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Abstract
A few-mode fiber Bragg grating (FM-FBG) inscribed in a few-mode fiber (FMF) can maintain multiple reflection peaks due to the stable multiple modes in FMF. This paper studies the sensing characteristics of multiple reflection peaks for a four-mode FBG (4M-FBG) and innovatively proposes a joint-peak demodulation method based on one FM-FBG to reduce measurement error in temperature or strain sensing. This joint-peak demodulation method, theoretically explained and experimentally verified, provides the possibility of generating miniature sensors with high measurement accuracy and stable measurement performance. The potential of 4M-FBG for simultaneous measurement of strain and temperature is studied in this paper. By measuring the changes of wavelength and intensity of the reflection peaks, temperature and strain can be measured effectively.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Optical fiber sensors are favored by researchers because of their small size, corrosion resistance, and easy realization of distributed sensing [1–3]. FBGs are based on fiber, so a diversity of fiber types can lead to a diversity of FBGs. Alessio Stefani et al. [4] innovatively used a polymer optical FBG to generate a high sensitivity (19 pm/g) accelerometer with a linear response range reaching 15 g. Yaofei Chen et al. [5] proposed a novel optical fiber sensor based on photonic crystal fibers and FBGs that can measure the magnetic field and temperature simultaneously. Most FBG sensors are based on single mode fibers. A single-mode FBG (SM-FBG) can only generate one reflection peak. In some measurement scenarios, such as the monitoring of railroad tracks during the passage of a high-speed train or the monitoring of a temperature field during an explosion, only a limited number of data can be recorded due to the rapid changes. In this case, random errors can cause a non-negligible effect on the SM-FBG measurement results. The development of FM-FBGs, that can generate multiple reflection peaks, provides a new approach to reduce the affect of random error.
In the field of fiber optic communications, mode-division multiplexing has been researched by scholars to further increase the capacity of long-haul transmission systems based on the stable few-mode characteristics of FMF [6–8]. Additionally, in the sensing field, sensors based on FM-FBGs have become a research topic of interest, providing an approach to composite sensors. Wang, Chongxi et al. [9] explored the temperature sensitivity of FBGs inscribed in polarization-maintaining FMFs. Their experimental results showed that with a temperature change of 100°C, the Bragg wavelength shift and reflected intensity decrease were 1 nm and 10 dBm, respectively. Xuekai Gao et al. [10] proposed a strain and temperature sensor based on FMF. The spectrum dips had different sensitivities of temperature and strain between spectrum dips. Chongxi Wang et al. [11] proposed polarization-maintaining FM-FBGs to measure both temperature and strain. The wavelength shift and the interrogated phase were also studied. Their sensitivities of temperature and strain were shown to be 10 pm/°C and 0.73 pm/µɛ and $- 3.2 \times {10^{ - 2}}$ rad/°C and $4 \times {10^{ - 4}}$ rad/µɛ, respectively. Yunhe Zhao et al. [12] proposed an FBG sensor based on FMF, which can measure directional curvature and temperature simultaneously. The resolutions for directional curvature and temperature measurement were $9.15 \times {10^{ - 4}}$ m-1 and 0.952°C, respectively. Yablochkin et al. [13] studied a few-mode regime and vibration sensor based on FMF. Ming Ding et al. [14] proposed a high-sensitivity thermometer using a multimode FBG. The thermometer had an average sensitivity of 266.25 pm/°C in a range of 80°C. Based on multireflected peaks having different responses to displacement, An Sun et al. [15] proposed displacement sensing using a multimode FBG.
The existence of multireflected peaks of FM-FBG provides not only a new approach to composite sensors but also a basis for the development of methods to improve sensing accuracy. In this study, the transmission mode and sensing characteristics of 4M-FBGs are analyzed theoretically and experimentally. Furthermore, a joint-peak demodulation method is proposed for the first time to achieve lower measurement error.
