# 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
These modes exhibit distinct propagation constants owing to differences in their electromagnetic fields and confinement within the fiber. Although the modes traverse the same physical distance, each accumulates a different phase, and this phase difference is critical to generating the interference pattern. Upon reaching the opposite end of the FMF and entering the output SMF, the modes recombine. The distinct phase of each mode, due to their differing propagation constants, results in constructive or destructive interference, producing the characteristic interference pattern that depends on the accumulated phase difference between the modes [15]. The FSR of the MZFI is given by the following equation:  
\[FSR=\frac{\lambda^{2}}{\Delta n_{eff}^{i}L} \tag{4}\]  
From Equation (4), it can be observed that the FSR decreases as the length of the FMF section increases. The FSR of the envelope is defined as [17]:  
\[FSR_{Envelope}=\frac{FSR_{Ref}FSR_{sen}}{\left|FSR_{Ref}-FSR_{sen}\right|} \tag{5}\]  
As seen in Equation (5), the envelope FSR is larger when the difference between the sensor FSR and the reference interferometer FSR is minimal. The \(M\) factor is the ratio between the FSR of the envelope and the FSR of the sensing interferometer:  
\[M=\frac{FSR_{Envelope}}{FSR_{Sen}}=\frac{\frac{FSR_{Ref}FSR_{Sen}}{\left|FSR_{ Ref}-FSR_{Sen}\right|}}{FSR_{Sen}}=\frac{FSR_{Ref}}{\left|FSR_{Ref}-FSR_{Sen}\right|} \tag{6}\]  
As described in previous works, harmonics occur when the OPL of the first interferometer increases by \(i+1\) times relative to the second interferometer, with an additional length mismatch factor. According to [18], this relationship is expressed as  
\[I_{S}=a+bcos(\varphi_{S});\ I_{R}=a+bcos(\varphi_{R}) \tag{1}\]0  
where \(\Delta\) represents the length mismatch factor, \"\(i\)\" is a positive integer corresponding to the harmonic order, and \(OPL_{Sen}\) and \(OPL_{Ref}\) are the optical path lengths of the sensor and reference interferometers, respectively. According to previous studies, Equation (7) can be related to the FMF lengths as follows:  
\[I_{S}=a+bcos(\varphi_{S});\ I_{R}=a+bcos(\varphi_{R}) \tag{1}\]1  
In a harmonic OVE, two types of envelopes arise: an external and an internal envelope. Regarding the external envelope, [PERSON] et al. [18] note that its FSR is a function of the FSRs of both interferometers (sensor and reference) and closely resembles the FSR of an OVE. Consequently, the harmonic OVE regenerates the external envelope with the same frequency and FSR as in the OVE:  
\[I_{S}=a+bcos(\varphi_{S});\ I_{R}=a+bcos(\varphi_{R}) \tag{1}\]2  
Due to the limitations highlighted in previous studies [14], the technique of tracking internal envelopes is applied. Therefore, by grouping the peaks of the harmonic spectrum,the maxima of the internal envelopes are obtained and classified into peaks of \(i+1\), where \"\(i\)\" represents the harmonic order. The intersections between these internal envelopes provide multiple points that can be used to track wavelength shifts. As the harmonic order \"\(i\)\" increases, the internal envelopes increase proportionally in number and FSR range, facilitating spectral change detection by monitoring these envelope intersections. The FSR of the internal envelope is defined as  
\[I_{S}=a+bcos(\varphi_{S});\ I_{R}=a+bcos(\varphi_{R}) \tag{1}\]3  
where \(FSR_{Ref}\) is the reference FSR; \(FSR_{sen}^{i}\) is the sensor FSR at the harmonic order \"\(i\)\", and \(FSR_{Ext}^{i}\) is the external envelope FSR at the harmonic order \"\(i\)\". For harmonic orders, the \(M\) factor of the internal envelope, \(M_{Int}\), is defined as  
\[I_{S}=a+bcos(\varphi_{S});\ I_{R}=a+bcos(\varphi_{R}) \tag{1}\]4  
In an MZFI, Equation (8) is used to determine the length of the sensor FMF. The OVE is achieved when \(i=0\), with a length mismatch factor \(\Delta\) added to the equation \(L_{Sen}=(0+1)L_{Ref}+\Delta\). The first harmonic is obtained when \(i=1\), doubling the reference MZFI length plus the length mismatch factor \(L_{Sen}=(1+1)L_{Ref}+\Delta\).