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Abstract
A high-precision and thermal-insensitive strain sensor based on two self-starting optoelectronic oscillators (OEOs) is proposed and experimentally demonstrated. Two OEOs are grouped into a cross-referencing structure by dense wavelength division multiplexing (DWDM); the two OEOs have the same characters and they are placed in the same environment. In this frequency encoded strain sensor, it converts the strain information of the single mode fiber to the frequency information, and the frequency information is acquired by measuring the intermediate frequency (IF) mixed by the two OEOs. The accumulative magnification effect at high-order resonant frequency modes makes the strain sensor achieve high sensitivity, which significantly improves the precision of the measurement strain. The cross-referencing structure of the two OEOs makes the influence of the environment, such as temperature, greatly reduced. In the experiments, measurement errors less than ± 0.3 με at a measurement range of 600 με have been realized, including a drift error due to a variation in the environment such as temperature. Furthermore, a quasi-distributed strain measurement system based on the proposed strain sensor has been designed.
© 2017 Optical Society of America
1. Introduction
Strain sensor based on fiber optics, which takes advantages of immunity to electromagnetic interference, compact size and the ability to make distributed measurements, is of fundamental importance in applications such as structural health monitoring, pipeline security monitoring, saving lives, aircraft structural health monitoring, ocean-bottom seismic system, and so on [1–5]. It has attracted a lot of attentions in the past decades, whether scientific research or commercial applications. In order to meet the growing demands in these areas such as high-precision, distributed deployment and lower cost, a large amount of conventional fiber optics methods for strain sensing have been designed and implemented, for instance, fiber Bragg grating (FBG) [6–10], optical interferometry [11–16], fiber bending attention [17], fiber Brillouin scattering [18, 19], fiber long-period grating [20], polarimetric [21], etc. Recently, optoelectronic oscillator (OEO) which is in advantages of high stability and low phase noise has been used in strain sensor, with the result that the speed and resolution are greatly raised [22–26].
FBGs are the mainstream elements for the measurement of strain. This is due to the advantages of the distributed deployment which are caused by the wavelength encoded technology make them suit to multiplexing, immunity to electromagnetic interference (EMI), being electrically passive, small and lightweight. However, limitations such as the cross-sensitivity to strain and temperature, the domain Gaussian noise source, and so on, limit the practical applications. For instance, it is impossible to differentiate the effect on the wavelength shift of strain from temperature when a single measurement of Bragg wavelength shift is considered. And the Gaussian noise interference in the sensing signals limits the static-strain resolution of a FBG-based Fabry-Perot (F-P) interferometer [8]. To overcome these limitations, a variety of methods have been developed. For instance, a reference FBG was used to calibrate the effect of temperature [6], Hilbert transform and cross third-order cumulant were used in a FBG-based Fabry-Perot (F-P) interferometer to reduce the influence of Gaussian noise [8], a line-by-line inscribed phase-shifted FBG was designed as a torsion sensor which measured the fiber torsion by monitoring the amplitude difference between two strain and temperature-insensitive polarization dependent peaks [10].
Optical interferometry is another research hotspot in the past decades, for example, in-fiber F-P interferometer [11,12], in-line fiber Mach-Zehnder (MZ) interferometer [13,14], Michelson interferometer [15], fiber mode interferometer such as signal-mode—multimode—signal-mode fiber structures [16]. There is no doubt that the optical interferometry is a high-resolution and high-sensitivity measurement method to strain sensor. Nevertheless, it is also beset by the cross effect of strain and temperature.
At the same time, many other methods of producing strain sensors have been proposed and implemented [17–21]. In [17], a fiber bending loss encoded strain-displacement sensor was realized through an optical time domain reflectometer (OTDR) which was used to measure the bending loss between two strain-displacement chips. An optical pulse coding simultaneous strain and temperature sensor based on spontaneous Brillouin scattering has been realized [18], in which the accurate Brillouin intensity and frequency shift can be measured because the optical pulse coding provides a significant receiver signal-to-noise ratio enhancement. What is more, another spontaneous Brillouin scattering based strain sensor was designed in [19]. A long-period grating written in a photonic crystal fiber was used to reduce the effect of temperature when measured the fiber strain, that is because the photonic crystal fiber is temperature-insensitive [20]. Fiber-optic polarimetric coding strain sensor was designed [21], where a high resolution was realized, however, the complicated demodulation mechanism limits the practical application.
