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Abstract
Fiber optic Fabry-Perot Interferometer benefits many requirements in pressure sensing. The variation of pressure sensitivity of the polymer-diaphragm-based Fabry-Perot pressure sensor with temperature is studied by investigating the thermal effect of the cavity air and the diaphragm separately. FP cavity vacuum treatment and multi-curvature diaphragm simulation and experimental studies are conducted. Experimental results show that the sensor pressure sensitivity decreases with increasing temperature by 0.46nm/(kPa·°C). The diaphragm’s thermal effect is the leading cause of temperature-induced inconsistency in pressure sensitivity, accounting for 0.43nm/(kPa·°C).
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1. Introduction
Numerous optical fiber sensors have been proposed to meet the demand for high-precision measurement in many fields, such as bridge structural health monitoring [1], oil and gas wells [2], aerospace [3], and biomedical applications [4–7]. Fiber optical sensors with a Fabry–Pérot (FP) cavity structure have attracted much interest due to their simple structure, low cost, fast response time, and high sensitivity [8,9]. Various fiber optical FP sensors have been shown in earlier investigations to be capable of measuring temperature and pressure [10–15]. Countless fiber optical FP pressure sensors have been created using long-period gratings [16], hollow microsphere cavities, [17] and hybrid fiber architectures [18]. The most popular construction is built on a silicon substance with the advantages of long-lasting and biocompatible, but the production process necessitates expensive machinery and exact supervision. The low Young's modulus and straightforward curing process of the FP pressure sensors with polymer diaphragms have made them a popular option for simple manufacture, preferential price, and high sensitivity [19].
However, polymer-diaphragm-based FP pressure sensors have a significant temperature cross-coupling effect [20], and a slight temperature change might result in inaccuracies in the pressure measurement. The temperature sensitivity of the fiber optical FP pressure sensors is attributed to the large thermal expansion of the material and air. Researchers have studied eliminating the temperature coupling effect and obtaining higher accuracy pressure measurements [21–24]. Among the many ways to reduce temperature sensitivity, simultaneously measuring the pressure and temperature and constructing a ventilation channel are the two most effective ways [25–29]. For example, Simon proposed an FP sensor consisting of two resonators to achieve pressure and temperature response [30]. Liu proposed a temperature-insensitive optical fiber FP sensor for liquid-level measurement based on borosilicate glass ferrules [31]. The above literature study focuses on the effect of temperature on the variation of FP cavity length. Among them, fiber Bragg gratings (FBGs) are also often used as temperature and stress sensors. D.Tosi proposed using FP and FBGs in cascaded configurations for simultaneously temperature and pressure sensing [32]. Due to the advantages of wavelength-encoded information and multiplexing capabilities, FBGs are especially desirable for multi-parameter measurement. However, FBGs are usually written inside the fiber, which may lead to errors in the case of tip measurement.
Although the pressure measurement error can be reduced by simultaneous temperature measurement, there is a problem of inconsistency of pressure sensitivity caused by temperature. Especially in the polymer diaphragm, the temperature increase can lead to variations in the elastic modulus [33]. Therefore, in the case of temperature disturbances, using a single sensitivity calibration may result in pressure measurement errors. Some researchers have discussed and tried to solve this problem when developing temperature-insensitive pressure sensors based on FBGs. For example, A.Leal-Junior presented pressure cycling of FBGs embedded in diaphragms at different temperatures, indicating that the pressure sensitivity varies due to the temperature-dependent elastic modulus of the diaphragm. [34].
In this paper, the temperature effects on pressure sensitivity consistency of FP pressure sensors are studied by analyzing the thermal effects of air and diaphragm separately. Pressure sensitivity in the range of 30 to 60°C was studied experimentally. The experimental results indicate that the pressure sensitivity tends to decrease slightly with increased temperature, mainly caused by the thermal effect of the diaphragm. To the best of our knowledge, the effect of temperature on the sensitivity of the polymer-diaphragm-based FP pressure sensor has not been studied before.
