## Abstract

Force/strain sensitivity and fringe contrast are important parameters of spheroidal Fabry-Perot interferometers (FPIs). A static structural model and a ray optics model are proposed in this paper for analyses of force/strain sensitivity and fringe contrast. The models proposed show that the sensitivity and fringe contrast of FPIs with spheroidal cavities can be controlled through the dimensions of the spheroids. To corroborate the analyses, three spheroidal FPIs are fabricated via a chemical etching method and static force experiments are carried out. The maximum relative errors of force sensitivity and fringe contrast are 5.2% and -6.4%, respectively. We believe that this research will contribute to improvements in the performance of spheroidal FPIs.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Fabry-Perot interferometer (FPI) has become an attractive choice for force/strain sensing owing to their advantages of small size, compact structure, high sensitivity, flexibility and small cross sensitivity [1–3]. FPI force/strain sensors are widely used in a variety of fields that range from industry to biochemistry, particularly suitable for narrow places where the force/strain monitoring is extremely important [4,5]. FPI cavities with different shapes, including a spheroid, a cube and a cylinder, that have been fabricated by femtosecond laser processing, special fiber splicing, chemical etching and electrical arc discharging [6–9]. Villatoro et al. proposed a spheroidal cavity formed by splicing a photonic crystal fiber with a conventional single-mode fiber [10]. Two spheroidal FPIs with different cavity radii of 26 μm, 58 μm and the fringe contrast in 8 to 12 dB range was fabricated. Through the strain experiment, they found that the interferometer with a larger cavity has a higher strain sensitivity. Wang et al. proposed a novel technology of making spherical cavities. Two single-mode fibers with liquid-coated ends were spliced together to form an air bubble [11,12]. Five FPIs with different cavity lengths were fabricated. However, it was found that the smaller the cavity is, the higher the sensitivity becomes.

A large number of scholars have done a lot of meticulous and profound research on new structures and processing technologies of FPIs [13–16]. Most authors ignored the influence of cavity size and shape on the strain sensitivity. Ecke et al. [17] proposed a high hydrostatic pressure sensor with a hollow glass sphere. They assumed the pressure on the sphere was iso-tropic and they analyzed the pressure-induced fringe shift in theory. They drew a conclusion that large diameter and thin-walled spheres have a greater sensitivity. Favero et al. [18] took the lead by theoretically analyzing the influence of shape parameters on the strain sensitivity. Through analyzing the volume of the section of the optical fiber that contains the cavity and its microscopic changes, they established the relationships between the polar radius, the equator radius and the Poisson’s ratio. They were able to point out that the spheroid with smaller polar radius and larger equator radius has a higher strain sensitivity. Favero et al. made a qualitative analysis of the relationship between the strain sensitivity and the shape parameters of the spheroidal cavity. But he did not take the stress concentration into consideration in his research. In addition, fringe contrast is an important factor that affects the performance of an FPI sensor. Maintaining a high fringe visibility is critical with these sensors as lower values result in a reduced signal-to-noise ratio [19]. To the authors’ best knowledge, no research on the relationship between dimension of the spheroidal cavity and fringe contrast of the interference has been proposed so far.

In this article, we presented a theoretical analysis of the force sensitivity and fringe contrast characteristics of spheroidal Fabry-Perot interferometers and also studied on effects of the cavity shapes and dimensions on the force sensitivity, strain sensitivity and fringe contrast. Furthermore, simulations and experiments were carried out, and the results, which supported the theory, were also presented.

## 2. Theoretical analysis and simulation

#### 2.1 Theoretical analysis of force sensitivity

Let us assume a Fabry-Perot interferometer (FPI) whose cavity is made of air, has a spheroidal shape and that is located on the central axis of the single-mode fiber whose radius is *r*, as is shown in Fig. 1.