Recently, OEO which has been developed since more than ten years [27,28] was applied to realized frequency-coding strain sensors [22–26]. Compared with the strain sensors mentioned in [6–21], the OEO based strain sensors have advantages of high precision and high speed. Since the high spectral purity and low phase noise of an OEO system make direct measurement with high precision possible. Furthermore, the frequency of a microwave can be measured by a digital signal processor at a high speed, which caused that the optical interrogation can be implemented at a high speed [29]. For instance, a femtometer-resolution wavelength interrogation of a phase-shifted FBG sensor was implemented by using an OEO [22], in which the FBG was regarded as a strain tuning bandpass filter, and the generated frequency was tuned by the center frequency of the bandpass filter. Nevertheless, the cross-sensitivity of strain and temperature in FBG was not solved in [22], to overcome this limitation, dual frequency OEO was designed [23,24,26]. However, the measurement range was limited in [22–24,26] by the response frequency of the optoelectronic devices, where the values of measurement range were less than 50 με. In [25], an OEO-based strain sensor with extended measurement range was implemented. The length of the fiber is changed by strain, which further leads to the change of frequency of the OEO. Whereas the problem cross-sensitivity of temperature and strain was not solved.
In this paper, we proposed and experimentally demonstrated a high-precision thermal-insensitive frequency-coding strain sensor with extended range based on two optoelectronic oscillators. Two severally self-starting OEOs with different wavelength were designed and the two signals were combined together into a long optical fiber in terms of dense wavelength division multiplexing (DWDM). And the two output frequencies were introduced to a frequency mixer, consequently, the strain change of target was measured by the frequency difference of two OEOs. Thanks to the cross-referencing structure realized by DWDM, the influence of surrounding such as temperature was overcome. Compared to the strain sensors mentioned in [6–26], our strain sensor eliminates the temperature effect through a cross-referencing structure of OEO and extends the measurement range through the strain-loop delay-frequency coding technology when maintains the advantages such as high speed and high precision of OEO based strain sensors. Furthermore, our OEO based strain sensor does not need a temperature control device nor a stability control system, which greatly reduced the complexity of the system. In addition, since the advantage of DWDM, the proposed strain sensor can be realized quasi-distributed measurement, it shows the great application potential in the structural health monitoring, pipeline security monitoring, ocean-bottom seismic system, and so on. And a measurement error of ± 0.3με at a measurement range of 600με was realized in this paper.
2. Principle
Figure 1 shows the schematic diagram of the presented high-speed and high-precision temperature insensitive strain sensor with extended measurement range. It consists of two self-starting OEOs which have the same oscillation frequency as well as loop length and come together as a cross-referencing architecture by DWDM. In both of the OEOs, a light from a LD is introduced into a MZM, of which the output is passed through a long optical fiber which is shared by two OEOs with DWDM and detected with a PD. The output of the PD is amplified and filtered and fed back to the MZM. Before fed back to the MZM, the signal is introduced to the output by an electrical divider. Then the output of reference loop is divided by a second electrical divider. One is introduced to an electric spectrum analyzer (ESA1) to measure the oscillation frequency and free spectral range (FSR) and the other one mixed with the output of target loop by a mixer. The intermediate frequency (IF) signal is analyzed by an electric spectrum analyzer (ESA2). It should be noted that the lengths of the two self-starting OEOs are equal at the initial state, which is easy to implement because the long fiber is shared by the two OEOs. The sensing fiber is a single mode optical fiber same as the other fibers in the OEO, and it is one portion of the oscillation loop, hence the proposed sensor has a flexibility of the sensor position.
Fig. 1 The schematic diagram of the proposed strain sensor. LD: laser diode; MZM: Mach-Zendher modulator; PD: photoelectric detector; DWDM: dense wavelength division multiplexing; BPF: band-pass filter; EA: electrical amplifier; ED: electrical divider; ESA: electrical spectrum analyzer.
