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Abstract
Fiber optic sensors are increasingly used in several fast-growing industries. Aerospace, energy storage, and the medical sector consider new implementations of optical fibers mainly for condition monitoring purposes. Applications using optical fibers entail measurements of distributed strains and temperatures. However, the spectral shifts of transmitted and reflected light are simultaneously sensitive to both of these influences. This coupled sensitivity can introduce large errors for signal interpretation. An accurate calculation model for signal decoupling is necessary to distinguish pure mechanical strains from pure thermal loading. Approaches where the spectral shift is assumed to vary linearly with temperature give large errors when the temperature variation is high. This investigation derives and validates a new temperature formula that is used for high precision strain and temperature discrimination. The non-linear temperature formula is deduced from physics-based models and is validated with Rayleigh backscattering based OBR measurements. Our calculation approach demonstrates improved accuracy over an extended temperature range. The relationship between strain and temperature effects in the coupled mechanical and thermal loading environment is further studied in detail.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
More than 60 different optical fiber sensor types have been developed as future sensors in previous decades [1,2]. Compared to conventional electrical sensors, optical fibers have several distinguishing advantages: (i) immunity to electromagnetic interference and environmental corrosion; (ii) small size and insignificant disturbance for the integrity of the structure; (iii) high temperature resistance; (iv) a lifetime exceeding 25 years [3–5]. In addition, optical fibers can multiplex a large number of sensors along a single fiber and set up a distributed optical fiber sensors (DOFS) network. By measuring the intrinsic backscattering variation in the fiber caused by external perturbations, the DOFS techniques based on multi fiber Bragg grating (FBG), Raman, Brillouin, and Rayleigh backscattering have been developed successfully. Nowadays, the DOFS are used for structural health monitoring in industrial infrastructure, in aerospace components, in architectural structures, and in human health applications.
Today, one of the most significant limitations for using the DOFS is their coupled sensitivity to both temperature and strain. For instance, temperature variations along the sensing fiber will introduce errors for the strain measurement. The ability to distinguish between strain and temperature effects is critical for the large-scale success of DOFS applications. Rayleigh backscattering spectrum has been adopted in DOFS to determine strain and temperature along the entire length of an optical fiber. Modeled as a weak FBG with a random period, changes in the refractive index or in the physical optical fiber length cause the Rayleigh backscattering spectral (RBS) shift in frequency. The RBS shift shows a cross-sensitivity to both strain and temperature similar to FBGs. In the field of DOFS techniques based on FBGs, different approaches have been proposed to discriminate strain and temperature [6–9]. These methods utilize well-designed optical fiber sensor systems which have different temperature and strain sensitivities. Once the temperature and strain coefficients of the optical fiber are known, the temperature and strain variation can be determined by using the inverse of the coefficients’ matrix. Most recent strain/temperature discrimination methods in Rayleigh backscattering based DOFS are analogous to the decoupling techniques of FBGs [10–14]. One solution utilizes the specral shifts of the fast and slow modes of birefringence by autocorrelation and by cross-correlation of RBS shifts in the polarization maintaining fiber (PMF) [10,11]. The distributed autocorrelations of the Rayleigh spectral signature are strongly related to thermal effects on the fiber, while distributed cross-correlations of the Rayleigh spectral signature are related to both thermal and strain effects. Another method uses stimulated Brillouin scattering and Rayleigh backscattering in a single-mode fiber (SMF) to discriminate between the temperature and strain changes [12,13]. The Brillouin frequency shift and the distributed RBS shift were induced by strain and temperature variations. Although both methods are sensitive to temperature and strain, their sensitivities are different and independent from each other; therefore, a complete temperature and strain discrimination can be realized by measuring a pair of spectral shifts along the PMF sensor. Although the methods above conveniently utilize a single fiber for dual-measurements of strain and temperature, they are also complicated and costly. Measuring Brillouin scattering and Rayleigh backscattering in a PMF requires two systems of measurement techniques. PMF is also more expensive compared to single-mode optical fibers for large applications. A simple way to achieve discrimination utilizes a SMF and a reduced-cladding SMF [14], because the temperature and strain coefficients of the two optical fibers are different. However, two types of SMFs must be attached closely side by side, and two interferometers are needed for measurements when using this system.
All previous strain/temperature discrimination methods for DOFS based on Rayleigh backscattering (developing from the techniques of FBGs) assume that strain- and thermally-induced effects are both linear with respect to RBS shifts. The superposition of strain and temperature induced spectral changes has also been found to apply well for FGBs. Rayleigh scattering-based DOFS have been successfully used to perform temperature measurements under thermal conditions [15,16], and coupled with irradiation [17], or humidity [18]. In these investigations, RBS shifts are modeled using linear relations to temperature with good accuracy within temperature ranges from ca. 0 °C to ca. 75 °C. However, obvious measurement errors occur for linear fitting at high temperatures (> 80 °C) and at cryogenic low temperatures (< -73 °C). Additionally, the temperature induced RBS shifts are shown to be humidity and low dose radiation independent. The review of existing literature for Rayleigh scattering-based DOFS shows clearly that superposition/decoupling of strain and temperature induced spectral changes has not been addressed with sufficient rigor thus far, especially for thermal conditions beyond typical inside and outside temperatures.
