# High-Sensitivity Displacement Sensor Using Few-Mode Optical Fibers and the Optical Vernier Effect
## 4 Results and Discussion
### Harmonic OVE Analysis  
The harmonic analysis of the OVE, as discussed in [PERSON] [18], involves detecting each harmonic peak's maxima, then grouping them into sets of \"\(i+1\),\" where \"\(i\)\" represents the harmonic order. Notably, the number of peak maxima coincides with the number of generated internal envelopes. As shown in Table 2, the first four harmonics of the three proposed configurations were obtained using Equation (8). Figure 11 clearly illustrates that both peaks and envelopes increase according to the proportion \"\(i+1\)\", with two peaks and internal envelopes in the first harmonic, three in the second harmonic, four in the third harmonic, and five in the fourth harmonic. The intersection points of the internal envelopes serve as critical markers for monitoring wavelength shifts in response to applied displacement. A displacement analysis was performed with the first device from Table 2. A linear displacement ranging from 0.4 mm to 0.9 mm, in 0.1 mm intervals, was applied. The spectral shifts are shown in Figure 12, and the sensitivities are shown in Figure 13. Figure 12 demonstrates that as the harmonic order increases, more intersection points of the internal envelopes are obtained. However, the downside is that sensitivity decreases, as observed in Figure 13. One factor to consider is that, due to the cyclical behavior, if an appropriate range is not selected, the displacement may be \"cut off\" within the OSA span, as occurred in Figures 12 and 13d. Further analysis of FMFs with varying lengths revealed an interesting observation. When the length of \(FMF_{R\!\!f}\) exceeds that of \(FMF_{Sen}\), the envelope shifts toward the shorter wavelengths (Blueshift). Conversely, when the length of \(FMF_{R\!\!f}\) is shorter than that of \(FMF_{Sen}\), the envelope shifts toward longer wavelengths (Redshift). This behavior was first reported in [19], where it was explained that when \(FMF_{Sen}\) is smaller than \(FMF_{R\!\!f}\), the envelopes of both the OVE and the individual interferometer shift in the same direction. Otherwise, they shift in opposite directions. This occurs because the \(M\) factor becomes negative when \(FMF_{Sen}\) is greater than \(FMF_{R\!\!f}\), causing the OVE and individual interferometer envelopes to shift in opposing directions. Figure 10: Slopes of the envelopes from table 1: (**a**) experiment 1, (**b**) experiment 2, (**c**) experiment 3, (**d**) experiment 4 and (**e**) experiment 5. The Blueshift or Redshift displacements are directly related to the ratio of the lengths of \(FMF_{R\!f}\) and \(FMF_{Sen}\). When \(FMF_{R\!f}\) is longer, the optical paths are extended, leading to a greater accumulated phase difference between the signals. In this case, the OVE amplifies the phase difference, causing a Blueshift in the envelope, as constructive and destructive interference occurs at higher frequencies (shorter wavelengths) due to the increased optical path difference. In contrast, when \(FMF_{R\!f}\) is shorter than \(FMF_{Sen}\), the phase difference is reduced, and interference occurs at lower frequencies (longer wavelengths), resulting in a Redshift. Figure 11: Specific characteristics of the first 4 harmonics of the 3 different devices described in table 2: (**a**) experimental setup 1, (**b**) experimental setup 2 and (**c**) experimental setup 3. It is observed that both the peaks and internal envelopes increase in proportion to \({}^{\prime\prime}i+1^{\prime\prime}\), where \({}^{\prime\prime}i^{\prime\prime}\) is the harmonic order. Figure 12: Spectral shift due to linear displacement. The red arrow indicates the wavelength shift: (**a**) first harmonic (one intersection); (**b**) second harmonic (three intersections); (**c**) third harmonic (six intersections); (**d**) fourth harmonic (multiple intersections).