The oscillation frequency of OEO can be given by [28] as fosc = k/τ, where k is a very large integer for the high-order mode of OEO, τ is the total group delay of the loop, including the physical length delay and the group delay resulting from dispersive components. When the lengths of the two self-starting OEOs are nearly equal, the initial state of the two OEOs can be written as fosc1 = k1/τ, fosc2 = k2/τ, Δfinitial = (k2-k1) /τ, where fosc1, k1 are the oscillation frequency and high-order mode of the target loop, fosc2, k2 are the oscillation frequency and high-order mode of the reference loop, τ is the total group delay of the loop. Δfinitial is the oscillation frequency difference between the two OEOs. In addition, the value of high-order can be obtained using the following equations, k1 = ⌊ fosc1/ fFSR ⌋, k2 = ⌊ fosc2/ fFSR ⌋, where fFSR is free spectral range of the two loops, which is the frequency spacing between two adjacent resonance peaks, and ⌊ ⌋ means rounding down to the nearest integer.
Thanks to the advantage of the accumulative magnification effect at high-order resonant frequency modes, the measurement accuracy of the strain sensor can be vastly improved. In an OEO, the high-order frequency can be many times higher than the fundamental frequency, to be more exact, the value of the high-order k can be 105-106. Therefore, the measurement error of the fundamental frequency can be reduced by measuring a high-order frequency. Scilicet, the measurement precision of the strain sensor is significantly improved.
When the loop length of the OEO is changed, the oscillation frequencies of two OEOs are shown below:
where Δf is the oscillation frequency difference between the two OEOs, which can be measured by electronic spectrum analyzer directly, and Δτ1 and Δτ2 are the delay variations caused by the length change.In addition, the delay variations are dependent on the fiber strain and temperature. If the fiber dispersion is not considered, the delay in the fiber can be expressed as τ = Ln/c, where L is the length of optical fiber, n is the refraction index of the optical fiber, c is the speed of light in vacuum. When consider the strain and temperature effects simultaneously, the variation of fiber-optic delay can be written as following equations, which have been analyzed in detail in [30].
where J is the elastic coefficient of fiber-optic delay which was calculated as 3.82ps/km٠με and measured as 3.78ps/km٠με in a standard SMF-28 type single-mode fiber [30], K is the thermal coefficient of fiber-optical delay which was measured as 39.2 ps/km·°C in a standard SMF-28 type single-mode fiber [30]. L is the fiber length, Σ is the strain of the fiber, ΔT is the temperature variation. In the proposed strain sensor, the sensing fiber with a length of Ltarget is a part of the oscillation loop, and the oscillation loop length of reference OEO is LREF. Hence the variation of fiber-optic delay in this strain sensor can be written as: in Eq. (3a), the strain of the fiber can be written as Σ = ΔLtarget/Ltarget, where ΔLtarget is the length change of the sensing SMF caused by the strain. In the proposed strain sensor, ΔLtarget << LREF, thus Eq. (3a) can be rewritten as Eq. (3c). According to Eq. (3a) and Eq. (3b), Δτ1 is equal to Δτ2 when the strain is zero, no matter how the temperature changes. Furthermore, When the initial oscillation frequency difference between two OEOs Δfinitial = 0, the frequency difference does not change with the changing temperature according Eqs. (1a) – (3b), it means that the frequency difference variation between the two OEOs is caused by the strain only. By using the Eqs. (1a) – (1c), the delay differential variation between the two OEOs, which is caused by the strain only, can be written as below:the free spectral range of the OEO fFSR = 1/τ, the delay differential variation Δτ thus can be rewritten as following equation:the loop delay of the two OEOs are expressed as τ2 = nLREF/c, τ1 = n(LREF + ΔLtarget)/c, where ΔLtarget is the length change of the target, n is the effective refractive index of the fiber during the measurement, and c is the speed of light in vacuum. Then, the measured absolute distance LREF and length change ΔLtarget caused by the strain only can be calculated using oscillation frequency fosc2, free spectral range fFSR and frequency difference between two OEOs Δf in the following formula: where LREF is the absolute distance of the loop of OEO, which includes the physical length delay and the group delay, and the fiber with length of over 1 km is the major factor. Furthermore, the oscillation frequency at tens of gigahertz is the k-order of the fundamental frequency which equals the free