2. Mathematical model and simulation results
FP sensors are created between two flat, partially reflective surfaces parallel to each other. An incident optical beam undergoes successive back-and-forth reflections between these two surfaces. Figure 1 shows the sensing FP cavity situated at the optical fiber end with a reflecting polymer diaphragm. When the elastic polymer diaphragm deflects under applied pressure, the cavity length changes, which can be observed on the spectrometer as the reflectance spectrum shifts.
The analytical expression for small deflections of a clamped circular diaphragm under uniform pressure is used. The change in cavity length is linearly related to the change in pressure ΔP. The maximum deflection Δlmax happens at the center of the diaphragm and can be expressed as
A certain contact angle will be formed against the hollow-core fiber wall due to the capillary force during the molding process. An uneven diaphragm shape in the radial direction is prepared. Therefore, the diaphragm thickness inhomogeneity error is taken into account in the mathematical model of deflection, and Eq. (1) can be modified as [35]
2.1 Pressure sensitivity drift due to air thermal effect
As shown in Fig. 2(a), the shape of the diaphragm before the air thermal effect is considered a straight line(Ay1B), and a curved line is used to resemble the deformed-shape diaphragm(Ay2B). P(x,y) is a general point on the spherical diaphragm surface. The radius of curvature of the spherical is given by. The FP cavity length at any general point P is, and the phase ϕ change is. For FP cavity length variation caused by the deformation of the diaphragm due to the thermal expansion of air, as follows [36]
Fig. 2. (a) Schematic diagram of the structure of the FP sensor with a deformed diaphragm (b)Simulated spectra after deformation at different temperature.
A three-dimensional model was developed with a sensitive diaphragm in the upper part, an air FP cavity in the middle, and single-mode fiber in the lower part, as shown in Fig. 3. Uniform cylindrical type is used with a thickness set to 15µm, radius 42.5µm and FP cavity length of 80µm. The initial ambient temperature is fixed at 25°C. Thermodynamic and solid mechanics modules simulate thermal expansion and force processes. Fluid-solid coupled multiphysics field is used to simulate the air thermal expansion process. By unchecking the diaphragm thermal-physical field effect, the diaphragm's thermal expansion effect is ignored.
The cavity length shift and the pressure sensitivity drift with temperature are also shown in Fig. 3. The simulation result implies, that the cavity length shift (black line in Fig. 3) is positively proportional to the temperature, and the absolute value of the bending curvature β is increased accordingly. However, the pressure sensitivity drift (blue line) is increased with a step of 0.14 nm/(kPa·°C) within 30°C to 60°C at 1kPa pressure due to the air thermal effect, which is contrary to the laws of mathematical models. We attribute this phenomenon to the change in the shape of the diaphragm, which thus no longer applies to the case described by Equation(2). More discussion of the higher sensitivity of plano-concave diaphragms over inner-concave diaphragms can be found in Ref. [37].
2.2 Pressure sensitivity drift due to diaphragm thermal effect
The thermal effect of the diaphragm is reflected in two aspects, one is the thermal expansion effect of the diaphragm thickness, and the other is the change of the modulus of elasticity of the diaphragm with temperature, as shown in Fig. 4(a)(b), separately. From the simulation results of the expansion effect in Fig. 4(a), a temperature increase of 1°C causes a decrease in sensor sensitivity of 0.82 nm/kPa from 30°C to 60°C, which is 5.9 times the effect of air in the FP cavity. This significant effect is explained by a third-order effect of thickness change on the deformation.
Fig. 4. (a) Variation of linearity of FP pressure sensor sensitivity with temperature caused by thermal expansion of the diaphragm (b) Variation of linearity of FP pressure sensor sensitivity with temperature caused by Young's modulus variation of the diaphragm.