In this paper, a simple and effective calculation model is realized for strain and temperature discrimination by using the OBR, a high resolution DOFS technology based on Rayleigh backscattering. Identical physically separated optical fibers are used to compensate for temperature effects. During the procedure, the measurement fiber experiences a coupled mechanical and thermal loading while the compensation fiber only the thermal loading. The temperature effect for the OBR is shown to be nonlinear due to the thermo-optic coefficient and its relationship with temperature. Improving state of the art, a polynomial temperature calculation formula is deduced from a physics-based model. High precision temperature measurements over an extended temperature range are carried out by using a customized test set-up where OBR measurements are taken inside the temperature chamber of a DMTA (dynamic mechanical thermal analysis) test instrument. The relationship between strain and temperature effects is studied in the coupled mechanical and thermal loading environment. A novel high-precision calculation model is applied to subtract temperature effects from the coupled measurement data.
2. Experimental setup
2.1 Interrogator devices and sensors
As shown in Fig. 1(a), a reflectometer OBR 4600 from Luna Instruments is used as the interrogator device for the DOFS measurements. The optical fiber SMB-E1550H from OFS Fitel with a silica/silica/polyimide structure and a core diameter 6.5 µm, a cladding diameter 125 µm, and a coating of 155 µm diameter is used for sensing strain and temperature. The independent temperature measuring system in the experiment consists of an electronic temperature sensor with accompanying data acquisition equipment. The temperature sensor is a special K-type wire thermocouple SRTC-TT-KI-40-1M and the data is read and stored by a DP9800-TC digital thermometer from OMEGA Engineering.
2.2 Mechanical and thermal loading set-up
A customized experimental set-up is used to investigate the combined loading effect from strain and temperature on the OBR RBS shift measurements. The set-up assembly (Fig. 2) consists of a temperature chamber, a clamper, a perforated tube, and a dead weight system. The temperature chamber from a NETZSCH Eplexor DMTA machine is used to control the temperature around the optical fibers. The chamber has capacity from -150 °C to 500 °C. However, a smaller range from -130 °C to 240 °C is used for this investigation. The clamper is attached to the ceiling of the chamber to fix the top of the optical fiber. The optical fiber then passes through a perforated tube which protects it from vibrations induced by the cooling fan. The mechanical loading on the fibers is created by a system of dead weights. One thermocouple is plugged into the perforated tube to provide accurate and independent temperature information about the optical fiber measurement region. By these small modifications to the DMTA temperature chamber, a controlled thermal and mechanical loading can be applied to individual fibers, or groups of optical fibers simultaneously.
2.3 Measurement procedure
Before each OBR measurement, the DMTA chamber temperature is maintained for two minutes to fully heat or cool down the optical fiber sensor and the measurement system around it. During the testing procedure the optical fibers are heated from room temperature to the highest temperature first and then cooled down to the lowest temperature of the range. The signals from the optical fibers and the K-type sensors are recorded simultaneously over ca. 5 seconds period at each temperature measurement. The average RBS shift over the measurement zone of the optical fiber is recorded, while the mean temperature of the K-type sensor is used as an independent temperature measurement. The K-type sensor data is used for the temperature values on the x-axes of figures shown in the following Sections. The instrumentation parameters for OBR 4600 and DP9800-TC digital thermometer are shown in Table 1.
Table 1. Equipment parameters.
3. Results and discussion
3.1 Thermal loading effects
In this Section, a polynomial formula is first deduced from the underlying physics. Thereafter it is applied for precise temperature measurements with the OBR. The non-linear formula is compared to the usual linear formula under pure thermal loading conditions over four temperature ranges. It has to be pointed out that FBGs are outside of the experimental scope of current investigation. However, the analyses approach described below is also applicable to these optical fiber sensors.
