1 Introduction

Low-pressure and liquid levels are essential parameters for the oil-gas, food, and chemical industries. Also, low-pressure is a crucial factor to be estimated in several medical procedures (Leitão et al. 2016; Marques et al. 2017; Sartiano and Sales 2017; Shin et al. 2019; Wolthuis et al. 1991). Therefore, many optical devices have been proposed to satisfy this demand (Escudero et al. 2019; Kim et al. 2018; Lacam and Chateau 1989; Leger et al. 1990; Musayev and Karlik 2003). Among these proposals, fiber-optic sensors have been demonstrated as a reliable alternative to conduct the independent or simultaneous detection of liquid levels and pressure.

One of the most common methods used to detect pressure is optical fiber Fabry–Perot Interferometer structure (FPI) (Abeysinghe et al. 2001; Bai et al. 2016; Spillman 1982; Vorathin et al. 2020; Wang et al. 2006a, b). These structures can be classified according to the cavity formation, such as intrinsic and extrinsic FPI sensors. The most common extrinsic Fabry–Perot fiber optic structure used for pressure sensing applications employs Micro Electro Mechanical Systems (MEMS) (Abeysinghe et al. 2001; Li et al. 2006; Qi et al. 2019; Wang et al. 2006a, b). The adequate performance of these structures has been widely demonstrated. However, its manufacturing process involves intricate micromachining and chemical processes. Meanwhile, some intrinsic Fabry–Perot fiber-optic structures have been proposed and demonstrated for pressure detection using the arc splice technique. Most of these techniques combine a solid core fiber and air-silica fiber optic structure to produce an intrinsic air cavity (Ma et al. 2011; Yu et al. 2015; Zhang et al. 2019). These devices are compact and versatile; however, the number of steps required to achieve the structure reduces its repeatability under the same splicing parameters. Other low-cost methods have been demonstrated using etched fiber optics and polymer adhesive (Chen et al. 2017), or by merely adding special adhesive at the tip of conventional fibers (Oliveira et al. 2019; Zhang et al. 2017), as well as combining long polymer membranes with high reflective material at the fiber’s tip (Zhu et al. 2017).

Currently, most of the proposed fiber optic sensors to estimate liquid level are based on inline, all-fiber structures, such as tapered fibers (Hsu et al. 2013), Mach–Zehnder Interferometers (Liu et al. 2020; Sun et al. 2015), Michelson Interferometers (Li et al. 2018) , and Multimode interferometers (Antonio-Lopez et al. 2011). These structures can be easily adapted to the sensing region. However, some of them present highly phase modulation responses for more than one parameter. As a result, crosstalk measurement and instabilities could occur.

It is worth stressing that some works have combined the benefits of extrinsic Fabry–Perot interferometers to detect liquid levels (Diaz et al. 2019; Lü and Yang 2007; Wang and Li 2014). It is necessary to notice that the membrane material is a key aspect of designing an FPI sensor for pressure detection. This parameter will govern the sensitivity, dynamic range, and temperature operation. A hard-solid membrane may reduce the possibility of monitoring the low-pressure range, considering that the low-pressure range is < 10 kPa (Srivastava and Chattopadhyay 2017).

In this work, an alternative method to detect pressure and liquid level is presented. The technique is based on an extrinsic Fabry–Perot structure produced by the junction between a multimode fiber optic and a Mylar® polymer film. The soft properties of the material allow the detection of low-pressure levels with high sensitivity and good resolution. The present technique is a cost-effective alternative to detect pressure and liquid levels, considering the wavelength operation and the materials involved.

2 Theory

The proposed low-pressure and liquid level multimode fiber-optic sensor is based on the refractive index interface generated by the junction between a thin Mylar® polymer film (MPF) and a multimode fiber-optic (MMF). The schematic interaction between the polymer and the multimode fiber optic is presented in Fig. 1.