The shape of the cavity in the cross section is an ellipse, which can be written as:

where*b*is the polar radius on the x-axis and

*a*is the equator radius on the y-axis. According to the differential geometric theory, the area of fiber materials remaining at sub-segment d

*x*can be calculated through: where

*r*is the radius of a single-mode fiber. Considering definitions of stress $\sigma \textrm{ = }F/S$ and strain $\varepsilon = \sigma /E$ [18], the elongation of the fiber at sub-segment d

*x*can be expressed as:

*E*is the Elastic Modulus of a single-mode fiber, $\varepsilon$ is the strain at sub-segment d

*x*and

*F*is the external force. The elongation of the whole cavity can be calculated through integral operation, which can be simplified into:

According to the definition of strain $\varepsilon = \Delta l/l$ and considering the stress concentration, the strain of a spheroidal cavity can be expressed as:

*l*=2

*b*is the initial cavity length, $\kappa (a,b)$ is the stress compensation factor which is a function of the polar and equator radius. The equivalent cross-sectional area can be recorded as $S^{\prime} = \pi a{({r^2} - {a^2})^{1/2}}/arctan[a/{({r^2} - {a^2})^{1/2}}]$. According to Eqs. (4) and (5), we can get $\partial l/\partial F = 2b/[E\pi a{({r^2} - {a^2})^{1/2}}] \cdot \arctan [a/{({r^2} - {a^2})^{1/2}}] \cdot \kappa (a,b) = 2b\kappa (a,b)/ES^{\prime}$. In addition, $\partial \lambda /\partial l = 2n/m$ can be obtained through $\lambda = 2nl/m$ [1]. When a force

*F*is applied on the FPI, force sensitivity of the peak wavelength shift can be calculated through

*m*

^{th}FPI peak and

*n*is the refractive index of air [1].

Stress concentration occurs where the shape of an object is drastically changed. An elliptical hole with a higher eccentricity will cause greater deformations in stress-concentration areas. In order to validate $\kappa (a,b)$, the stress compensation factor, elongations of air cavities with different dimensions were analyzed in ANSYS (the Elastic Modulus of 73 GPa and the Poisson’s Ratio of 0.17) (Fig. 2).

Figure 2 shows the simulated elongations at the central axis of the fiber. In order to obtain an accurate stress compensation factor, we took the average elongation of the central axis and edge of a fiber as an effective elongation. The stress compensation factor can be calculated through

where $\Delta {l_s}$ is an effective elongation of simulation and $\Delta l$ is calculated through Eq. (4). Stress compensation factors for different cavities were calculated through Eq. (7) (Table 1).It’s interesting that the stress compensation factor is linearly related to the equator radius, as is shown in Table 1. Slope and intercept of the fitting line are functions of the polar radius. The expression of compensation factor is obtained through curve fitting:

It’s necessary to explain that units of the equator and polar radius in Eq. (8) are μm. Shortening polar radius *b* or increasing equator radius *a* will aggravate changes in ellipse and produce a higher stress concentration, which is consistent with Eq. (8). After compensating the stress concentration, the elongation error is no larger than 5%.

Bring Eq. (8) into Eq. (6), then force sensitivity can be recalculated. In Eq. (6), it’s obvious that the force sensitivity is affected by the *m*^{th} interference fringe, equator radius and polar radius.

The relationship between dimension of the cavity and force sensitivity of the *m*^{th} interference fringe (for instance, *m*=100) is plotted in Fig. 3.

As can be seen in Fig. 3(a), force sensitivity increases exponentially with equator radius. In Fig. 3(b) the sensitivity increases linearly as polar radius increases. We find out that the force sensitivity (of the same *m*^{th} interference fringe) is positively correlated with both equator radius and polar radius.

Usually, we are more interested in force sensitivity at a certain wavelength, for instance, a wavelength that is near 1550 nm. The relationship between dimension of the cavity and force sensitivity of the wavelength that is near 1550 nm is plotted in Fig. 4.

As is shown in Fig. 4(a), force sensitivity increases exponentially with equator radius. Whereas, in Fig. 4(b), the force sensitivity decreases exponentially as polar radius increases. Therefore, we draw a conclusion that force sensitivity (with a wavelength near 1550 nm) is positively correlated with equator radius and negatively correlated with polar radius.

In Eq. (6), force sensitivity is affected by equator radius *a*, polar radius *b* and interference order *m*. The central wavelength of the *m*^{th} FPI peak is related to polar radius and not affected by equator radius. When polar radius is a constant, interference fringe remains unchanged, and force sensitivity is only related to equator radius. Therefore, force sensitivities of Fig. 3(a) and Fig. 4(a) show the same trends.