2. Joint-peaks demodulation method
COMSOL was used in this research to analyze the modes of FMF. The effective indexes of fiber core and cladding are 1.4457 and 1.4378, respectively. For a fiber with a core diameter of 18.5 µm, there are four transmission modes in the core—$\textrm{L}{\textrm{P}_{01}},$ $\textrm{L}{\textrm{P}_{11}},$ $\textrm{L}{\textrm{P}_{21}},$ and $\textrm{L}{\textrm{P}_{02}}$—as shown in Fig. 1. As a type of weakly guiding conduit, the characteristic equation of the propagation constant ($\beta $) of FMF can be expressed as:
Fig. 1. Transmission modes in FMF. (a) $\textrm{L}{\textrm{P}_{01}},$ (b) $\textrm{L}{\textrm{P}_{11}},$ (c) $\textrm{L}{\textrm{P}_{21}}$ and (d) $\textrm{L}{\textrm{P}_{02}}.$
When $m = 0,$ $\textrm{H}{\textrm{E}_{11}}$ or $\textrm{H}{\textrm{E}_{12}}$ exists in the fiber core, which corresponds to $\textrm{L}{\textrm{P}_{01}}$ in Fig. 1(a) or $\textrm{L}{\textrm{P}_{02}}$ in Fig. 1(d), respectively. When $m = 1,$ $\textrm{T}{\textrm{E}_{01}},$ $\textrm{T}{\textrm{M}_{01}},$ and $\textrm{H}{\textrm{E}_{21}}$ exist in the fiber core, which corresponds to LP11 in Fig. 1(b). When $m = 2,$ $\textrm{E}{\textrm{H}_{11}}$ and $\textrm{H}{\textrm{E}_{31}}$ exist in the fiber core, which corresponds to $\textrm{L}{\textrm{P}_{21}}$ in Fig. 1(c).
The higher-order mode $\textrm{L}{\textrm{P}_{02}}$ is more prone to loss during transmission [16,17] compared with other modes. In this experiment, to be cost-effective and improve the possibility of engineering applications, single-mode fibers (SMFs) were used to introduce light from the light source into the FMF, and the reflected light of 4MF-FBG was introduced into a demodulator through the SMFs, as shown in Fig. 2. The SMF and the FMF were connected by fusion splicing. Core diameter difference between FMF and SMF increases the loss of $\textrm{L}{\textrm{P}_{02}}$ mode. According to the research of Dan Su et al. [18], when the 4MF-FBG is only 0.5 cm away from the splicing point, four reflection peaks related to four modes can be observed. However, when the 4MF-FBG is over 10 cm from the splicing point, the reflection peak related to $\textrm{L}{\textrm{P}_{\textrm{02}}}$ disappears. In our experiment, because the 4MF-FBG was 15 cm away from the splicing point, the $\textrm{L}{\textrm{P}_{02}}$-related peak has been lost and only three reflection peaks can be observed, as shown in Fig. 3.
The expression for the corresponding reflection peaks is shown as Eq. (4).
In FBG measurement, due to the measurement accuracy and resolution of the demodulation system, it is unavoidable to introduce random errors, ${\delta _{\textrm{ran}}},$ in the measurement. The random error is normally distributed, as shown in Eq. (5). The random errors contained in each reflection peak are independent of each other.
where $\mu$ represents the mean of the measurement data and $\sigma$ represents the standard deviation of the measurement data.The operation shown in Eq. (6), which we call a joint-peak demodulation method, is performed on the three peak reflection measurements of the 4M-FBG.
where ${x_1},$ ${x_2},$ and ${x_3}$ represent the measured values of ${P_1},$ ${P_2},$ and ${P_3},$ respectively; ${P_{2 - 1}}$ represents the difference between the true values of ${P_2}$ and ${P_1};$ and ${P_{3 - 2}}$ represents the difference between the true values of ${P_3}$ and ${P_2}.$ When enough measurements are available, the means ${\mu _1},$ ${\mu _2},$ and ${\mu _3}$ of the peaks can be used to approximate ${P_{2 - 1}}$ and ${P_{3 - 2}}:$ ${P_{2 - 1}}\textrm{ = }{\mu _2}\textrm{ - }{\mu _1},$ ${P_{3 - 2}}\textrm{ = }{\mu _3}\textrm{ - }{\mu _2}.$Based on the operational relationship between independent normal distributions, the mean and standard deviation of ${x_\textrm{c}}$ can be expressed as Eq. (7) and (8). For the convenience of description, ${x_\textrm{c}}$ is regarded as the measured value of a virtual reflection peak ${P_\textrm{c}}.$ ${\mu _\textrm{c}}$ and ${\sigma _\textrm{c}}$ represent its mean and standard deviation, respectively.