3.1.1 Polynomial temperature formula
As described by Wang [19], the distributed Rayleigh backscattering along the length of the optical fiber can be modeled as a weak FBG with a random period. Thus the RBS shifts induced by the temperature change are identical to FBGs, expressed by [20]:
Thereby, KT also becomes expressed as a quadratic function of temperature:
Finally, inserting Eq. (10) into Eq. (3), the latter can be rewritten and rearranged as:3.1.2 Comparison between linear and polynomial formulas
A reference OBR test was implemented at Tr = 30.6 °C, and then a series of measurements were performed over a temperature range from –130 °C to 200 °C. Five independent optical fibers (A, B, C, D and E) were positioned simultaneously in the temperature chamber (see Fig. 2) without any mechanical loading. The measured RBS shift was fitted by the usual linear model and the new non-linear model separately to compare the difference between the two approaches. The measured data and the parameters of linear and non-linear curve fits are shown in Fig. 4 and in Table 2, respectively. The linear model is described by $\mathrm{\Delta }{v_T} = a + b{T_m}$, where${\; \Delta }{v_T}$ is the RBS shift induced by thermal loading, Tm is the temperature of the measurement scan, b = $- \frac{c}{{\bar{\lambda }}}{K_T}$, and a has the value of $\mathrm{\Delta }{v_T}$ when Tm = 0 °C. The non-linear model is expressed by Eq. (11), where Tr = 30.6 °C (reference temperature). By using the linear curve fitting formula, the value of b = -1.225 GHz/°C was obtained, similar to $- \frac{c}{{\bar{\lambda }}}{K_T}$ = -1.248 GHz/°C predicted previously in Section 3.1.1. The results in [15–18,27] show that linear fitting is reasonable within a small temperature range, however the discrepancies of using a linear formula become evident with temperatures far away from the reference temperature. The measurement data and the fitting curves of four different ranges symmetric to room temperature (see Fig. 4) are shown in Fig. 5 for visual comparison. In the smallest Range 1 (from -5 °C to 45 °C), both the linear and polynomial fitting curves agree with the measurement data, almost overlapping each other. With the expansion of the temperature range, the linear and polynomial fitting curves start to diverge from each other as shown for Range 2 (from -25 °C to 65 °C) and for Range 3 (from -85 °C to 125 °C). In the widest Range 4 (from -125 °C to 165 °C) only the non-linear curve is still acceptable for representing the measurement data.
Fig. 5. The RBS shifts from the OBR and two alternative fitting curves when expanding the temperature range.
Table 2. Parameters obtained for linear and non-linear curve fitting models.
The adjusted R-squared metric is a modified version of the R-squared that is often used to evaluate the goodness-of-fit. High adjusted R-squared ≤1 indicates a good fit for the model. The adjusted R-squared is calculated for different temperature ranges and the results are shown in Fig. 6(a). The linear and non-linear fitting curves have very similar R2 values for Range 1. When the temperature range becomes wider, as expected, the polynomial formula shows a greatly improved goodness-of-fit compared to the linear formula. Figure 6(b) dispalys the relative errors for the OBR temperature measurements (%error) when using the polynomial and the linear formulas:
where TK is the temperature measured by the K-type thermocouple (Fig. 2) and TOBR is the temperature measured by the OBR using the optical fiber. When the temperature remains within Range 1, there is little difference between the errors of polynomial and linear formulas (mostly < 5%). When the temperature exceeds Range 1, the polynomial formula shows a better temperature measurement precision over the whole measured temperature range. This experiment clearly demonstrates how the polynomial formula deduced from physics-based models improves the temperature measurement accuracy over an extended temperature range.Fig. 6. The goodness of linear and polynomial fitting formulas with increasing temperature range: (a) Adjusted R-squared metric, (b) Relative error in the temperature measurement.
3.2 Mechanical loading effects
In the absence of temperature change, the mechanical strain relationship to the RBS shift can be written as [19]:
where Kɛ is the strain coefficient and c is the speed of light. Previous research in our group has shown that Kɛ is a constant with a value of 0.787 [28]. Thus, Eq. (13) yields the conversion factor: ɛ = (-6.67 µɛ /GHz) ΔνM for this OBR system.3.3 Decoupling of mechanical and thermal loading effects
In this Section, the optical fibers are exposed to loading environments where a constant mechanical loading is coupled with varying thermal loads, and a constant thermal loading is coupled with varying mechanical loads. The relationship between strain and temperature effects for the RBS shift is experimentally investigated. Both, the linear and the non-linear (polynomial) formulas are applied for decoupling the mechanical and thermal loading effects. The relative errors of the two methods are compared.
3.3.1 Coupled mechanical and thermal loading
The optical fiber (OF1) was measured before and after it was loaded by a dead weight (60.23 g) at room temperature. This measurement gives the pure mechanical loading induced RBS shift (ΔvPM). The mean value of ΔvPM over the measurement zone (Fig. 2) was calculated -95 GHz, shown by a green dashed line in Fig. 7(a). Subsequently, four additional independent optical fibers (OF2, OF3, OF4 and OF5) were placed into the temperature chamber. The measurement temperature of the chamber was varied from –125 °C to 240 °C. The RBS shift of OF1 (ΔvT&M) was induced by coupled mechanical and thermal loading, while the other four fibers sensed only the same thermal loading without any mechanical loading. The ΔvT is the average RBS shift of four optical fibers over their measurement zones. The ΔvT&M from OF1 shows the same nonlinear trend as ΔvT but at values uniformly shifted below from ΔvT. The mechanical loading induced RBS shift ΔvM (blue dotted line) is achieved by subtracting ΔvT from ΔvT&M. As evident from Fig. 7(a), the ΔvM shows almost no variation with temperature and is in excellent agreement with ΔvPM.