Polar radius *b* has a great influence on the central wavelength of the *m*^{th} FPI peak. If interference order *m* is assumed to be 100, wavelength $\lambda$ will be changed from 400 nm to 2000nm when the polar radius is changed from 10 μm to 50 μm. In Fig. 3(b), wavelength $\lambda$ has a major influence on the force sensitivity, which increases linearly as the polar radius increases. However, when wavelength $\lambda$ is assumed to be a constant (about 1550 nm), *S’* and $\kappa (a,b)$, namely the equivalent cross-sectional area and the stress compensation factor, have major influences on force sensitivity. Equivalent cross-sectional area *S’* is not affected by polar radius, but stress compensation factor $\kappa (a,b)$ decreases exponentially as polar radius increases. Therefore, the force sensitivity in Fig. 4(b) is negatively correlated with polar radius.

Set $\kappa (a,b) = 1$ in Eq. (6), then, we get a force sensitivity (with a wavelength near 1550 nm) without considering stress concentration. When wavelength $\lambda$ (about 1550 nm) and stress compensation factor $\kappa (a,b)$ in Eq. (6) are constants, the force sensitivity is only related to equivalent cross-sectional area *S’*. In Fig. 5(a), the FPI with a long equator radius *a* has a small equivalent cross-sectional area *S’*, which results in a high force sensitivity. FPIs with the same equator radius *a*, have the same equivalent cross-sectional area *S’*. Thus, in Fig. 5(b), the FPI exhibits a similar force sensitivity with the same equator radius *a*. The wavelength is not exactly equal to 1500 nm, so the force sensitivity fluctuates slightly with increases in the polar radius.

Compared with Fig. 5(a), there are significant differences in force sensitivities at different polar radii in Fig. 4(a). That is because a shorter polar radius will produce a higher stress concentration factor, making the FP cavity more easily to deform. For the same reason, there are differences between Fig. 4(b) and Fig. 5(b).

#### 2.2 Theoretical analysis of strain sensitivity

When such FPI sensors are tightly attached/bonded to an external object, the expansion or contraction of that object can be measured from the shift of the transmission spectrum. The size of the FPI is too small (only a few hundred microns) for the external object, so the strain of the object where the FPI is attached is considered uniform. We assume that the strain of the FPI is also uniform, because stress concentration is not considered in the analyses of strain sensitivity.

Considering the definitions of stress $\sigma = F/S^{\prime}$ and strain $\varepsilon = \sigma /E$ [18], there is $F = \varepsilon ES^{\prime}$. If stress compensation factor $\kappa (a,b)$ is ignored, force sensitivity in Eq. (6) can be re-written as

We arrive to the following expression:

This indicates that strain sensitivity could be enhanced solely by choosing a longer wavelength. It’s necessary to explain that the units of $\lambda$ and $\partial \lambda /\partial \varepsilon$ in Eq. (10) are μm and μm/ε.

For example, Fig. 6(a) shows reflection spectra at different strains (0 ε, 0.001 ε, 0.002 ε and 0.003 ε) of a spheroidal FPI whose initial cavity has a length of 60 μm. The reflection spectra are calculated in MATLAB according to the expression ${I_r} = 2{R_s}{I_i}[1 + cos({{4\pi nl} / \lambda })]$ [11]. ${R_s} = 0.035$ is the reflective index of the air/silica interface, ${I_i} = 1\textrm{ W/}{\textrm{m}^\textrm{2}}$ is the intensity of incident light, *n*=1 is the refractive index of the air, *l*=6000 nm is the cavity length and $\lambda$ is the wavelength from 1530 nm to 1575 nm.

As shown in Fig. 6(a), initially, there are two peaks (peak1 = 1538.46 nm and peak2 = 1558.44 nm) within the range of 1530 nm to 1575 nm. It’s definitely clear that peaks of the interference spectrum evenly shift towards the long-wavelength region as the strain increases. A linear fitting is performed and the strain sensitivities are 1538 nm/ε (at peak1) and 1558 nm/ε (at peak2), as is plotted in Fig. 6(b). The result is consistent with the theoretical derivation.

#### 2.3 Theoretical analysis of fringe contrast

Interference peaks in the reflection spectrum are generated from the interference of light reflected by front and back air/silica-faces of the cavity [20]. According to the interference theory, reflected light intensity at the back end-face of the cavity is a key factor that affects fringe contrast. For a spheroidal FP cavity, dimension of the cavity is critical to the reflected light intensity.

