# High-Sensitivity Displacement Sensor Using Few-Mode Optical Fibers and the Optical Vernier Effect
## 6 Conclusions  
This work presented a comprehensive analysis of the OVE using FMF fibers, which were applied for the first time to a linear displacement sensor. This study highlighted  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Year** & **Configuration** & **Application** & **Measurement Range** & **Sensitivity** & **M** & **Factor** & **Ref.** \\ \hline
2019 & Cascading an SI and an FMF & Strain & 0–300 \(\mu\)c & 65.71 pm/\(\mu\)c & 20 & [4] \\ \hline
2020 & Structure SMF-TTFMF-SMF & Temperature & 25–60 \({}^{\circ}\)C & \(-\)3.348 nm/\({}^{\circ}\)C & 11.3 & [5] \\ \hline
2020 & Structure SMF-FMF-SMF & Gas Pressure Temperature & 0–10 MPa & 4.072 nm/MPa & 21 & [6] \\ \hline
2020 & Structure SMF-FMF-SMF & Temperature & 25–40 \({}^{\circ}\)C & 1.753 nm/\({}^{\circ}\)C & 40.4 & [7] \\ \hline
2020 & Fabry–Perot in parallel with two access holes milled in the sensing FPI & & & \(-\)568 nm/RIU & 1 & \\  & & & \(-\)28,496 nm/RIU & 50.2 & [7] \\ \hline
2021 & FMF with cascaded microcavities using a PDMS coating & Temperature & 35–45 \({}^{\circ}\)C & 4.7 nm/\({}^{\circ}\)C & 4.9 & [8] \\ \hline
2021 & FMF and PDMS & Temperature & 40–56 \({}^{\circ}\)C & 3.98 nm/\({}^{\circ}\)C & 4.76 & [9] \\  & & &  
previously underexplored aspects, such as the direct relationship between FMF lengths and the displacement direction of the envelopes, spatial frequency, and harmonic order.  
It was demonstrated that to achieve well-defined envelopes in the OVE, the MZFI lengths required a tuning factor of \(\pm 1\) cm. With this adjustment, sensitivities of up to 39.07 nm/mm and \(M\) factors of up to 50.09 were achieved. To achieve the harmonic OVE, two conditions must be met: the MZFI lengths must follow a proportionality of \"\(i+1\)\", and the length mismatch factor must be \(\pm 1\) cm. Under these conditions, sensitivities of up to 520.8 nm/mm were achieved using the second harmonic. Furthermore, it was shown that when the \(MZF_{R\!f}\) lengths are greater than the \(MZF_{S\!m}\) lengths, the envelopes shift to shorter wavelengths (Blueshift), while the opposite leads to a shift toward longer wavelengths (Redshift).  
The FFT analysis confirmed the relationship \(f=1/\)FSR and identified three key peaks in the transmission spectra of the envelopes: two from the individual MZFIs and one from the envelope.  
Unlike other sensors with more complex and specialized configurations, this approach achieved highly precise and sensitive results using a simple yet robust SFS structure. This configuration optimized the detection of small displacements with high precision, enhancing the sensor's adaptability for industrial and scientific applications.  
Conceptualization, G.A.-S. and A.M.-R; methodology, all authors; validation, G.A.-S. and A.M.-R.; formal analysis, all authors; investigation, all authors; resources, G.A.-S. and A.M.-R.; data curation, L.E.G.-R., M.C.J.-M., G.A.-S. and J.A.M.-V.; writing--original draft preparation, G.A.-S., A.M.-R. and M.C.J.-M.; writing--review and editing, all authors.; visualization, all authors.; supervision, G.A.-S.; project administration, G.A.-S.; funding acquisition, G.A.-S. and A.M.-R. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Consejo Nacional de Humanidades, Ciencias y Tecnologias (CONAHCYT) in Mexico.  
Not applicable.  
The raw data supporting the conclusions of this article will be made available by the authors upon request.  
All authors declare that they have no conflicts of interest.