spectral range fFSR. Usually the value of k is between 105 and 106, which means, compared to the direct detecting the fundamental frequency, the measured error of the proposed method would be reduced by the factor 1/k. Note that the oscillation frequency fosc2, free spectral range fFSR and frequency difference Δf between two OEOs are measured simultaneously, which reduce the environmental influences such as temperature, mechanical vibration, etc. Thus, the strain including a drift error due to the variation of environment such as temperature can be written as the following formula:when the initial oscillation frequency difference between two OEOs Δfinitial = 0, the Eq. (7) can be rewritten as Eq. (8), where the measured frequency difference Δf will began at Δf = 0 Hz.according the Eqs. (7) – (8), the measurement error of strain can be written as: where dΔLtarget and dLtarget are the measurement error of length change of the sensing fiber and the measurement length error of sensing fiber respectively. In this proposed strain sensor, dΔLtarget << d Ltarget ٠Σ, thus the measurement error is mainly from the measurement error of length change of the sensing fiber as shown in Eq. (9b), which is caused by the uncertainty of frequency measurement according to Eqs. (6a) – (6b). And the measurement error of length change dΔLtarget can be written as bellow when considering the initial oscillation frequency difference between two OEOs Δfinitial = 0.where dΔf and dfosc2 are the measurement error of the Δf (the frequency of IF signal) and oscillation frequency fosc2 respectively, dLREF is the measurement error of the loop length, which is caused by the measurement error of the free spectral range fFSR according to Eq. (6b). Thus Eq. (10) can be rewritten as Eq. (11a), furthermore, it can be rewritten as Eq. (11b) when considering the condition Δf<< fFSR: according to Eq. (11b), the measurement error of the Δf (the frequency of IF signal) limits the measurement accuracy of length change of the sensing fiber, further the measurement accuracy of strain is limited according to Eq. (9b). And the measurement error of the Δf may be caused by the relative loop length of the two OEOs, the frequency accuracy of the electrical spectrum analyzer, or the phase noise of the OEOs, which are determined by the actual conditions and environments.During the actual operation, the dispersive effect of the fiber and the relative value of the loop length of the two OEOs would lead a drift to the frequency difference Δf due to the temperature fluctuation, which will cause a measurement error of strain according to Eqs. (9a) – (11b). The relative value of the loop length of the two OEOs can be maintained within ± 0.5ps in our lab, including the dispersive effect of the SMF. Considering the thermal coefficient of fiber-optical delay is 39.2 ps/km·°C, under the conditions of the relative length of the two OEOs is 10ps at loop length of 4 km and the oscillation frequency is 10.6 GHz, if the unstressed length of sensing fiber is 4m and the temperature fluctuation is 20 °C, the error of frequency difference Δf would calculated as 0.84Hz according to Eqs. (1a) – (1c), which would lead an error of 0.056 με to the strain measurement according to Eqs. (9a) – (11b). Besides, the frequency accuracy of the electrical spectrum analyzer can be 1 Hz usually, and the uncertainty of frequency caused by the phase noise may be less or larger than the frequency accuracy of the electrical spectrum analyzer, up to now, an OEO with 1 Hz uncertainty of frequency have been realized [31], it means a measurement error of the Δf is 1Hz can be realized and the measurement strain error of 0.067 με can be reached at the conditions given above. Thus, the influence of the temperature and relative length of the two OEOs to the strain sensor can be ignored in the actual applications.
Besides, a quasi-distributed strain sensor could be realized by DWDM using several basic sensor units, because the designed strain sensor is cascade-able. Figure 2 presents the schematic structure of the OEO based quasi-distributed strain sensor system, in which the LD1 of Fig. 1 is replaced by a tunable light source (TLS) whose wavelength is λi (i = 1- N) and a pair of WDMs being placed beside each sensing area. In addition, by changing the output wavelength of the TLS, each sensing area could be measured in sequence.
Fig. 2 The OEO based quasi-distributed strain sensor. LD: laser diode; TLS: tunable light source; MZM: Mach-Zendher modulator; PD: photoelectric detector; DWDM: dense wavelength division multiplexing; BPF: band-pass filter; EA: electrical amplifier; ED: electrical divider; ESA: electrical spectrum analyzer.