In the following analysis, an ellipse rotated around its minor axis, with a size of $2a \times 2b(a > b)$ will be assumed, where *a* is the equator radius on the y-axis and *b* is the polar radius on the x-axis (Fig. 7).

The elliptic equation is expressed as Eq. (1) and the radius of curvature at the end of the minor axis can be calculated through:

The incident light is assumed to be parallel with a uniformly intensified distribution, which propagates along the positive direction of x-axis. According to Snell’s law of refraction, the angle of refraction can be expressed as:

In Eq. (12), $\alpha = b{y_1}/{a^2}$ is the angle of incidence, ${y_1} = 4\textrm{ }\mathrm{\mu}\textrm{m}$ is the radius of fiber core, ${n_1} = 1.468$ is the refractive index of fiber core (SMF-28e Corning) and ${n_2} = 1$ is the refractive index of air cavity. The increment of displacement in y-direction at the right end-face of the cavity can be expressed as:

In Eq. (13), $\gamma = \beta - \alpha$ is the difference angle between refractions and incidences. Then it can be obtained from geometric relations:

where $\theta$ is the angle of light reflection, which is reflected on the right end-face. ${y_2} = {y_1} + \Delta y$ is the height of refracted light on the right end-face.The position of reflected light on the left end-face can be calculated through ${y_3} = l\tan (\varphi ) - {y_2}$. $\varphi = \gamma + 2\theta = [4({n_1} - 1){b^2}/{a^2} + (3 - {n_1})]b{y_1}/{a^2}$ is the incidence angle of reflected light. Substitute all variables and we arrive to the following expression:

Considering the paraxial approximation, the tangent value can be approximated by its radian. Equation (15) can be simplified into:

When the following two conditions are met, the reflected light can propagate in the fiber.

In Eq. (18), the Numerical Aperture *NA*=0.12 (SMF-28e Corning). There is a position *y’* satisfying Eq. (17) and Eq. (18), which ensures that the light is reflected to the fiber core and meets conditions of total internal reflection. Then, we obtain the expression of the reflected light intensity:

*I*is the incident light intensity,

_{i}*R*is the reflective index of the air/silica interface, $\eta = y^{\prime}/{y_1}$ is the incidence rate of reflected light and ${y_1} = 4\textrm{ }\mathrm{\mu}\textrm{m}$ is the radius of fiber core.

_{s}Ray optics models were built in COMSOL to verify the above theoretical analysis. The detailed parameters are shown in Table 2 and the simulation results (*a*=30 μm, *b*=20 μm) are illustrated in Fig. 8.

As shown in Fig. 8(a), all reflected rays are converged inside the fiber core. In Fig. 8(b), the intensity of reflected light exhibits a Gaussian distribution. The position of reflected light is 2.81 μm, which is consistent with the result (2.74 μm) of Eq. (16). Various spheroidal cavities with different dimensions were built and simulated in COMSOL. The reflection position and reflected light intensity were recorded and calculated.

In Fig. 9(a), the simulated value is consistent with the inferential value, and the reflection position rises rapidly with equator radius. The error between the calculation (Eq. (16)) and the simulation is no more than 0.43 μm (in fact, the error decreases as the equator radius increases, when *a*=100 μm and *b*=40 μm, the error is only 0.001 μm).

Figure 9(b) shows the intensity of reflected light through simulation and calculation of Eq. (19). When the polar radius is a constant, the intensity of reflected light rises with increases in equator radius until reaching the maximum. The maximum light intensity ${I_2} = {I_i}{(1 - {R_s})^2}{R_s} \approx 32.6\textrm{W/}{\textrm{m}^\textrm{2}}$ is calculated through Eq. (19). The maximum error between the calculation of Eq. (19) and the simulation is 2.6 W/m^{2}.

In fact, there are losses when light propagates in the FP cavity. As is shown in Fig. 10, $\eta$ is the incidence rate of reflected light, $\upsilon$ is the transmission loss factor at the reflection surface and $\zeta$ is the loss factor of gas cavity [8,21].

