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By both simulation and experiment, we studied the relationship of the measurement accuracy and the birefringence of the distributed simultaneous strain and temperature sensor using polarization-maintaining fiber Bragg gratings (PANDA-FBGs). The PANDA-FBGs were applied to an optical frequency domain reflectometry (OFDR) which is capable of distributed measurement at high spatial resolution and sampling rate. The simulated results had agreement with the experimental results that the measurement accuracy of both strain and temperature were improved by increasing the birefringence. Additionally, the efficiency of the accuracy improvements decreased when accuracy increased.
© 2017 Optical Society of America
1. Introduction
Recently, optical fiber sensors (OFSs) for strain measurement have been widely used in many fields including structural health monitoring (SHM) [1–3]. They offer the capability of distributed sensing, and can be conveniently embedded in structures with minimum perturbation to the structural strength. Among OFSs for SHM, fiber Bragg grating (FBG) sensors have attracted many attentions [4–6]. FBG can give absolute measurement of variation of environment (strain, temperature, etc.) by detecting the wavelength shift caused by them [7, 8]. Although, FBG sensors were essentially used for point sensing, in 2008, Igawa et al demonstrated distributed strain measurement using a long length FBG [9]. It has proven the capability of FBG sensors for full distributed measurement. Furthermore, FBG sensors have sufficient accuracy at high spatial resolution of millimeter or sub millimeter order, which is important for SHM [10–12].
However, for strain measurement, large temperature variations are common in practical applications. The measurement accuracy will be reduced by the cross-sensitivity of temperature and strain in the sensing head. Thus, a compensation of the influence caused by temperature variation is required. Conventionally, in addition to the strain sensor, which is sensitive to both strain and temperature, a reference FBG in stress-free condition is used as temperature sensor [13]. Since the reference FBG is parallel located in the same environment as the strain sensor, the error of strain induced by temperature variation can be compensated by subtracting the wavelength shift of temperature sensor from the wavelength shift of the strain sensor. However, it is complicated to keep the reference FBG free-stress. And the reference fiber will degrade the strength of host structure for embedded sensing. What’s worse, for distributed sensing, it is very difficult to avoid the misplacement between the reference FBG and strain sensing FBG that will introduce extra sensing errors, especially when high spatial resolution is required. Hence, techniques of simultaneous strain and temperature with one single FBG are under development. Echevarria et al discriminated strain and temperature by measuring the first and second order diffraction wavelength of an FBG [14]. Sudo et al inscribed an FBG into a polarization maintaining (PM) fiber and realized simultaneous point sensing of strain and temperature [15]. In 2011, Wada et al applied a PANDA-FBG to an optical frequency domain reflectometry (OFDR), and succeeded in simultaneous distributed measurement of strain and temperature [16], which was an important advance for SHM.
In this paper, we studied the properties of PANDA-FBGs for simultaneous strain and temperature distribution measurement, and established a numerical model for the estimation of measurement accuracy. Meanwhile, we applied multi-types of PANDA-FBGs to the measurement and evaluated their performance practically with specific equipment. According to the results, the simulation and experiments had agreement in the relationship between the measurement accuracy and the birefringence of PANDA fibers. This work might help in the design of this type of FBG sensors. Once the accuracy required for a specific application is known, the model helps to know which should be the value of birefringence for achieving the desired result.
2. Principle of simultaneous strain and temperature measurement using PANDA-FBG
PANDA fiber is a type of polarization maintaining optical fiber, where the propagating mode in it will split into two orthogonal polarized mode (fast and slow modes) due to the property of birefringence [17]. The strength of birefringence is defined as
where ns and nf are the refractive indices for fast and slow modes respectively [18]. The PANDA-FBG indicates the FBG inscribed on the PANDA fiber, as shown in Fig. 1. Due to the birefringence, it has two distinct Bragg wavelength which can be expressed as where λs,f are the Bragg wavelength of slow and fast modes, ns,f is the effective refractive index of slow or fast mode, Λ is the period of the FBG, respectively [19].Since each Bragg wavelength has different response to external variation of strain and temperature, we can determine the strain and temperature changes at the same time by measuring the two Bragg wavelength [15, 20]. Their relationship is expressed as
where Δλf and Δλs are the wavelength shifts of fast and slow modes, Δε and ΔT are the strain and temperature variations respectively. K represents the coefficient of strain or temperature sensitivities, and the subscripts f, s, ε and T correspond to the fast mode, slow mode, strain and temperature respectively. Once the matrix K is determined, the strain and temperature variation can be yielded from the two wavelength shifts, which is expressed as3. Numerical model of simultaneous strain and temperature measurement using PANDA-FBGs
In order to investigate the influence of birefringence in the measurement accuracy of strain and temperature we made a numerical model and simulated the simultaneous measurement, as shown in Fig. 2. The simulation consists the calculation of matrix K, the simulation of calibration, the simulation of simultaneous measurement, and the estimation of measurement accuracy.