3. Experimental results and discussions
A proof-of-concept experiment is carried out based on the setup in Fig. 1. A laser source with wavelength of 1550.12 nm is applied as LD1 and LD2 has a wavelength of 1550.92nm, both of the two laser sources (Yenista optics TLS-AG) have a power of 10 dBm. The modulators are single-drive intensity LiNbO3 modulator (OCLARO F-10) which have a 3-dB bandwidth of 11 GHz and insertion loss of 6 dB. Both of the PDs (Kang Guan KG-PD-10G) have responsivity of 0.8 A/W and 3-dB bandwidth of 11GHz. The EAs of two OEOs have response of 40GHz. The two BPFs have same center frequency 10.6GHz and same 3-dB bandwidth 13MHz. The electrical spectrum of the reference loop and the intermediate frequency (IF) signal are obtained by electrical analyzers ESA1 and ESA2 respectively. The target is a piezoelectric ceramics-based optical fiber stretcher (Core Morrow N01) with a controlling voltage range of 0-120 V and a radial deformation resolution of 4 nm, it is used to realize the generation and control of strain in this sensor. The sensing fiber has a length of 4 m. Note that the length of the two self-starting OEOs are equal at the initial state, which is actually less than 1mm in this demonstrative experiment.
Figure 3 shows the measured spectrum of OEOs with a loop length of 4 km, and the oscillation frequency of the two OEOs are same as 10.6 GHz approximately. The measured oscillation frequency is 10664440910 Hz and the side-mode suppression ratio is 19.87 dB as shown in Fig. 3(a), and the fundamental frequency as free spectral range fFSR is 49963 Hz, then the value of k2 is calculated as k2 = ⌊10664440910/49963⌋ = 213446. The simulate reference fiber length can be calculated as 4119380.320mm according to Eq. (6b), where c = 299792458 m/s and n = 1.4566. Figure 3(b) shows the initial frequency difference between reference loop and target loop, which is measured as 81.691 kHz. The full width at half-maximum of IF signal is 10Hz when measured by the electrical spectrum analyzer (Anritsu MS2725C) at a RBW of 1Hz and span of 100Hz, the uncertainty of the frequency measurement of IF signal is the limited of the proposed strain sensor. The Fig. 3(c) shows the electrical spectrum of the IF signals at different strain states, it indicates that the shape of the IF signals does not change and the only change is the frequency, which provides convenience for mensuration.
Fig. 3 Measured electrical spectrum of OEOs with loop length of 4km. (a) the initial reference oscillation frequency. (b) the intermediate frequency (IF) signal. (c) the IF signal at different strain state.
Then OEOs with loop length of 500 m in the proposed strain sensor was designed. Figure 4 shows the measured electrical spectrum of OEOs with loop length of 500m. It shows that the measured oscillation frequency is 10663747273 Hz and the side-mode suppression ratio is 52.68 dB. The fundamental frequency as free spectral range fFSR is 380000 Hz and the value of k2 is calculated as k2 = ⌊10663747273/380000⌋ = 28062. The simulate reference fiber length can be calculated as 541622.628mm. Figure 4(b) shows that the initial frequency difference between reference loop and target loop is measured as 25 kHz approximately. The full width at half-maximum of IF signal is 30Hz when measured by the electrical spectrum analyzer (Anritsu MS2725C) at a RBW of 1Hz and span of 100Hz.
Fig. 4 Measured electrical spectrum of OEOs with loop length of 500m. (a) the initial reference oscillation frequency. (b) the intermediate frequency (IF) signal.