The intensity of interference light can be expressed as ${I_r} = {I_1} + {I_2} + 2{({I_1}{I_2})^{1/2}}\cos (4\pi nl/\lambda )$. ${I_1} = {I_i}{R_s}$ and ${I_2} = {I_i}{(1 - {R_s})^2}{(1 - \upsilon )^2}(1 - \zeta )\eta {R_s}$ are intensities of light reflected from the front and back air/silica-faces of the cavity [21]. Obviously, the maximum and minimum intensities of interference fringe are ${I_M} = {I_1} + {I_2} + 2{({I_1}{I_2})^{1/2}}$ and ${I_m} = {I_1} + {I_2} - 2{({I_1}{I_2})^{1/2}}$. Then the fringe contrast *F* can be expressed as:

The fringe contrast is related to parameters ${R_s},\upsilon ,\zeta ,\eta$, therefore, it can be controlled through dimensions of the cavity.

## 3. Experiment and discussion

#### 3.1 Experimental system set-up

An optical system was built as illustrated in Fig. 11. An ASE-C + L light source was used to provide the broadband light from 1525 nm to 1625 nm. A three-port circulator was used to guide light from the light source to the FPI and transmit the reflected light to an OSA. The reflection spectrum was analyzed in an optical spectrum analyzer (ANDO AQ6317B, 600∼1750nm optical input, 15 pm resolution) and transmitted to a computer through a data acquisition board (ADLINK USB-3488A).

Chemical etching is a simple and cost-effective method of processing micro-cavity structures [22,23]. In the previous article [22], the etching process of spheroidal FP cavity has been introduced. Three FPIs with different shapes were fabricated by controlling corrosion time. Figure 11 shows the image of the FP cavity, which is magnified 200 times under a microscope. Then dimensions of the FP cavity are obtained and shown in Table 3. We would like to point out that our technique does not allow the fabrication of cavities with any arbitrary shapes or dimensions.

#### 3.2 Static force experiment

As is described in Fig. 11, an optical fiber is fixed on a bracket, whose weight is suspended at the end of the fiber. The force in vertical direction increases from 0 N to 1 N with the weight changes from 0 g to 100 g at intervals of 20 g. Smoothing filtering are applied to the collected spectra of the FPI with a cavity size ($a \times b$) of $35.38\textrm{ }\mathrm{\mu}\textrm{m} \times 25.38\textrm{ }\mathrm{\mu}\textrm{m}$, as is shown in Fig. 12(a).

In Fig. 12(a), it’s definitely clear that peaks of the interference spectrum evenly shift towards the long-wavelength region as the force increases. The light source is ASE-C (1525∼1565 nm) + L (1565∼1625 nm), so there is a peak around 1565 nm. Four loading force experiments are implemented. In order to avoid influences of sudden changes near 1565 nm, peaks within the range of 1548 nm to 1560 nm are traced and used to calculate the force sensitivity, as is plotted in Fig. 12(b). Sensitivities of FPI1, FPI2 and FPI3 are respectively 4.89 nm/N (R-squared is 0.99), 4.99 nm/N (R-squared is 0.98) and 4.53 nm/N (R-squared is 0.99). The maximum standard deviations are 0.09 nm (FPI1), 0.13 nm (FPI2) and 0.16 nm (FPI3). Although the linearity of FPI2 (R-squared is 0.98) is not as good as that of FPI1 and FPI3, it is acceptable.

Shape parameters and experimental sensitivities of the three FPIs are listed in Table 3. The fiber radius is shorter than 62.5 μm due to corrosion behaviors. In order to compensate for the stress concentration caused by the shorter diameter, the equator radius can be approximately recalculated through $a^{\prime} = a \times 6.25 \times {10^{ - 5}}/r$. All parameters are substituted into Eq. (6), then the force sensitivity is calculated, as is listed in Table 3.

The maximum relative error is 5.2%. Force sensitivity of a spheroidal FPI can be effectively calculated through Eq. (6).

#### 3.3 Experimental contrast

Initial reflection spectra of the three spheroidal FPIs were recorded and plotted in Fig. 13.

There is a jump near 1565 nm, so we only record intensities of some peaks and valleys. Average fringe contrasts are calculated and listed in Table 4.