Fig. 2 Simulation to estimate the accuracy of simultaneous strain and temperature distribution measurement.
Theoretically, the strain response arises because of both the physical elongation of the sensor, and the change in effective refractive index, neff, due to photo-elastic effects. Meanwhile, the temperature response arises due to the inherent thermal expansion of the optical fiber material and the temperature dependence of neff [18]. The shift in Bragg wavelength with strain and temperature for single mode FBG can be expressed using
where P11, P12 are the Pockel’s coefficients, ν is the Poisson’s ratio, and α is the thermal expansion coefficient. dneff/dT is the rate of refractive index changes during the annealing process in the fabrication of the fiber [4].For PANDA-FBGs, we assumed that the strain and temperature response of fast and slow mode are independent. Thus, in Fig. 2: Step 1, the elements of matrix K were calculated as
where the subscripts s and f represent the slow and fast mode respectively. The relationship of birefringence and refractive index changes is described by where Tanl is the annealing temperature in the fabrication.Meanwhile, as shown in Fig. 3, the PANDA fiber consists of core, cladding and stress applying parts (SAP). Thus, the total thermal expansion coefficient can be calculated as
where rcl = 62.5 µm, rsap = 21 µm and rco = 6 µm are the radius of cladding, SAP parts and core, α(co,cl) and αsap are the thermal expansion coefficients of core, cladding and SAP.The values of the parameters in the numerical model were based on FUJIKURA Co., Ltd. PANDA fiber (SRSM15-PS-Y15), as given in Table 1. In this study, we assumed that the refractive index is linear related to temperature in the annealing process. Thus, dneff/dT were considered as constant.
Table 1. The values of the parameters in the numerical model.
In the model of calibration, Fig. 2: Step 2, we set the variations of strain and temperature as
And the corresponding wavelength shift were calculated asThen, we introduced random errors to strain, temperature and Bragg wavelength shift, which are expressed as
where Eε, ET and Eλ are the errors of strain, temperature and Bragg wavelength shift respectively. The values are shown in Table 2. They are based on the experimental equipment in this research which will be introduced in Section 4.Table 2. The random errors of the parameters in the numerical model.
Then we conducted linear fitting in order to obtain the error introduced matrix K which is denoted by
where the subscript j means the jth calculation. In total, the simulation of calibration was repeated for 2000 times.In Fig. 2: Step 3, we simulated the process of simultaneous measurement. Firstly, we input the strain (Δεi) and temperature (ΔTi) variation into Eq. (3), and calculated the corresponding wavelength shifts as
where the subscript i means the ith calculation.Then, we introduced random errors to the wavelength shifts and recovered the output strain and temperature changes by the error introduced matrix (Eq. (19)), , which is expressed as
Finally, in Fig. 2: Step 4, the estimated accuracy of strain (±aε) and temperature (±aT) were calculated as
where N = 1000 is the number of calculation for each .In this simulation, we input Case1: Δεi = 100 με, ΔTi = 95°C, Case2: Δεi =500 με, ΔTi = 130°C, and the variation from Case 1 to Case 2: Δεi = 400 με, ΔTi = 35°C, respectively. Then, we increased birefringence from 3.0 × 10−4 to 3.2 × 10−3 by the interval of 1 ×10−4 and obtained the simulated relationship of measurement accuracy (absolute value) and birefringence, as shown in Fig. 4. The simulation showed that the measurement accuracy was improved by increasing the birefringence of PANDA fiber.
Fig. 4 Relationship of simulated measurement accuracy and birefringence. With larger birefringence, the accuracy of both strain and temperature became better. Worse accuracy was obtained when larger Δεi and ΔTi were input. The error bars show the errors caused by the matrix K.