According to the measurements of Figs. 3 and 4, the fFSR of OEOs with 500 m loop length is 7.6 times the fFSR of OEOs with 4 km loop length. Then according to Eqs. (1a) – (1c) the frequency difference of OEOs with 500 m loop length would be 7.6 times the OEOs with 4 km loop length. The insets of Fig. 5 are the response of displacement of the piezoelectric ceramics-based optical fiber stretcher to the increase and decrease in controlling voltage. The experimental results demonstrate that its radial displacement is not linear with controlling voltage and the radial displacement at each testing voltage point is different during the rising and falling processes of controlling voltage, furthermore, the displacement curve during the falling process of voltage has a bigger curvature. And Fig. 5 shows the response of frequency difference to the increase in controlling voltage and decrease in controlling voltage, which is consistent with the response of displacement of the piezoelectric ceramics-based optical fiber stretcher as shown in the insets of Fig. 5, it indicates that the proposed strain sensor has a good response to strain, which is a frequency encoded sensor by converting the strain changes to frequency changes. Comparing Figs. 5(a) and 5(b), the frequency difference in Fig. 5(b) is 7.6 times frequency difference in Fig. 5(a) approximately, which agree well with the analysis at the beginning of this paragraph.
Fig. 5 Measured relationship between frequency difference and different controlling voltage. (a) loop length L = 4 km. (b) loop length L = 500m.
What more, to verify the stability of the proposed strain sensor, each controlling voltage was maintained 20 minutes and measurement was repeatedly conduced several times during this period. During the measurement period, the sensor was placed in an open environment, which caused the sensor to be continuously affected by the unstable and random environment temperature fluctuations. Figure 6 shows the measured frequency residuals during the measurement period, the frequency residuals of strain sensor with 4 km loop length are less than ± 10 Hz as shown in Figs. 6(a) and 6(b) shows the frequency residuals of strain sensor with 500 m loop length are less than ± 30 Hz. Based on the above discussion, the uncertainty of frequency measurement more likely caused by the full width at half-maximum of IF signal and could be eliminated by averaging the multiple measured values.
Fig. 6 Measured residuals of frequency difference. (a) loop length L = 4 km. (b) loop length L = 500m.
The effective strain for the sensor was obtained by measuring the frequency shifts when it was subjected to controlled strain. To achieve this, the strain calibration was performed by using a purpose-built calibration rig where the axial stress of the sensing fiber is provided by the gravity of the standard weights. Silicon dioxide (SiO2) is a major material for the fiber, and the SiO2 is an isotropic linear elastic material. According to Hook’s law, the strain of fiber is linearly dependent on the axial stress of the fiber when the fiber is within the range of elasticity, and it can be written as σ = YΣ, where σ is the axial stress of the fiber, Y is the Young’s modulus of the fiber, Σ is the strain of the fiber. And the axial stress σ is also can be written as σ = F/S, where F and S are the axial tension and the cross-sectional area of the fiber respectively. Thus, the strain of fiber can be rewritten as
When the axial tension is provided by the gravity of the standard weights, the axial tension can be written as F = mg, where m is the weight, g is the acceleration of gravity. The relationship between the length change and the axial stress of the fiber (SMF-28e) in Fig. 7 was measured by using an optical delay measurement system with a vector network analyzer (VNA, Anritsu 37369D), where the axial stress is provided by standard weights, and the acceleration of gravity is 9.7913m/s2 (experimental site: 30.67 degrees north latitude, 500 meters in elevation). The metrical data in Fig. 7 indicates that the tensile elongation increases linearly with axial tension of the fiber, which agrees well with the analysis in Eq. (12). When the initial length of fiber is 1m, the increment of the tensile elongation is 946.9μm/N. And it is 142μm/N when the initial length of fiber is 0.15m, which means that a calibration rig with a 1.39μm spatial resolution can be obtained when the standard weights with one gram are used.
The strain coefficients of both OEO based sensors which have loop length of 4 km and 500m respectively were obtained through calibration and shown in Fig. 8. It indicates that the strain sensor with loop length of 4 km has a sensitivity of 10.35 Hz/με as shown in Fig. 8(a) and the strain sensor with loop length of 500 m has a sensitivity of 78.29Hz/με as shown in Fig. 8(b). And the sensitivities of the strain sensors with loop lengths of 4 km and 500 m are calculated as 10.355 Hz/με and 78.754 Hz/με respectively, by substituting the measured results of Figs. 3 and 4 into Eq. (8). Clearly, the theoretical values of measurement sensitivities are very close to the experimental values. In addition, the inset in Fig. 8(a) shows that the residuals of the strain sensor with loop length of 4 km are less than ± 1 με, and the inset in Fig. 8(b) shows that the residuals are less than ± 0.3 με in the strain sensor with loop length of 500m. In this sense, the proposed strain sensor has a high measurement accuracy. Furthermore, we tested the stress tolerance of the sensing fiber, which is the limited of the measurement range of the proposed strain sensor, by loading more weights to the sensing fiber. And the results show that the fiber would exceed the line elastic range and be snapped after more than 2% of the fiber length increase. It means that the measurement range of the proposed measuring principle can reach 20000με.