Bringing all dimension parameters into Eq. (17), Eq. (18) and Eq. (19), incidence rates $\eta$ of reflected light FPI1, FPI2 and FPI3 are respectively calculated to be 0.597, 0.785 and 0.738. The transmission loss factor at reflection surface $\upsilon$ is 0.4 [21] and the loss factor of gas cavity $\zeta$ is 0.3 [14]. Theoretical fringe contrasts of the three FPIs can be obtained through Eq. (20), as is shown in Table 4.

In Table 4, the relationship among incidence rates of reflected light, theoretical contrasts and experimental contrasts of the three FPIs is FPI1 < FPI3 < FPI2. The three FPIs have the same reflected light intensity (from the front air/silica-face) ${I_1}$ and loss parameters $\upsilon ,\zeta$, so the FPI with a high incidence rate $\eta$ shows a strong fringe contrast. The fringe contrast with a maximum relative error of -6.4% can be effectively estimated by the model proposed.

## 4. Conclusion

In conclusion, a research was carried out on influences of the size of a spheroidal cavity on its force/strain sensitivity and fringe contrast. Three spheroidal FPIs were fabricated through chemical etching method and static force experiments were carried out. Theoretical calculations well conformed to the experiments results with a maximum relative error of 5.2%, and both of them indicated that sensitivity (wavelength near 1550 nm) was positively correlated with equator radius and negatively correlated with polar radius. In addition, strain sensitivity was dictated by polar radius which could be enhanced by choosing a longer wavelength. Furthermore, theoretical fringe contrasts were conformed to the experimental values with a maximum relative error of -6.4%. Force/strain sensitivity and fringe contrast of spheroidal FPIs could be effectively analyzed based on the theory proposed in this paper, which would contribute to improvements in the performance of spheroidal FPI sensors.

## Funding

National Natural Science Foundation of China (61973194); Fundamental Research Fund of Shandong University (2018JC031); Major Scientific and Technological Innovation Project of Shandong Province (2019JZZY010427); Shenzhen Fundamental Research and Discipline Layout project (JCYJ20190806155616366).

## Disclosures

The authors declare no conflicts of interest.

## References

**1. **Y. Liu, C. Lang, X. Wei, and S. Qu, “Strain force sensor with ultra-high sensitivity based on fiber inline Fabry-Perot micro-cavity plugged by cantilever taper,” Opt. Express **25**(7), 7797–7806 (2017). [CrossRef]

**2. **J. Wu, M. Yao, F. Xiong, A. Ping Zhang, H. Y. Tam, and P. K. A. Wai, “Optical fiber-tip Fabry-Perot interferometric pressure sensor based on an in situ μ-printed air cavity,” J. Lightwave Technol. **36**(17), 3618–3623 (2018). [CrossRef]

**3. **J. Ma, J. Ju, L. Jin, and W. Jin, “A compact fiber-tip micro-cavity sensor for high-pressure measurement,” IEEE Photonics Technol. Lett. **23**(21), 1561–1563 (2011). [CrossRef]

**4. **P. Huang, N. Wang, J. Li, Y. Zhu, and J. Zhang, “Fiber Fabry-Perot force sensor with small volume and high performance for assessing fretting damage of steam generator tubes,” Sensors **17**(12), 2899–2910 (2017). [CrossRef]

**5. **D. Tosi, P. Saccomandi, E. Schena, S. Silvestri, D. B. Duraibabu, S. Poeggel, G. Leen, and E. Lewis, “Intra-tissue pressure measurement during laser ablation with fiber-optic extrinsic Fabry-Perot sensor,” Sensors **16**(4), 544–555 (2016). [CrossRef]

**6. **T. Han, Y. Liu, Z. Wang, Z. Wu, S. Wang, and S. Li, “Simultaneous temperature and force measurement using Fabry-Perot interferometer and bandgap effect of a fluid-filled photonic crystal fiber,” Opt. Express **20**(12), 13320–13325 (2012). [CrossRef]

**7. **M. Quan, J. Tian, and Y. Yao, “Ultra-high sensitivity Fabry-Perot interferometer gas refractive index fiber sensor based on photonic crystal fiber and Vernier effect,” Opt. Lett. **40**(21), 4891–4894 (2015). [CrossRef]

**8. **P. Chen and X. Shu, “Refractive-index-modified-dot Fabry-Perot fiber probe fabricated by femtosecond laser for high-temperature sensing,” Opt. Express **26**(5), 5292–5299 (2018). [CrossRef]