Fig. 7. The RBS shifts of optical fibers under coupled mechanical and thermal loading conditions, measured by the OBR.
Finally, another optical fiber (OF6) was subjected to a changing mechanical load by using different dead weights at 20.6 °C and at 30.1 °C separately. Figure 7(b) shows the ΔvT&M of OF6 at these two different temperatures while increasing the dead weight. The temperature difference induced ΔvT is calculated by subtracting ΔvT&M at 20.6 °C from ΔvT&M at 30.1 °C at the same weight. Figure 7(b) shows that, at constant temperature, the mechanical loading affects the RBS shifts in the OBR linearly with applied dead weight (i.e., mechanical strain). The constant small difference between ΔvT&M (empty and solid circles) comes from the small temperature difference. The ΔvT remains mechanical loading independent. These results show that mechanical and thermally-induced effects can be considered independent from each other in the OBR measurements. Thus, the strain and the temperature effects can be added/subtracted by superposing the RBS shifts under coupled loading conditions.
3.3.2 Application of the polynomial formula
Mechanical loading is typically of main concern in the analysis of structures. That means the strain measurements are usually considered as the primary outputs from optical fiber sensors in engineering applications. The environmental temperatures can be obtained from separate thermal sensors and the influence of temperature is then subtracted from the overall loading effect. The linear and polynomial formulas are hereby applied to obtain pure mechanical strains from the coupled loading on the optical fiber. A new optical fiber (OF7) is loaded by using a dead weight of 29.32 g. The temperature data is obtained from an independent K-type thermal sensor. The pure ΔvM is decoupled by the same subtraction of OBR measurements as seen in Fig. 7(a) and as described in Section 3.3.1. Figure 8 shows the ΔvT&M of OF7 while changing the surrounding temperature from –100 °C to 240 °C. The pure mechanical loading induced RBS shift ΔvRM of OF7 was measured by the OBR at reference temperature (31.4 °C) before and after loading by the dead weight. The ΔvM-P is the mechanical loading induced RBS shift when subtracting the ΔvT calculated by the polynomial formula (Table 2) from ΔvT&M, while the ΔvM-L uses the linear formula (Table 2). As shown in Fig. 9(a), both the values of ΔvM-P and ΔvM-L are in very good agreement with the ΔvRM from 0 °C to 50 °C. The difference between the ΔvM-P and ΔvM-L increases rapidly when the temperature is higher than 60 °C or lower than 0 °C.
Fig. 9. A comparison between linear and polynomial models for an engineering application: (a) decoupled mechanical loading RBS shifts, (b) relative errors of RBS shift measurements.
The relative error ($\%error$) between linear and polynomial formulas is also compared in Fig. 9(b):
where ΔvM can be ΔvM-P or ΔvM-L based on the formula applied. When the temperature shift is small (e.g., as in Range 1 in Figs. 4–6), there is little practical difference between the measuring errors of the polynomial and the linear formulas (< 3%). When the temperatures exceed the range from ca. -25 °C to 75 °C, the linear model becomes poorly suited for decoupling mechanical and thermal loading effects in the OBR measurement. The polynomial model works well over the whole temperature range. Thus, the polynomial formula is a more accurate model to account for thermal loading effects. It enables to decouple RBS shifts with a higher precision over a wider range of temperatures.4. Conclusions
- 1. Thermal loading induced RBS shifts of the OBR measurements are proven to be nonlinear with temperature due to the relationship between the thermo-optic coefficient and temperature.
- 2. A polynomial formula is deduced from physics-based models for accurate temperature measurements with the OBR. The polynomial formula shows a higher temperature measurement precision than a linear formula over an extended temperature range.
- 3. Pure mechanical loading induced RBS shifts are shown to be linear with strain. Under coupled OBR measurement conditions, the mechanical and thermal loading effects are shown to be independent from each other. Pure mechanical strains can be obtained by subtracting the temperature effect (e.g. modelled by the polynomial formula) from the coupled RBS shift.
Funding
European Union’s Horizon 2020 research and innovation programme and Hydrogen Europe and Hydrogen Europe Research (Grant ID: 826262).
Acknowledgments
This project has received funding from the Fuel Cells and Hydrogen 2 Joint Undertaking under grant agreement No. 826262, https://thor-fch2.eu/.
Disclosures
The authors declare no conflicts of interest.
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