**1. Introduction**

With the evolution of technology, the impact of sensors on human life has increased. The significant impact of optical systems is due to the emergence of fiber optics in 1960 [1]. The features such as low attenuation and insensitivity to electromagnetic fields have allowed optical systems to replace electronic communications. In addition, they have enabled the design of ultra-sensitive optical sensors with applicability in nanotechnology [2].

The cleaved-tip optical sensors are the simplest [3]. When used alone, they are only intensity sensors. However, when they are coupled to reflective surfaces [4] or when coatings are deposited on the tip [5], in addition to the intensity operation, they can also operate at wavelength because they allow the implementation of Fabry–Perot interferometers (FPI) [6]. The operation of an FPI is unidirectional where, initially, part of the beam is reflected, and another part is transmitted forming two beams. The transmitted beam is then reflected travelling a distance that is twice the length of the interferometer. Finally, the two beams overlap resulting in the interference pattern. Although in certain areas there are sensory architectures with higher sensitivity and the possibility of multiple measurements [7–9], the cleaved-tip sensors are the easiest to control and manufacture in addition to covering a higher number of research areas.

Despite the vast advances in the development of optical sensors, it was only in 2011 that the Vernier effect was applied in interferometry [10]. This phenomenon is based on the optical waves beat where two waves appear, the envelope and the carrier [11]. Normally, measurements are based on the envelope wave because it shows the highest sensitivity. In the case of interferometry, the effect is in the wavelength dimension, so spectral optical waves are used.

Currently there are several interferometric displacement sensors. From these, the Mach–Zehnder with a sensitivity of 1.53 nm/μm [12] stands out for a wide range. For

**Citation:** Robalinho, P.; Frazão, O. Giant Displacement Sensitivity Using Push-Pull Method in Interferometry. *Photonics* **2021**, *8*, 23. https:// doi.org/10.3390/photonics8010023

Received: 15 December 2020 Accepted: 18 January 2021 Published: 19 January 2021

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a narrow range, the application of surface plasmon resonance (SPR) with a sensitivity of 10.32 nm/μm for a micrometric range [13] and a sensitivity of 31.45 nm/nm for a nanometric range [14] stand out. Recently a new strain sensor architecture based on a push-pull deformation method was reported [15].

In this work, the enhanced Vernier effect combined with the push-pull method is presented. This sensor consists of two FPIs formed by two cleaved tips and a mirror. The signal from the two interferometers are overlapped by means of a 3 dB fiber coupler, resulting the Vernier effect. In this research, the sensitivity of the single interferometer is compared with the sensitivity of the enhanced Vernier effect envelope, both with the same free spectral range (FSR). In addition, it also presents the results regarding the maximization of displacement sensors with enhanced Vernier effect.

#### **2. Materials and Methods**

The Vernier effect consists on the overlapping of two optical waves, a concept similar to the beat of two sound waves [16]. This optical phenomenon results in the formation of two new waves: the envelope and the carrier. Usually, optical sensors are based on the traditional Vernier effect, where one of the interferometers is referenced and another is the sensing probe. However, in this paper, the enhanced Vernier effect is presented, and it uses two sensing interferometers which move in opposite directions [17]. One of the major problems with this type of Vernier effect is the lack of sensors with symmetrical sensitivities. Thus arises the application of the push-pull method in interferometry, allowing symmetrical variations of two equal interferometers. This results in two sensors with symmetric sensitivities allowing the maximization of the enhanced Vernier effect [15].

The architecture of the two interferometers is presented in Figure 1 and it is composed by two cleaved fibers and a double-sided reflecting surface. The two interferometers are linked together with a 3 dB coupler. The operating method consists in dividing the light beam into two beams that through the cleaved fibers and the reflecting surface form two Fabry–Perot interferometers (FPIs).

**Figure1.**Sensorarchitectureconcept.

The optical signal of the two interferometers is overlapped by means of the coupler forming the output signal of the sensor. The intensity of a FPI can be written as:

$$I\_{FPI} = \left[ A \cos(\Delta \phi) + I\_0 \right]^2 \tag{1}$$

where Δ*φ = <sup>π</sup>nL/λ*, *n* is the refractive index, *L* is the interferometer length, *λ* is the wavelength and *I*0 is the difference in intensity of the optical paths. If the optical signal is coupled between fibers, a π/2 phase must be added to Δ*φ*. Thus, the sensor output signal is described by:

$$I\_{V\varepsilon} = \left[2\cos\left(\frac{\Delta\Phi\_1 - \Delta\Phi\_2}{2}\right)\cos\left(\frac{\Delta\Phi\_1 + \Delta\Phi\_2}{2} + \frac{\pi}{2}\right) + I\_0\right]^2\tag{2}$$

where the indices 1 and 2 allow identify the two interferometers, the components Δ*φ1* − Δ*φ2* and Δ*φ1 +* Δ*φ2* correspond to the envelope and the carrier, respectively. Figure 2 presents the simulation of Equation (2) with two cavities-one with 500 μm and the other with 600 μm-and a refraction index of 1.00027316. The length of the interferometers can be written as *L* = *L*0 + Δ*L*, where *L*0 is the initial length and Δ*L* is the mirror displacement. Therefore, taking into account that the interferometers are submitted to a symmetric displacement, Equation (2) can be rewritten as follows:

$$I\_{V\varepsilon} = \left[2\cos(\gamma[2\Delta L])\cos\left(\gamma[L\_{01} + L\_{02}] + \frac{\pi}{2}\right) + I\_0\right]^2\tag{3}$$

where *γ = πn/2 λ*. As can be seen, the envelope of the enhanced Vernier effect has a sensitivity two-fold that of the one achieved for a single FPI. Furthermore, the envelope does not depend on the length of the interferometers allowing the result of ultra-sensitive displacement sensors with macroscopic FPIs to be replicated.

**Figure 2.** Vernier effect simulation.

To evaluate the efficiency of the implementation of the Vernier effect, the *M*-factor is considered and is given by [16]:

$$M = \frac{\Delta\lambda\_{\rm env}}{\Delta\lambda\_1} \tag{4}$$

where Δ *λenv* is the free spectral range (FSR) of the envelope and Δ *λ*1 is the FSR of a single interferometer. With the appearance of the enhanced Vernier effect, new factors have emerged to evaluate the performance of the Vernier effect [18]:

$$M\_{\text{sens}} = \frac{\text{sens}\_{\text{env}}}{\text{sens}\_{\text{carr}}} \tag{5}$$

where *sensenv* and *senscarr* are the envelope and the carrier sensitivities, respectively,

$$M\_{FSR} = \frac{\Delta\lambda\_{cuv}}{\Delta\lambda\_{carr}}\tag{6}$$

where Δ *λcarr* is the *FSR* of the carrier and,

$$M\_{Vrinir} = \frac{M\_{\text{sens}}}{M\_{FSR}}\tag{7}$$

where *MVernier* is = 1 for the traditional Vernier effect, < 1 for reduced case and > 1 for the enhanced case.