**9. **S. Pevec and D. Donlagic, “Miniature fiber-optic sensor for simultaneous measurement of pressure and refractive index,” Opt. Lett. **39**(21), 6221–6224 (2014). [CrossRef]

**10. **J. Villatoro, V. Finazzi, G. Coviello, and V. Pruneri, “Photonic-crystal-fiber-enabled micro-Fabry–Perot interferometer,” Opt. Lett. **34**(16), 2441–2443 (2009). [CrossRef]

**11. **S. Liu, Y. Wang, C. Liao, G. Wang, Z. Li, Q. Wang, J. Zhou, K. Yang, X. Zhong, J. Zhao, and J. Tang, “High-sensitivity strain sensor based on in-fiber improved Fabry–Perot interferometer,” Opt. Lett. **39**(7), 2121–2124 (2014). [CrossRef]

**12. **C. Liao, S. Liu, L. Xu, Y. Wang, Z. Li, Q. Wang, and D. N. Wang, “Sub-micron silica diaphragm-based fiber-tip Fabry-Perot interferometer for pressure measurement,” Opt. Lett. **39**(10), 2827–2830 (2014). [CrossRef]

**13. **S. Wu, L. Wang, X. Chen, and B. Zhou, “Flexible optical fiber Fabry–Perot interferometer based acoustic and mechanical vibration sensor,” J. Lightwave Technol. **36**(11), 2216–2221 (2018). [CrossRef]

**14. **T. Nan, B. Liu, Y. Wu, J. Wang, Y. Mao, L. Zhao, T. Sun, and J. Wang, “Ultrasensitive strain sensor based on Vernier effect improved parallel structured fiber-optic Fabry-Perot interferometer,” Opt. Express **27**(12), 17239–17250 (2019). [CrossRef]

**15. **M. S. Ferreira, J. Bierlich, J. Kobelke, K. Schuster, J. L. Santos, and O. Frazao, “Towards the control of highly sensitive Fabry-Perot strain sensor based on hollow-core ring photonic crystal fiber,” Opt. Express **20**(20), 21946–21952 (2012). [CrossRef]

**16. **D. Duan, Y. Rao, Y. Hou, and T. Zhu, “Microbubble based fiber-optic Fabry–Perot interferometer formed by fusion splicing single-mode fibers for strain measurement,” Appl. Opt. **51**(8), 1033–1036 (2012). [CrossRef]

**17. **J. P. Dakin, W. Ecke, K. Schroeder, and M. Reuter, “Optical fiber sensors using hollow glass spheres and CCD spectrometer interrogator,” Opt. Laser. Eng. **47**(10), 1034–1038 (2009). [CrossRef]

**18. **F. C. Favero, L. Araujo, G. Bouwmans, V. Finazzi, J. Villatoro, and V. Pruneri, “Spheroidal Fabry-Perot microcavities in optical fibers for high-sensitivity sensing,” Opt. Express **20**(7), 7112–7118 (2012). [CrossRef]

**19. **X. Xu, J. He, M. Hou, and S. Liu, “A miniature fiber collimator for highly sensitive bend measurements,” J. Lightwave Technol. **36**(14), 2827–2833 (2018). [CrossRef]

**20. **Y. Liu, S. Qu, W. Qu, and R. Que, “A Fabry–Perot cuboid cavity across the fiber for high-sensitivity strain force sensing,” J. Opt. **16**(10), 105401 (2014). [CrossRef]

**21. **M. Tian, P. Lu, L. Chen, D. Liu, and M. Yang, “Micro-multicavity Fabry-Perot interferometers sensor in SMFs machined by femtosecond laser,” IEEE Photonics Technol. Lett. **25**(16), 1609–1612 (2013). [CrossRef]

**22. **G. Wei, Q. Jiang, and T. Zhang, “A flexible force sensor based on spheroidal Fabry-Perot microcavity,” Optik **181**, 483–492 (2019). [CrossRef]

**23. **S. Pevec and D. Donlagic, “Miniature all-fiber Fabry-Perot sensor for simultaneous measurement of pressure and temperature,” Appl. Opt. **51**(19), 4536–4541 (2012). [CrossRef]