Additionally, the influence to the accuracy of other independent parameters were investigated by the numerical model. Taking FUJIKURA Co., Ltd. PANDA fiber SRSM15-PS-Y15 (B = 5.2 × 10−4) as reference, we assumed that a FBG of which the period is 531 nm was inscribed with phase mask and UV beam. Then we introduced the perturbation from −20% to 20% to birefringence (B), period (Λ), and thermal expansion coefficient (α), respectively, and calculated the corresponding changes of the estimated accuracy. According to the results, the increase of birefringence and period can make efficient improvements of the accuracy, as shown in Fig. 5. Considering the accuracy and attenuation, Λ = 531 nm of which the Bragg wavelength (~ 1550 nm) is in the C band (1530 ~ 1565 nm) was chosen. Within C band, the maximum period of ~ 536 nm can only bring the improvement of ~ 0.7 µε and ~ 0.1°C, compared with the reference fiber. However, it is available to increase the birefringence to over 10 × 10−4 with current technology [21]. As estimated in Fig. 4, the potential accuracy improvements by increasing birefringence are more than ~ 22 µε and 2.4°C, more efficient than increasing the period. In this study we examined the dependence of measurement accuracy on the birefringence in experiments.
Fig. 5 The comparison between the influence of multi-parameters to the measurement accuracy. the positive values at Y axis mean increase of the error while negative values mean decrease of the error.
4. Experiment
In order to evaluate the measurement accuracy of PANDA-FGBs in simultaneous strain and temperature measurement practically, we conducted a series of experiments using various PANDA-FBGs. All the FBGs were tested under similar external environment.
4.1. Experimental arrangement
In the experiment, we applied quantifiable strain and temperature variation to the fiber under testing (FUT). The principle of the experimental equipment is to heat the FUT at sensing part and stretch it with special tools. At the same time, we monitor the applied strain and temperature simultaneously by sensors other than OFS.
In the research, we employed a cylinder heater to apply temperature variation to FUT and applied strain by a stationary stage and a translation stage, as shown in Fig. 6.
The stages were fixed on the optical table. During the experiments, FUT was located on the mid axis and fixed by adhesive. The length of FUT, LFUT, was 1m. The displacement of translation stage, Ds, was measured by the laser displacement sensor. Thus the applied strain can be described as
The cylinder heater was located between the stages and fixed on the optical table as well. It consists of two semi-cylinder parts (Part A, Part B), and can be open and closed from the side, as shown in Fig. 7. In the heater, we installed 9 channels of thermocouples. Among them, Ch-0 was used for PID temperature control, the other 8 channels from Ch-I to Ch-VIII were used for monitoring applied temperature distribution. The position of Ch-I was set to be zero point. And the distance between adjacent channels were set to be 12 mm.
As shown in Fig. 8, the FUT was fixed to the stages, and the middle part of the PANDA-FBGs were inside of the heater, while the rest of them were outside of it. z represents the position. The area between Ch-II to Ch-VII was the Application Area where the characteristics and performance of PANDA-FBGs were tested. The specification of the equipment are shown in Table 3.
Table 3. The specification of the experimental equipment.
In the experiment, the uniform strain and temperature distribution were applied to the FUT in the Application Area of which the length is 60 mm. When the errors during the experiment are introduced, from Eq. (4) we can obtain
where δε is the error of strain, δT is the error of temperature, δKinv is the error of inversed matrix K, δλf is the error of wavelength shift of fast mode and δλs is the error of wavelength shift of slow mode. Substituting Eq. (3) and Eq. (4) into Eq. (26), we can obtainFrom the second term of Eq. (27) at right hand side, we know that the measurement accuracy are related to the applied strain and temperature variation, and will be amplified by the matrix K. Thus, in order to obtain a general accuracy of distributed measurement and compare between different types of PANDA-FBG sensors, uniform strain and temperature should be applied. Otherwise, for each sensing point, the accuracy will be different.