The black curve represents the variation of the modulus of elasticity of the diaphragm with temperature in Fig. 4(b), which shows a nonlinear variation during the temperature rise from 30°C to 60°C. The modulus of elasticity shows a downward trend, then reverses at 55°C. The minimum modulus of elasticity appears at 55°C, and its value is 1.926MPa. The change in sensitivity is indicated by the blue curve of about 0.32nm/(kPa·°C) within 30°C to 60°C, which is the exact opposite of the change in modulus and less than half of the effect of diaphragm expansion.
2.3 Combined effect
Combining the thermal effects of both the air and the diaphragm, the simulation result of the FP sensor pressure sensitivity drift with temperature is shown in Fig. 5. The pressure sensitivity shows a decreasing trend with a drift of 0.86nm/(kPa·°C) by fitting the curve with the quadratic term. It is noted that pressure sensitivity drift caused by curvature radius is calculated to be 0.126 nm/(kPa·°C·µm). This pattern of the sensitivity value is believed to be mainly due to the thermal effect of the diaphragm, significantly the increase of the diaphragm thickness. The effect of thermal expansion of the HCF as the cavity supporting-structure is also simulated, and the results show that the 80µm initial cavity length changes by 0.002µm when the temperature changes from 30°C to 60°C. Compared with 1.4µm (air) and 0.3µm (diaphragm), supporting structure expansion is three and one order of magnitude smaller, respectively. Therefore, the effect can be negligible.
3. Experimental results
3.1 Verification experiments on the effect of thermal expansion of each part
The detailed preparation process of the FP pressure sensor is illustrated in Ref. [37]. The dip-coating method to manufacture polymer-diaphragm-based FP pressure sensors is shown in Fig. 6(a). The fused SMF-HCF structure is fixed to the dip coater by clamps. FP cavity length, dipping process parameters (including dipping time, solution viscosity, dipping angle, and speed of lifting and impregnation), and curing process parameters are unified to produce pressure sensors with the same thickness diaphragm. Figure 6(b). shows that the experimental system comprises a light source provided a wavelength range of 1520–1580nm, a circular FP pressure sensor, and an optical spectrum analyzer with a resolution of 0.2nm used to monitor and normalize the reflected output spectrum.
Fig. 6. (a) Schematic of dip coater (b)Schematic of an experimental device for measuring pressure sensitivity at different temperature.
In the pressure sensitivity experiment, the fiber tip was placed horizontally in a deep container filled with water, and pressure changes were obtained by the variable distance between the FP pressure sensor and the water surface (0 and 1kPa); A heating coils device provides the change in water temperature on the outside of the measuring cylinder. The shift in peaks for the FP sensor can be seen concerning both pressure and temperature change.
A conventional temperature sensor was used as the reference, which was fixed inside the water container. In the experiments, water was first heated to 70°C and was cooled automatically below room temperature (25°C) to ensure identical water temperature. The experiment is started when the water temperature is lowered to 60°C, and the spectrum of the sensor at 0kpa and 1kpa is recorded with a step of 5°C within 30 to 60°C.
The resulting spectrum graphs are shown in Fig. 7(a). It can be seen that the spectrum moves to longer wavelengths, and the optical power is gradually becoming smaller as the temperature increases. A 0.26dB reduction in optical power can be observed with 10°C growth. A comparison of the spectral changes of the sensor before and after pressurization at different temperatures is shown in Fig. 7(b). Δλ1, Δλ2, Δλ3, and Δλ4d indicate the spectral shift at 30, 40, 50, and 60°C, respectively, and the calculated sensitivity results are shown in Table 1. and Fig. 8.
Fig. 7. (a) Interference spectra at different temperatures before pressurization (b)Comparison of spectral changes at different temperatures before and after pressurization.
Fig. 8. Experimental results on the variation of sensor pressure sensitivity at different temperatures.
