambda}L\bigg{]}\Delta\lambda_{S}+\frac{\partial\varphi_{S}}{ \partial c}\Delta c \tag{3}\]  
A thermoplastic polyurethane (TPU) tubular structure was designed using computational modeling software to form a 'U' shape to facilitate the controlled bending required for linear displacement. The structure has a cross-section with an outer diameter of 4 mm and an inner diameter of 2 mm. On the one hand, it features a 0.8 mm wide slot for inserting the optical fiber, ensuring a precise fit and preventing damage during deformation. The U-shape has an arc radius of 25 mm, connected laterally by two 50 mm tubes. This structure ensures that the linear displacement is precisely correlated to the curvature of the optical fiber, which is achieved using a transducer, as shown in Figure 2.  
The mechanical properties of TPU are a tensile strength of 52 MPa, a tensile elongation of 500%, a density of 1.12 g/cm\({}^{3}\), and a Shore A hardness of 85A. These properties suggest that the material would have good mechanical strength, especially in applications requiring flexibility and the ability to withstand repeated stresses without fracturing. The tensile elongation and Shore A hardness are ideal for situations where the material is subjected to both tensile and yield stresses, and the tensile strength of 52 MPa is sufficient for many industrial and commercial applications. Figure 1: Schematic diagram of the cascade MZFIs. Figure 2: Transducer made of TPU that correlated the phase difference induced by the curvature of the optical fiber with the linear displacement. In an MZFI composed of a two-mode FMF spliced between two SMFs, interference arises due to the differences in propagation constants between the modes transmitted through the FMFs. As light propagates from the input SMF into the FMF, the two modes are excited.