Fig. 1
figure 1

Illustration of the junction between the Mylar® polymer film and the multimode fiber optic

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When the light in the MMF arrives at the MPF, one part is transmitted, and the rest is reflected. The refractive index interface causes the first reflection between n1 (1.468 − MMF) and n2 (1.67 − MPF). It is worth noting that the reflection can be estimated by the Fresnel relation, R1= (n1 − n2/ n2 + n1)2. The transmitted light portion goes through the MPF cavity and reaches the second refractive index interface, generated between n2 and n3 (surrounding environment). Here, the following reflection occurs: R2= (n2 − n3/n3 + n2)2. The one-layer Fabry–Perot expression governs the interaction of these reflections (Van De Stadt and Muller 1985; Zhao et al. 2018):

$${I}_{R}\left(\lambda \right)={R}_{1}+{(1-\alpha )}^{2}{(1-\beta)}^{2}{(1-{R}_{1})}^{2}{R}_{2}+2\left(1-\alpha \right)\left(1-\beta\right){\left({R}_{1}{R}_{2}\right)}^{\frac{1}{2}}\,{\text{cos}}\left(4\pi\varDelta nd/\lambda \right)$$
(1)

where α and β represent the transmitted and the cavity losses, respectively, the round-trip phase is described in terms of d-cavity distance (thickness of the MPF), nc cavity’s refractive index, and λ wavelength operation.

However, when pressure is applied at one lateral face of the MPF, the membrane is deflected. As a result, the material is stressed. In our case, when water pressure was applied in the opposite direction of the MMF, the Fabry–Perot cavity was altered; then, the pumped light into the MMF reached over the stressed material (see Fig. 2a).

Fig. 2
figure 2

a Deflection membrane by water pressure applied and b cylindrical system coordinates for the MPF and MMF interaction

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The deflection generated by the water pressure can be analyzed using a cylindrical coordinate system (see Fig. 2b). Hence, the stress in the radial (σr) and angular (σθ) components can be described by the following relations (Reddy 2006):

$${\sigma }_{r}(r,z)=\frac{3q{a}^{2}z}{4{d}^{3}}\left[(1+\nu )-(3+\nu ){r}^{2}/{a}^{2}\right]$$
(2a)
$${\sigma }_{r}(r,z)=\frac{3q{a}^{2}z}{4{d}^{3}}\left[(1+\nu )-(1+3\nu ){r}^{2}/{a}^{2}\right]$$
(2b)

here, d is the cavity thickness, a is the membrane’s radii, and q is the transversal load applied by the water pressure. The material properties are represented by Poisson’s ratio (\(\nu\)). It can be seen that both stresses will depend on the polymer displacement (Z), the water pressure applied (expressed by the load q), as well the coordinates (r, Z). Furthermore, both stresses decreased as the radii (a) increased. As a result, a stress difference occurred in the radial directions (r, θ, Z), presenting an anisotropy optical medium. Then, the induced birefringence can be estimated by (Urbańczyk and Pietraszkiewicz 1988):

$${\Delta }n=C{\Delta }\sigma$$
(3)

In the last relation, C is the polymer’s elastic-optic coefficient, and ∆σ is the stress difference between the analyzed components; for this case, ∆σ is the difference between σr and σz. This refractive index variation modifies the round trip presented in Eq. 1. It is worth noting that the water pressure and liquid level are related by P = ρgh, in which the well-known acceleration of gravity (g) and water density (ρ) constants are involved; thus, it is possible the estimation about the liquid level that originates the pressure.

3 Experimental setup

The experimental setup shown in Fig. 3 was employed to demonstrate the water pressure and liquid level detection. An interrogation multimode fiber optic setup was implemented. The spectrometer and an incandescent lamp were interconnected using a 50/50 multimode fiber coupler (TM105R5F1A).

Fig. 3
figure 3

a Polymer spectrum response after and before water interaction. b Computed reflection response of polymer Fabry–Perot cavity

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A container was also manufactured to fix a commercial 3.6 μm thickness polymer film used to detect the pressure and liquid level (see Fig. 4). The container was a cube with a 5-inch edge length. A hole was drilled in one face of the container, where the Mylar® film was fixed using special glue; the total membrane diameter is 12.7 mm. At the same position, on the opposite face, a manometer was set to monitor the pressure. At this point, a valve was located to control the liquid level. A pipe with a scale was placed at the top of the box/container. At this point, the liquid level was monitored as the water was added. As the liquid level increased, the pressure at the bottom of the container increased; Here, the manometer monitored the pressure at the same position as the polymer. The water increment caused the soft polymer to deflect and to be stressed. An optical fiber FC/PC connector was used to incite the MMF and MPF interaction. This connector was set over a translation stage and fixed using a fiber holder.