3. Experimental verification
To verify the joint-peak demodulation method, temperature and strain test benches were set up separately in this study, as shown in Fig. 4. In the strain test, the strain loaded on the 4M-FBG was adjusted from 0 µɛ to 2300 µɛ by the translation stages. In the strain test, the thermostat temperature was adjusted from 50°C to 200°C.
The results of the strain and temperature measurements are shown in Fig. 5. Due to random errors, which reflection peaks (${P_1},$ ${P_2},$ ${P_3},$ and ${P_\textrm{c}}$) has the maximum error is uncertain. For example, at position 1 in Fig. 5(a), ${P_2}$ has the maximum error among the three reflection peaks. However, at position 2 and 3, ${P_3}$ has the maximum error. ${P_3},$ which is obtained by the joint-peak demodulation method, considers all three peaks and can always stay at a low value. The errors of ${P_1},$ ${P_2},$ ${P_3},$ and ${P_\textrm{c}}$ are recorded as ${P_{\textrm{1 - err}}},$ ${P_{\textrm{2 - err}}},$ ${P_{\textrm{3 - err}}},$ and ${P_{\textrm{c - err}}},$ respectively, and they are listed in Table 1. Additionally, the maximum values of ${P_{\textrm{1 - err}}},$ ${P_{\textrm{2 - err}}},$ and ${P_{\textrm{3 - err}}}$ appeared in different points. For example, in Fig. 5(b), the maximum value of ${P_{\textrm{1 - err}}}$ $({M_{\textrm{1 - err}}})$ appears at position 1; the maximum value of ${P_{\textrm{2 - err}}}$ $({M_{\textrm{2 - err}}})$ appears at position 2; and the maximum value of ${P_{\textrm{3 - err}}}$ $({M_{\textrm{3 - err}}})$ appears at position 3. Moreover, the largest value among ${M_{\textrm{1 - err}}},$ ${M_{\textrm{2 - err}}},$ and ${M_{\textrm{3 - err}}}$ is uncertain, as shown in Fig. 6(a). Values of ${M_{\textrm{1 - err}}},$ ${M_{\textrm{2 - err}}},$ ${M_{\textrm{3 - err}}},$ and ${M_{\textrm{c - err}}}$ from the 52 strain measurements are obtained. Every experiment was measured over the full measurement range. ${M_{\textrm{c - err}}}$ represents for the maximum error of ${P_\textrm{c}}.$ The statistical analysis show that in two experiments out of the 52 experiments, the ${M_{\textrm{1 - err}}}$ is the largest among ${M_{\textrm{1 - err}}},$ ${M_{\textrm{2 - err}}},$ and ${M_{\textrm{3 - err}}}.$ In the other 50 experiments, there were 25 times when ${M_{\textrm{2 - err}}}$ or ${M_{\textrm{3 - err}}}$ was the largest, respectively. We connected these maxima with a curve to form an envelope to compare with ${M_{\textrm{c - err}}},$ as shown in Fig. 6(b). It can be seen that ${M_{\textrm{c - err}}}$ is smaller than the maxima in all experiments, which indicates that ${M_{\textrm{c - err}}}$ can significantly reduce the maximum measurement error. This conclusion is also confirmed in the temperature experiments, as shown in Fig. 6(c) and (d). The numbers for ${M_{\textrm{1 - err}}},$ ${M_{\textrm{2 - err}}},$ and ${M_{\textrm{3 - err}}}$ to be the largest are 5, 16 and 31, respectively. The averages and standard deviations of ${M_{\textrm{1 - err}}},$ ${M_{\textrm{2 - err}}},$ ${M_{\textrm{3 - err}}},$ and ${M_{\textrm{c - err}}}$ are shown in Table 2. It can be seen that ${M_{\textrm{c - err}}}$ have the smallest average and standard deviation in both strain and temperature measurement experiments. This indicates that the error of ${P_\textrm{c}}$ is more stable and lower than that of ${P_1},$ ${P_2},$ and ${P_3},$ making it more suitable for strain and temperature sensing.
Fig. 6. Maximum errors of ${P_1},$ ${P_2},$ ${P_3},$ and ${P_\textrm{c}}.$ (a) Strain measurement errors. (b) ${M_{\textrm{c - err}}}$ and maximum value among ${M_{\textrm{1 - err}}},$ ${M_{\textrm{2 - err}}},$ and ${M_{\textrm{3 - err}}}$ in strain measurement. (c) Temperature measurement errors. (d) ${M_{\textrm{c - err}}}$ and maximum value among ${M_{\textrm{1 - err}}},$ ${M_{\textrm{2 - err}}},$ and ${M_{\textrm{3 - err}}}$in temperature measurement.