Table 1. Comparison of pressure sensitivity at different temperatures
It can be seen that the sensor pressure sensitivity exhibits a linear relationship (R2 = 0.97316) over the entire tested temperature range. Based on linear regression analysis of the measured data, the pressure sensitivities were calculated to be 0.46 nm/(kPa·°C). This result is half the value obtained from the numerical simulations (0.86 nm/(kPa·°C)). The limitation of the thermal expansion effect induced by the accumulation of polymer along the wall's inner side is possibly the reason for lower drift. The simulated and experimental sensitivity-temperature variation trends coincided, regardless of the different rates of low-temperature change, and thus indicated the correct prediction of the important effect of diaphragm expansion on sensitivity drift.
Further tests involving the decomposition structure impact were proposed. The air expansion effect is investigated by vacuum cavity treatment. Therefore, FP sensors were evacuated and sealed, and the sensitivity was tested at different temperatures based on the above experimental setup. Figure 9(a) shows the room temperature interference spectra before and after evacuation. Multiple replicate experiments have been conducted to verify the repeatability and accuracy of the spectra in vacuum experiments. The spectral data of each evacuation are shown in Table 2. The average value of Δλ1 (before evacuation) and Δλ2 (after evacuation) are 17.476nm and 17.462nm, respectively. The same spectrum can be maintained in all three evacuations because of the similar Δλ. According to the results, a linear relationship can be observed with R2 = 0.95896, and the variation of sensor sensitivity with temperature after vacuum treatment can be obtained as 0.43nm/(kPa·°C), as shown in Fig. 9(b). A slight difference of merely 0.03nm/(kPa·°C) is observed by comparing the results with the atmospheric pressure state(0.46 nm/(kPa·°C)). It is evident from the tiny gap that air expansion effects can cause only minimal sensitivity drift.
Fig. 9. (a) Interference spectra before and after repeated evacuation (b) Experimental results on the variation of sensor pressure sensitivity at different temperatures after evacuation.
Table 2. Comparison of pressure sensitivity at different temperatures
Moreover, FP sensors with different diaphragm curvatures radii of 123.387, 93.334, 74.409, and 66.581µm were performed, as illustrated in Fig. 10(a). The obtained interference spectra of these FP sensors in the natural state are shown in Fig. 10(b). Red circles are used to highlight the inner surface contours of the diaphragm. The bright white part on the left side of Fig. 10(a) is SMF; the right side is the FP cavity structure with a polymer diaphragm attached to the top. After a uniform production process, the diaphragm thickness of the first three FP sensors is measured to be the same at 4.9µm. The diaphragm thickness of sensor 4 is increased to 26.5µm by prolonging the dipping time during the manufacturing process, as indicated in Fig. 10(a) in the front of the gray and white curved FP cavity structure. The change in curvature of the diaphragm is accomplished by adjusting the curing temperature. The light intensity of the interference spectra exhibits a diminishing trend in diaphragms with increasing curvature, which is consistent with the simulation results.
Fig. 10. (a) Sensors with different diaphragm curvature (b)Unpressurized interference spectra of different curvature sensors.
The intensity of light emitted from the FP cavity and the pressure sensitivity of these sensors were recorded separately, as shown in Table 3. FP cavity sensitivity has been standardized by cavity length values. It is noted that a flatter diaphragm contributes to higher pressure sensitivity, and sensitivity drift was measured to be 0.24nm/(kPa·°C·µm). The experimental results of sensor 4 show that for a 5.4 increase in diaphragm thickness, the sensitivity is severely decreased by a factor of 65. This also indicates that diaphragm expansion (the thickening of the diaphragm) is proven to have a more significant influence on pressure sensitivity drift than air expansion(diaphragm bending radius) and modulus changes.
Table 3. Comparison of interference cavities with different curvature sensors
3.2 Temperature compensation experiment
In order to attenuate the effect of temperature on the pressure sensitivity of the sensor, the temperature-measuring FP cavity mentioned in Ref. [38] was used to monitor the ambient temperature in real-time and compensate for the drift value. The temperature measurement performance of this FP cavity is shown in Fig. 11(a), and the result of the temperature drift compensation is shown in Fig. 11(b).