Fig. 4
figure 4

Experimental setup

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The junction between the MPF and MMF produced the interference spectrum shown in Fig. 3a. This spectrum is achieved before the water is added into the container. The power intensity increment was generated by the Mylar’s high refractive index (1.67). However, the reflection R2 decreased around five times, when the surrounding refractive index went from n3 = 1.0(air) to n3 = 1.3329 (water), resulting in a decline of all the interference signals. Despite this decreased power level, the interference spectrum was significantly observed. Moreover, it is worth mentioning that no significant liquid level was evident when the membrane was completely covered by water. Furthermore, by considering the interference spectrum when the MPF is surrounded by air, it can be assumed that α and β from Eq. 1 can be ignored. The total power reflection generated by the MPF and MMF interaction is computed and presented in Fig. 3b. The presented spectrums correspond to the surrounding medium of water and air. These interference spectrums present minimal and maximal visibility (V) of 3 and 9 dB, respectively. The Free Spectral Range (FSR) of both spectrums vary from 21 to 31 nm. This linear increment response is related to the following expression:

$$FSR=\left({\lambda }_{1}-{\lambda }_{2}\right)= \frac{{\left({\lambda }_{1}{\lambda }_{2}\right)}^{2}}{2{n}_{c}d}$$
(4)

where λ1 and λ2 represent the adjacent deep wavelength points of an interference peak. Considering the following parameters of the air reflection spectrum: λ1a = 578.7, λ2a = 608.9, cavity’s refractive index n2 = 1.67, and cavity thickness of d = 3.6 μm, the FSR can be estimated around 29.3 nm. However, considering the difference between the adjacent points, the FSR is around 30.2 nm; this FSR corresponds to a Fabry–Perot cavity length close to 3.493 μm. The difference between the manufacture data thickness and the thickness computed in this work can be attributed to the stress applied at the moment of fixing the polymer; moreover, the interaction between the MPF and multimode fiber connector also induces minimal stress. Both aspects produce a difference between the analyzed FSR. However, when the polymer is completely covered by water, the FSR is altered; and moreover, a wavelength shifting is presented.

The following points of the water reflection spectrum are shown in Fig. 3b: λ1w = 569.1 and λ2w = 600.1; an FSR around 31 nm is computed. By a constant cavity’s refractive index (1.67), the cavity’s length can be estimated at around 3.298 μm; However, this implies a considerable reduction of the initial Fabry–Perot cavity length (3.493 μm). Nevertheless, considering the cavity length as a constant (3.493 μm), a cavity’s refractive index of 1.5767 is computed. This alteration is more suitable for the experiment, and the refractive index change can be related to mechanical properties alteration when the polymer when is covered by water (Volynskii and Bakeev 2018).

Moreover, according to the Mylar® properties, the absorption value is close to 0.8 % after immersing the polymer in water for 24 h (http://usa.dupontteijinfilms.com/wp content/uploads/2017/01/Mylar_Chemical_Properties.pdf). It is essential to notice that this absorption is the maximal saturation value. By using the cavity length, the refractive index of the cavity, and the Fresnel reflections analyzed above; it is possible to analyze and compare the Fabry–Perot response for water and air surrounding medium; these data are used in Eq. 1 for both cases (see Fig. 5).

Fig. 5
figure 5

Computed reflection response of Fabry–Perot cavity surrounded by air and water

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As can be appreciated in Fig. 5, the power level and the FSR are very similar to the data presented in Fig. 3b. The visibility variation is related to the wavelength refractive index dependence and the stress presented in both cases. Furthermore, a Finite Element Model (FEM) was carryout by using ANSYS software. Here, the membrane dimensions and the following parameters are used: E = 4.8956 MPa and \(\nu\) = 0.38; as a result, it is possible to know the stress in the cylindrical coordinates; the stress values as well the thickness membrane variation are presented in Table 1.

Table 1 Stress and thickness ANSYS model response for low-pressure variation
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The ANSYS model reveals a significant stress difference between σr and σz; moreover, the radial and theta directions’ stress are the same. It is important to notice that the refractive index alterations caused by the stress changes are limited by the elastic-optic coefficient (C), which is not available for the polymer sample. However, considering Eq. 3, a refractive index alteration can be expected when the membrane is stressed by external low-pressure. Moreover, it can be assumed that the sensor’s operations are mostly governed by the birefringence effect generated by the stress difference in the radial and displacement coordinates. Hence, it is crucial to notice that the maximal thickness variation (17 nm) represents a minimal variation and contribution to the phase variation.

4 Results

Once the system operation is validated, the water liquid level is increased in the container; consequently, the pressure also increases, the mentioned relation, P = ρgh link both. Indeed, considering the manometer’s pressure, the liquid level can be compared to the scale set at the pipe container.

