**4. Dual FSI concept for large range measurements**

As mentioned before, as we increase the measurement range, accuracy and complexity are typically linked together. This is due to the fact that in FSI the measurement uncertainty increases with distance, as a consequence of the uncertainty propagation of the synthetic wavelength measurement (directly related to the FP performance). In order to achieve the required uncertainty one would have to increase the FP stability (thermal and mechanical) and improve the calibration and resonance detection characteristics (i.e. using Pound-Drever-Hall technique to lock the laser into FP resonances). As a consequence, the expected impact on sensor complexity would be critical. For large ranges, this increase in uncertainty can be considered the main drawback of the technique.

To overcome this problem, maintaining the low complexity associated to short distances, the measurement process for larger ranges can be reduced to the close range case, by limiting the OPD in the interferometer. This can be achieved by increasing the reference arm with a long reference fibre and introducing the concept of dual FSI mode, where an ancillary interferometer is used to measure (calibrate) continuously the fibre length (Cabral et al, 2009).

By adding a long reference fibre in the reference arm, the measured OPD is reduced to the real distance minus half fibre optical path length. Using an additional interferometer, the fibre is calibrated with the same accuracy but with lower requirements. This assumes that the fibre length is constant, during a certain time interval, and therefore the calibration will be the result of averaging for that period (uncertainty is reduced by a factor proportional to the square root of the number of measurements).

In this dual FSI approach, the final uncertainty is the sum of two components:


In addition, during the calibration period, we must guarantee that the change in fibre OPL due to thermal variation is negligible compared with the required calibration uncertainty. It must be noted that the change in fibre length due to thermal variation is not the only limiting factor to the number of measurements that can be averaged for fibre calibration. Beyond a certain reduction factor, i.e. after a certain averaging period, the uncertainty

As mentioned before, as we increase the measurement range, accuracy and complexity are typically linked together. This is due to the fact that in FSI the measurement uncertainty increases with distance, as a consequence of the uncertainty propagation of the synthetic wavelength measurement (directly related to the FP performance). In order to achieve the required uncertainty one would have to increase the FP stability (thermal and mechanical) and improve the calibration and resonance detection characteristics (i.e. using Pound-Drever-Hall technique to lock the laser into FP resonances). As a consequence, the expected impact on sensor complexity would be critical. For large ranges, this increase in uncertainty

To overcome this problem, maintaining the low complexity associated to short distances, the measurement process for larger ranges can be reduced to the close range case, by limiting the OPD in the interferometer. This can be achieved by increasing the reference arm with a long reference fibre and introducing the concept of dual FSI mode, where an ancillary interferometer is used to measure (calibrate) continuously the fibre length (Cabral et al, 2009). By adding a long reference fibre in the reference arm, the measured OPD is reduced to the real distance minus half fibre optical path length. Using an additional interferometer, the fibre is calibrated with the same accuracy but with lower requirements. This assumes that the fibre length is constant, during a certain time interval, and therefore the calibration will be the result of averaging for that period (uncertainty is reduced by a factor proportional to

uncertainty in the measurement of the reduced OPD (twice the absolute distance minus

In addition, during the calibration period, we must guarantee that the change in fibre OPL due to thermal variation is negligible compared with the required calibration uncertainty. It must be noted that the change in fibre length due to thermal variation is not the only limiting factor to the number of measurements that can be averaged for fibre calibration. Beyond a certain reduction factor, i.e. after a certain averaging period, the uncertainty

In this dual FSI approach, the final uncertainty is the sum of two components:

calibration uncertainty of the Reference Fibre Optical Path Length (OPL);

5.10-6 Hz/Hz

5.10-6 m/m

5.10-6 Hz/Hz

*Parameter Value Uncertainty* Frequency sweep range Δν = 150 GHz δΔν = 750 kHz

Synthetic wavelength Λ = 2 mm δΛ = 10 nm

Synthetic fringes δ*N* = 1/360 FP FSR *FSR* = 1 GHz δ*FSR* = 5 kHz

Number of FSR *r* = 150 δ*r* = 3.10-6 Refractive index *n* = 1,00027 δ*n* = 1.10-6

Table 1. FSI parameters used in the simulation presented in Fig. 3.

**4. Dual FSI concept for large range measurements** 

can be considered the main drawback of the technique.

the square root of the number of measurements).

Reference Fibre length);

component related to the FP FSR calibration uncertainty becomes dominant and there is no benefit in increasing the calibration period.

Besides the measurement dispersion, there are other uncertainty components that can influence the final measurement accuracy. The resolution in the fringe phase and resonance position measurements is currently much smaller than the measurement dispersion, and thus its contribution is not significant. The uncertainty in the calibration of the FP FSR is the most important one for large distance measurements in a single FSI configuration but, in the dual approach, it is possible to achieve a very small contribution considering the value of the dispersion component.

In order to maintain low hardware complexity, the optical setup for the implementation of this Dual FSI concept makes use of the same laser and FP for the two interferometers, and also some common beam splitters (BS). For that, a Mach-Zehnder configuration was selected, and is presented in Fig. 4, where the two interferometers are shown separately (note that BS1, BS2 and the reference fibre are common to both interferometers).

Fig. 4. Dual FSI interferometers setup, measurement (a) and reference (b).

The measurement interferometer evaluates the OPD between the OPL of twice the distance in measurement (due to the round trip) and the OPL added by the reference fibre, as shown in Fig. 4(a). The reference interferometer, shown in the Fig. 4(b), will measure the OPL added by the reference fibre that cause the reduction of the OPD in the measurement interferometer.
