**3. Results**

The setup used to characterize the sensor is shown in Figure 3. The mechanical part consists of a double-sided reflective silver surface coupled to a piezoelectric with a displacement of 180.0 ± 0.2 nm/V. Also, a standard optical fiber (SMF28), an erbium broadband source, centered at 1550 nm and a bandwidth of 90 nm, and an optical spectral analyzer (OSA, "YOKOGAWA AQ5370C") with a resolution of 0.02 nm, were used. All the splices in the system were made with a conventional splice machine ("Sumitomo Electric—Type-72C", Osaka, Japan). The overlap of the interferometers signals is obtained with a 3 dB fiber coupler. The fiber cleavage was performed by "Fiber Cleaver FC-6RS" which provides a cleavage angle of less than 1◦ (Figure 3c).

**Figure 3.** Experimental setup: (**a**) scheme, (**b**) picture, and (**c**) fibers cleavage angle obtained with "Sumitomo Electric—Type-72C".

The sensor characterization involved nano-step measurement provided by the piezoelectric device. To determine the efficiency of the displacement sensor, a single interferometer with a length of 73 ± 2 μm was initially developed, which allowed only three fringes in the bandwidth supplied by the erbium source (Figure 4a) to be obtained. Varying the distance between the fiber and the reflecting surface resulted in Figure 4b where a sensitivity of 29.2 ± 0.3 nm/μm with a *r*2 of 0.9995 was obtained.

Following, the Vernier effect was investigated. Since the accuracy of the envelope measurement increases with the increase of the carrier frequency, the sensitivity of a single interferometer with a length of 15.4 ± 0.5 mm was first studied (Figure 5), where a sensitivity of 0.95 ± 0.05 nm/μm with a *r*2 of 0.995 was achieved. Thus, the interferometers used to generate the Vernier effect have a lower sensitivity than the reference single interferometer. In this case, the FSR is 0.07 nm.

Now, a second fiber has been added allowing to obtain the Vernier effect. Although both interferometers have the same length (15.4 ± 0.5 mm), they differ by a value close to 70 μm. Hence an envelope with only three fringes appears (Figure 6). This last feature is important because it allows us to compare the envelope of this sensor with the first singular interferometer that is present in this paper.

**Figure 4.** The single interferometer with a length of 73 ± 2 μm: (**a**) Spectrum at the beginning of measurements and (**b**) intensity as a function of the distance between the optical fiber and the reflecting surface and as a function of wavelength.

**Figure 5.** The single interferometer with a length of 15.4 ± 0.5 mm: (**a**) spectrum at the beginning of measurements and (**b**) intensity as a function of the distance between the optical fiber and the reflecting surface and as a function of wavelength.

**Figure 6.** The two interferometers with a length of 15.4 ± 0.5 mm: (**a**) spectrum at the beginning of measurements and (**b**) intensity as a function of the distance between the optical fiber and the reflecting surface and as a function of wavelength.

In this case, the envelope sensitivity was 60 ± 1 nm/μm with a *r*2 of 0.998 (which is shown in Figure 8) and the carrier had a negligible sensitivity. The envelope sensitivity was two-fold that obtained by the reference single interferometer. the *FSR* for the envelope and carrier was 18 nm and 0.07 nm, respectively. Therefore, the *M*-factor is 257, *Msens* = 360, *MFSR* = 257, *MVernier* = 1.4.

The last characterization was to optimize the proposed setup in order to obtain the maximum sensitivity of this configuration. In this case, we used two interferometers with a length of 6.1 ± 0.2 mm each and by reducing the difference between the two interferometers, an envelope with only one fringe was achieved, as depicted in Figure 7. The sensitivity obtained was 254 ± 6 nm/μm with an *r*2 of 0.9990 (which is shown in Figure 8) for the envelope, and 0.160 ± 0.005 nm/μm with a *r*2 of 0.992 for the carrier. Also, the *FSR* for the envelope and the carrier was 75 nm 0.09 nm respectively. Therefore, the *M*-factor was 1071, *Msens* = 1587, *MFSR* = 833, *MVernier* = 1.9. Table 1 summarizes all the values.

**Figure 7.** The two interferometers with a length of 6.1 ± 0.2 mm: (**a**) spectrum at the beginning of measurements, and (**b**) intensity as a function of distance to the reflecting surface and as a function of wavelength.

**Figure 8.** Wavelength shift for a single interferometer, the Vernier effect compared to single interferometer and the Vernier effect maximized. The error bars associated with each value are the same dimension as the geometry of the point that represents it.

**Table 1.** Parameters values that characterize the implementation of the enhanced Vernier effect maximized.