Table 1. Errors of ${P_\textrm{1}}\textrm{,}$${P_\textrm{2}}\textrm{,}$${P_\textrm{3}}\textrm{,}$ and ${P_\textrm{c}}$
Table 2. The averages and standard deviations of ${M_{\textrm{1 - err}}}\textrm{,}$${M_{\textrm{2 - err}}}\textrm{,}$${M_{\textrm{3 - err}}}\textrm{,}$ and ${M_{\textrm{c - err}}}$
4. Simultaneous measurement of strain and temperature
During strain or temperature measurement, the reflection peak of the 4F-FBG not only changes in wavelength, but also changes in intensity. Therefore, the 4F-FBG has the potential for simultaneous measurement of temperature and strain. For the 4F-FBG used in this experiment, the intensity of the reflection peak ${P_2}$ changed linearly in the temperature range of 50-110°C or the strain range of 0-900 µɛ, as shown in Fig. 7. Therefore, the intensity of ${P_2}$ was used to demodulate signals. $\varDelta {I_{{P_2}\textrm{ - }T}}$ and $\varDelta {I_{{P_2}\textrm{ - }\varepsilon }}$ represent the intensity changes of ${P_2}$ caused by temperature and strain, respectively. $\varDelta {\lambda _{{P_\textrm{c}}\textrm{ - }T}}$ and $\varDelta {\lambda _{{P_\textrm{c}}\textrm{ - }\varepsilon }}$ represent the wavelength changes of ${P_\textrm{c}}$ caused by temperature and strain, respectively. The wavelength changes of ${P_\textrm{c}}$ was used to demodulate signals because of its lower error.
Fig. 7. Changes of wavelength and intensity of reflection peaks. (a) $\varDelta {\lambda _{{P_\textrm{c}}\textrm{ - }T}}$ and $\varDelta {I_{{P_2}\textrm{ - }T}}$ change with temperature. (b) $\varDelta {\lambda _{{P_\textrm{c}}\textrm{ - }\varepsilon }}$ and $\varDelta {I_{{P_2}\textrm{ - }\varepsilon }}$ change with temperature.
The equations of the wavelength changes of ${P_\textrm{c}}$ and the intensity changes of ${P_2}$ are shown as:
$T$ and $\varepsilon$ can be obtained by solving Eq. (9)-(11):
Fig. 8. Experiment set up for simultaneous measurement of strain and temperature. (a) The fixation of the fiber. (b) Demodulation of 4M-FBG. (c) Inside the thermostat.
Fig. 9. Experiment results of simultaneous measurement of strain and temperature. (a) Wavelength change of ${P_\textrm{c}}.$ (b) Intensity change of ${P_2}.$
Based on Eq. (12) and the measurement results, the temperature and strain were calculated. The maximum errors of temperature and strain were 8.46% F.S. and 8.71% F.S., respectively. The errors are mainly caused by the measurement error of the reflected peak intensity. Compared with the wavelength change, the sensitivity of the intensity change is much lower. Its measurement is more susceptible to random errors, which can be caused by the fluctuation of light source intensity, bending of optical fiber, or even the disturbance of air flow in the thermostat.
5. Conclusions
In this paper, a joint-peak demodulation method is proposed based on multiple reflection peaks of FM-FBGs to reduce the nonlinear error in temperature or strain measurements. The method reduces the smallest standard deviation of the 4M-FBGs reflection peaks by 36%. This method was explained theoretically and has been verified experimentally. Because of the random error, the peak with the maximum error among multiple FM-FBG peaks is random. ${P_\textrm{c}},$ a virtual reflection peak obtained by this method, has a stable and lower error and is more suitable for temperature or strain measurement. With multiple reflection peaks, 4M-FBG has the potential for simultaneous measurement of strain and temperature. By measuring the changes of wavelength and intensity of the reflection peaks, a simultaneous measurement method is provided in this paper.
Funding
National Natural Science Foundation of China (51720105016, 51805421, 51890884, 91748207).
Acknowledgments
We thank the funding from National Natural Science Foundation of China (Nos.51805421, 51890884, 91748207, 51720105016). We also thank the support from the International Joint Laboratory for Micro/Nano Manufacturing and Measurement Technologies.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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