# High-Sensitivity Displacement Sensor Using Few-Mode Optical Fibers and the Optical Vernier Effect
## 4 Results and Discussion  
### Fast Fourier Transform Analysis  
To analyze the spectra and identify the signals generated by both the \(MZFI_{R\ell}\) and \(MZFI_{Sen}\), as well as confirm their overlap with the signals in the OVE, the Fast Fourier Transform (FFT) was applied to each of the proposed experimental setups. The results verified that all systems followed the expected trend, where the FSR decreased as the MZFI length increased. The relationship between spatial frequency and wavelength, expressed as \(f=1/\text{FSR}\), was also consistent with previous studies [13]. The reference signal frequency \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Number of** & \(L_{R\ell}\) & \(L_{Sen}\) & \(FSR_{R\ell}\) & \(FSR_{Sen}\) & \(FSR_{Envelope}\) & \(M\) Factor \\
**Experiment** & **(cm)** & **(cm)** & **(cm)** & **(cm)** & **(nm)** & **(nm)** & **(nm)** & **M Factor** \\ \hline
1 & 10 & 10.5 & 11.12 & 10.36 & 60.51 & 5.84 \\
2 & 20 & 20.5 & 5.96 & 5.84 & 63.53 & 10.87 \\
3 & 30 & 30.5 & 4.2 & 3.96 & 67.93 & 17.15 \\
4 & 40 & 40.5 & 3.1 & 2.94 & 70.08 & 25.19 \\
5 & 50 & 50.5 & 2.45 & 2.31 & 115.72 & 50.09 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Main parameters of the experiments. \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & **Order** & \(L_{R\ell}\) &
\begin{tabular}{c} **Detuning** \\ **(cm)** \\ \end{tabular} & \(L_{Sen}\) & \(L_{Sen}\) & \(FSR_{R\ell}\) & \(FSR_{Sen}\) & \(FSR_{Ext}^{l}\) & \(FSR_{Ext}^{l}\) & \(M_{Int}\) \\  & & **(cm)** & **(cm)** & **(nm)** & **(nm)** & **(nm)** & **(nm)** & **Factor** \\ \hline \multirow{4}{*}{Experimental setup 1} & First Harmonic & & 21.5 & & 5.83 & 46.46 & 92.92 & 15.93 \\  & Second Harmonic & & 32 & 3.78 & 39.96 & 119.88 & 31.71 \\  & Third Harmonic & & 42.5 & 42.5 & 2.79 & 36.13 & 144.52 & 51.8 \\  & Fourth Harmonic & & & 53 & & 2.34 & 18.09 & 90.45 & 36.65 \\ \hline \multirow{4}{*}{Experimental setup 2} & First Harmonic & & & 26 & & 4.55 & 300.3 & 600.6 & 132 \\  & Second Harmonic & & & 38.5 & 3.19 & 89.32 & 267.96 & 84 \\  & Third Harmonic & & & 51 & & 2.4 & 61.6 & 246.4 & 102.66 \\  & Fourth Harmonic & & & 63.5 & & 1.98 & 27.72 & 138.6 & 70 \\ \hline \multirow{4}{*}{Experimental setup 3} & First Harmonic & & & 32 & & 3.78 & 1425.06 & 2850.12 & 754 \\  & Second Harmonic & & & 48.5 & & 2.54 & 239.39 & 718.185 & 282.75 \\  & Third Harmonic & & & 65 & & 1.81 & 45.49 & 181.96 & 100.53 \\  & Fourth Harmonic & & & 81.5 & & 1.45 & 37.7 & 188.5 & 130 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Main parameters of the experiments for the harmonics. was defined as \(Freq_{Ref}\), the sensor signal frequency as \(Freq_{Sen}\), and the envelope frequency as \(Freq_{Emendor}\). Figures 4-8 illustrate the frequency analysis for the corresponding devices listed in Table 1. In each figure, panels (a), (c), and (e) present the transmission spectra for \(FMF_{Ref}\), \(FMF_{Sen}\), and the superimposed spectrum (envelope), respectively. It is noteworthy that the \(FSR_{Ref}\) and \(FSR_{Sen}\) must be closely matched to generate an envelope with a significantly larger \(\mathrm{FSR}_{\mathrm{Envelope}}\) when superimposed. Panels (b), (d), and (f) in Figures 4-8 display the spatial frequencies corresponding to the transmission spectra. Panels (b) and (d) in Figures 4-8 highlight the dominant frequencies produced by the \(MZFI_{Ref}\) and \(MZFI_{Sen}\), characterized by their regularity and consistency. In panels (f) in Figures 4-8, three key peaks are observed: one representing the superimposed spatial frequency of the envelope  
Figure 5: (**a,c**) show the transmission spectra of \(MZFI_{Ref}\) and \(MZFI_{Sen}\), respectively. (**b,d**) represent their corresponding FFT spectra. (**e**) is the superimposed transmission of two MZFIs, and (**f**) is the FFT of the superimposed spectrum. Figure 6: (**a,c**) show the transmission spectra of \(MZFI_{Ref}\) and \(MZFI_{Sen}\), respectively. (**b,d**) represent their corresponding FFT spectra.