As the water is added to the container through the pipe, the interference spectrum exhibited a clear wavelength shift to the blue region (see Fig. 6). Then, the interference spectrum shifted, the visibility varied in a nonlinear manner. Thus, this parameter was not analyzed for sensing purposes. When the maximal pressure (6.20 kPa) was applied, a total signal shifting of around π/2 was observed, and some interference points changed the color region. In order to analyze the sensitivity and color change, the peaks (P1 = 555 nm and P2 = 528.7 nm) and depths (D1 = 568.78 nm and D2 = 541.22 nm) presented in Fig. 6 were analyzed and compared in Fig. 7a.

Fig. 6
figure 6

Interference spectrum response for water pressure variation

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Fig. 7
figure 7

a Wavelength shifting of the peaks and depths: P1 = 555 nm, P2 = 528.7 nm, D1 = 568.78 nm, and D2 = 541.22 nm. b Hysteresis analysis of the analyzed points

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These points exhibited very similar responses. Moreover, a maximal wavelength shifting of around 13.2 nm (P1) was achieved; Thus, the following maximal sensitivities can be expected by P1: 2.72 nm/kPa, and 26.77 nm/m. Despite its adequate power level and visibility, this point’s shifting did not present a color region change, desired to reduce its cost for future implementation. On the other hand, D1 presented a sensitivity of 2.49 nm/kPa and 24.5 nm/m. Therefore, given the spectrometer’s resolution (1 nm), a competitive resolution of around 0.40 kPa and 0.04 m was achieved. Even with the sensitivity and the power level presented, this point (D1) presented a significant color change from yellow to green. Moreover, this peak exhibited a quadratic polynomial fitting with an adjusted R-squared of 0.9539.

To evaluate the hysteresis, we slowly removed the water from the container. Then, the interference points P1 and D1 were analyzed and compared in Fig. 7b. As can be appreciated, depth D1 presented minimal error; here, the trajectory had a path difference of 1 nm. Furthermore, the interference was monitored to validate the sensing setup; here, constant pressure of 1.37 kPa was applied. The power and wavelength fluctuations presented by D1 are presented in Fig. 8. As can be observed, the maximal wavelength fluctuation is around 1 nm, then the maximal error measurement during the time can be estimated around 0.17%. The power fluctuations can be ignored because of the minimal variation, and modulation analyzed in this work.

Fig. 8
figure 8

Stability analysis

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It is important to stress that the analyte refractive index variation is limited by the container design. However, if a refractive index change is presented, it can be expected a visibility variation; this is assumed by considering the Fresnel reflection equation. It is important to point out that the results described above were achieved in a room temperature of 22 °C. Even though the MPF has a melting point close to 254 °C (Sanaâ et al. 2016), there could be some thermal effects that could affect the adhesive used to fix the polymer, as well possible degradation of the polymer chains. Therefore, it is suggested that the sensing temperature operation should be carried out at room temperature. Despite these challenges, the proposed system offers a competitive sensitivity and resolution. The results are compared with prior works in Table 2 (pressure) and Table 3 (liquid level).

Table 2 Comparative pressure sensitivity results considering prior works
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Table 3 Liquid level sensitivity comparison with prior works
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5 Conclusions

The interaction between a multimode fiber-optic and a Mylar® polymer film, using a pumped visible light spectrum, is demonstrated as a low-cost fiber optic sensor for pressure and liquid level monitoring. This union functions as a one-layer Fabry–Perot cavity. Here, its response is governed by the stresses applied over the polymer film. The water liquid level controls these stresses in the sensing setup, in which a scale and a manometer, respectively, were used to monitor the liquid level and water pressure. Moreover, both of them are compared by the pressure and density relationship for fluids. When the liquid level and water pressure are altered, the spectrum undergoes a wavelength shift to a shorter wavelength. The best wavelength operation was presented by a depth centered at 568.78 nm, where sensitivities of 2.49 nm/Pa and 24.5 nm/m were achieved. As a result, high resolutions of around 0.40 kPa and 0.04 m are obtained for pressure and level, respectively. The sensing setup presents a quadratic polynomial fitting with an adjusted R-squared of 0.9539. Furthermore, The hysteresis and stability analyses reveal minimal error measurements (1 nm and 0.17% , respectively). This sensor offers a cost-effective implementation, as well as high resolution and sensitivity. Monitoring low-pressure is strongly related to medical procedures; the results achieved can be an alternative for drug infusion monitoring, hyperbaric therapy, and medical ventilators.