In the experiments, the distributed wavelength shift were determined with the optical frequency domain reflectometry (OFDR) [9, 11, 16], as shown in Fig. 9. It is a combination of two interferometers. All the optical fibers and couplers were polarization maintaining. The clock signals observed by Detector 1 was used to trigger the sampling of Detector 2. And, the observed signals at Detector 2 were demodulated by short time Fourier transform (STFT). Since a polarization splitter was employed between Coupler 3 and Detector 2, the system was able to determine the wavelength of fast and slow mode individually. In the experiments, the settings of the OFDR are shown in Table 4. The sampling rate of OFDR was set to be 800 sps, the spatial resolution was about 5 mm, and the maximum measurement length was 1 m. Figure 10 illustrates the spectrogram of the full length of a PANDA-FBG under applied strain and temperature.
Table 4. The settings of OFDR system
Fig. 10 The spectrogram of a 300 mm PANDA-FBG under applied strain and temperature. In this case, uniform strain (~ 500 µε) and non-uniform temperature (inside: ~ 150°C; outside: room temperature; gradual change of temperature through the heater wall) distribution were applied. The Y-axis represents the relative positions (Li) to the Reflector 3 in Fig. 9.
4.2. Calibration of strain and temperature sensitivities of PANDA-FBGs
In this research, four types of PANDA-FBG were calibrated before being applied to simultaneous strain and temperature measurement. Since different amount of B2O3 were doped in the SAPs of the PANDA fibers, they have different birefringence [21]. The FBGs were inscribed with the same phase mask that the Bragg wavelength of slow mode were approximately the same. The dimensions of PANDA-FBGs are given in Table 5.
Table 5. The dimensions of different types of PANDA-FBGs.
The PANDA-FBGs were calibrated under the similar environments by the same experimental equipment. In the calibration of strain sensitivities, we applied the strain variation to the FUT from ~ 50 µε to ~ 550 µε by the interval of ~ 50 µε at the constant temperature of 99.6°C (average value by thermocouple Ch-IV and Ch-V), as shown in Fig. 11(a). Meanwhile, the wavelength shifts were monitored by OFDR. For each strain, the corresponding wavelength shifts were averaged from 100 samplings at 800 sps of 13 positions between Ch-IV (36 mm) and Ch-V (48 mm) with the interval of 1 mm.
Figure 12 illustrates the strain response of PANDA-FBGs. As strain increases, the Bragg wavelength shift to longer wavelength. The strain sensitivities, as shown in Table 6, are indicated by the slopes from linear fitting of measured data. The results show good linear relationship between applied strain and Bragg wavelength during the experiment.
Table 6. Strain sensitivities of different types of PANDA-FBG
In the calibration of temperature sensitivities, we kept the FUT under stress-free condition, and changed the temperature at midpoint (average value by thermocouple Ch-IV and Ch-VI) from ~ 100 °C to ~ 160 °C by the interval of ~ 20°C. The temperature condition generated stable temperature distribution, as shown in Table 3, at the same time, reduced the influence of nonlinear sensitivity [22]. The monitored temperature and strain are shown in Fig. 11(b).
The temperature response of PANDA-FBGs are shown in Fig. 13. The temperature sensitivities are also indicated by the slopes from linear fitting of measured data, as given in Table 7. The result shows good linear relationship between applied temperature and Bragg wavelength during the experiment.
Table 7. Temperature sensitivities of different types of PANDA-FBG.
4.3. Simultaneous measurement of strain and temperature
We respectively applied PANDA-A, B, C, D to the OFDR system and conducted experiments. In the experiment, we applied the strain and temperature variation simultaneous, as shown in Table 8. The relative strain and temperature from Case 1 to Case 2 (denoted by Case 2 - Case 1) were also examined. Meanwhile, the measured strain and temperature variation were recovered from the wavelength change by
where Δλfz, Δλsz are the measured wavelength shift of fast and slow modes at the position of z by OFDR. Δεmz, ΔTmz represent the measured strain and temperature at position z. And represent the inversed matrix K of PANDA-A, B,C,D respectively. The value of matrix K were obtained from the calibration, as given in Tables 6 and 7. In this research, the measured data points were readout by the interval of 1 mm within Application Area (12 mm ~ 72 mm, as shown in Fig. 8). For each measurement, we obtained the wavelength distribution for 2 s at the sampling rate of 800 sps. The measurement results for 1 sampling of PANDA-B and PANDA-D are shown in Fig. 14.Table 8. The applied variation of strain (Δεa) and temperature (ΔTa) in Application Area measured by experimental equipment in simultaneous measurement.