Fig. 11. (a)Temperature measurement performance of the FP sensor (b) Pressure sensitivity after temperature compensation.
The mean value of pressure sensitivity after temperature compensation is 109.661nm/kPa and the maximum error occurs at 35°C with a difference of 0.62nm/kPa. The effect of temperature drift on the accuracy of pressure sensitivity can be significantly reduced by temperature compensation, which can compensate the sensor pressure sensitivity from 95.7nm/kPa to 109.39nm/kPa at 60°C. However, this approach requires dual FP cavities and a dual demodulation system at the same observation point. For space-compact monitoring needs, this approach needs further improvement.
3.3 Lower thermal expansion material diaphragm
To verify that the pressure sensitivity of the FP cavity based on a lower thermal expansion material diaphragm is less affected by temperature, UV adhesive, and graphene-doped PDMS material were chosen for the experiment. The CTE (coefficient of thermal expansion) of pure PDMS and UV adhesive is 300 ppm/°C and 73ppm/°C, respectively. Doped graphene PDMS has been shown to have a lower CTE because of the negative CTE of graphene at room temperature [39]. The CTE of graphene-doped PDMS is about 220 ppm/°C [40]. A ratio of 15% by mass was chosen to prepare graphene-doped PDMS, and ultrasonic stirring excitation was used to ensure uniform doping. The two diaphragms were prepared the same way as the previous experiments, with a different curing method. The prepared FP cavities are shown in Fig. 12(a)(b).
The change in pressure sensitivity was tested at 30 to 60°C, and the results are shown in Fig. 13(a)(b). As can be seen in Fig. 13(a), the pressure sensitivity of the UV diaphragm-based FP sensor shows a slight increase with temperature, from 40.85 nm/(kPa·°C) to 43.5nm/(kPa·°C) within 30°C. The UV adhesive has a large elastic modulus (about 137 MPa), and the diaphragm thickness is controlled at less than 2µm to ensure high-pressure sensitivity. Therefore, this opposite trend could be due to the thinning of the diaphragm thickness, causing the shape change to be the dominant factor or the change in the elastic modulus with temperature. Figure 13(b) illustrates the temperature drift of the FP sensor pressure sensitivity of the graphene-doped PDMS diaphragm. Compared with pure PDMS, the temperature drift was reduced from 0.46 nm/(kPa·°C) to 0.28 nm/(kPa·°C), with an overall decrease of about 19 nm/kPa. The conclusion that the error in the sensitivity temperature drift can be reduced by using a material with a lower coefficient of thermal expansion can be derived from these experiments.
Fig. 13. (a) Experimental results on the variation of UV diaphragm-based FP sensor pressure sensitivity at different temperatures (b) Experimental results on the variation of graphene-doped PDMS diaphragm-based FP sensor pressure sensitivity at different temperatures.
4. Conclusion
In this study, the temperature-induced inconsistency in pressure sensitivity of polymer-diaphragm-based FP pressure sensors was investigated. The variation in diaphragm shape caused by the thermal effect of the air and the diaphragm is believed to result in inconsistent pressure sensitivity at different temperatures. All the measured responses coincide and indicate that the leading cause of the sensitivity drift is the diaphragm’s thermal expansion effect by comparing a vacuum-treated cavity and multi-curvature radius diaphragm shapes. Experimental results have shown that the pressure sensitivity decreases by 0.46nm/(kPa·°C) with increasing temperature in the designed temperature range. The use of materials with a low coefficient of thermal expansion is verified to be an effective strategy for limiting the drift of polymer-diaphragm-based pressure sensor sensitivity with temperature.
Funding
National Natural Science Foundation of China (61935002).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this paper may be available from the corresponding author upon reasonable request.
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