Fig. 14 Measured strain and temperature distribution along 300 mm PANDA-FBGs. The datas measured by PANDA-B and PANDA-D from 1 sampling at 800 sps are shown. In the heater wall, the gradual change of temperature can be observed. Close to the edge of the heater walls, larger variation of measured values may caused by the convection of air flow.
In this research, we assumed that the applied strain were uniform along the FUT. Meanwhile, we assumed that the temperature distribution between two adjacent channels were linear. Then, we used root mean square deviation (RMSD) at positions to represent the accuracy of simultaneous distributed strain and temperature measurement in Application Area, which are expressed as
where Δεmz, ΔTmz, Δεaz and ΔTaz represent the measured strain, measured temperature, applied strain and applied temperature variations at position z respectively. The accuracy were calculated from 1600 samplings for each type of PANDA-FBGs. In the experiment, the PANDA-FBGs have different birefringence while the other independent parameters were made to be the same during the manufacturing. Thus the relationship of measurement accuracy (absolute value) and birefringence in the experiments can be obtained, as shown in Fig. 15. According to the results, the accuracy of both strain and temperature measurement were improved while the birefringence was increased.Fig. 15 Relationship of experimental measurement accuracy and birefringence. In both cases, the accuracy (indicated by RMSD) became smaller while the birefringence was increased. However, worse accuracy was obtained when larger Δεa and ΔTa was applied.
4.4. Results and discussion
We plotted the simulated and experimental results in the same figure, as shown in Fig. 16. According to the figure, the simulation has agreement with experiment that higher birefringence leads to better measurement accuracy. Among the testing, in Case 2 - Case 1, under small temperature span, the measurement accuracy had the best agreement with the simulation. Relative bad accuracy in Case 2 might be caused by the amplification effect by the second term of Eq. (27) at right hand side. Additionally, for the large temperature span in Case 2 (~ 130°C), errors from the nonlinear temperature sensitivities could also lead to some disagreement between the experimental results and simulation based on linear approximation. Also, with higher temperature in Case 2, larger errors might be introduced by the instability of air flow. Although in this study, the influence of nonlinear sensitivities is limited, it is worth noting that when the measurement accuracy becomes higher, the ratio of nonlinear error to the total error will gradually become significant. Thus, high order coefficients should be taken into account in high accuracy measurement, and the linear K matrix method should be replaced by nonlinear analysis.
Fig. 16 Dependence of measurement accuracy on birefringence: simulation and experiment. In Case 2, the experimental accuracy larger than the simulated accuracy.
Additionally, we described the efficiency of improving accuracy by increasing the birefringence as
where ηε and ηT represent the efficiency of strain and temperature measurement accuracy improvement, Δαε and ΔαT are the changes of strain and temperature measurement accuracy, ΔB was the corresponding change of birefringence, respectively. As shown in Fig. 17, while the birefringence increases, the efficiency of the improvement for both strain and temperature decrease. Additionally, the efficiency are much lower at high birefringence than low birefringence. For example, the efficiency at B = 1.4 × 10−3 is about 1/10 of the efficiency at B = 4 × 10−4.In the experiment, PANDA-D achieved the best accuracy at the spatial resolution of 5 mm and the sampling rate of 800 sps in measuring the variation from Case 1 to Case 2 (Case 2 - Case 1: ±16.7 µε and ±1.9°C).
5. Conclusion and future works
In this work, we have built up a numerical model to estimate the accuracy of simultaneous distributed strain and temperature measurement PANDA-FBG sensors and study its dependence on the birefringence. The simulated results had agreement with the experimental results that the measurement accuracy of both strain and temperature were improved by increasing the birefringence. Meanwhile, we have found that the efficiency of the improvement were reduced when the birefringence were increased. Further more, the numerical model has been validated by the experiments. According to the results, once the accuracy required for an specific application is known, the model helps to know the birefringence for achieving the desired result.
In the future, we will investigate the influences of other parameters to the measurement accuracy by simulation and validate them by experiments. Additionally, under large temperature span, nonlinear analysis are expected to bring better measurement accuracy instead of linear K matrix method.
Acknowledgments
We would like to thank Mr. Makoto KANAI for his support in the preparation of the experiments.
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