Fig. 8 Response of the frequency difference as a function of the applied strain on the sensing fiber with a step of 35.05 με. (a) loop length L = 4 km. (b) loop length L = 500m.
Figure 9 shows the sensor outputs when subjected to varied temperature at constant strain. And the thermal-insensitive of the proposed strain sensor was clearly evident. The measured strain variation is within ± 2 με when the loop length of the strain sensor is 4km as shown in Fig. 9(a), and Fig. 9(b) shows that the measured strain variation is ± 0.5 με when the loop length of the strain sensor is 500m. From the above, temperature fluctuations will no longer a problem in practical applications.
Fig. 9 Sensor outputs when subjected to varied temperature. (a) loop length L = 4 km. (b) loop length L = 500m.
According to Eqs. (6a) – (8) and the measured frequency difference as Fig. 5, the strain has been measured. Figure 10 shows the comparison between the multiple measured strain and calibration value, and the residuals of the measured strain are shown as insets of Fig. 10. Figures 10(a) and 10(b) are the measured strain when the loop length of OEO are 4 km and 500 m separately. Note that each state was maintained 20 minutes and measurement was repeatedly conduced several times during this period. It is observed that measured range of 600 με has been realized in both the strain sensors. And the measured error of the strain sensor with loop length of 4 km is ± 1 με, the measured error of the strain sensor with loop length of 500 m is ± 0.3 με. Further, the relationship of strain error between the two sets of strain sensors should equal the relationship between the loop length of two sets of OEOs according to Eqs. (6a) – (8). However, the phase noise of the OEO would be increased when the loop length is shorter, which may reduce the accuracy of the measurement. Thus, the measured error of the strain sensor with loop length of 4 km is not 7.6 times the measured error of the strain sensor with loop length of 500 m as the measured relationship between the loop length of the two strain sensors. Based on above analysis, two strain sensors with measured error of ± 1 με and ± 0.3 με respectively have been realized at the range of 600 με, including a drift error due to the variation of environment such as temperature.
Fig. 10 Measured strain and residuals of the measured strain with range of 600 με. (a) loop length L = 4 km. (b) loop length L = 500m. (c) comparison of measured strain between the strain sensors with loop length of 4 km and loop length of 500 m.
To further prove the correctness of the measurements, the measured strains of the two strain sensors with loop length of 4 km and loop length of 500 m are compared as Fig. 10(c). Theoretically, the measured strains of the two strain sensors should be the same because the sensing part of the two strain sensors is the same device, however, the repeatability of the piezoelectric ceramics is not absolutely, which lead that the measured strains have a slightly difference practically. The measurements show that the Pearson correlation coefficient between the two strain sensors with loop length of 4 km and loop length of 500 m is 0.99985 which is significant at the 0.01 level (2-tailed), it indicates that the two sets of measurements are matched. And the inset of Fig. 10(c) shows the measurement difference between the two strain sensors, it shows that the average measurement difference between the two strain sensors was 4.6 με while the standard deviation was less than 5 με, which may be caused by the repeatability of the piezoelectric ceramics and it can be decreased by averaging multiple measurements. Thus, the matched measurement results demonstrate the correctness of measurements.
4. Conclusion
We proposed and realized a high-precision thermal-insensitive frequency encoded strain sensor based on two self-starting OEOs. The OEO-based system takes advantage of the accumulative magnification effect at high-order resonant frequency modes to achieve high sensitivity. Two self-starting OEOs were established and combined as a cross-referencing structure by DWDM and the initial states of the two OEOs were same in both oscillation frequency and loop delay, one for probing and one for referencing, and the change in frequency is measured in real time, which makes the sensor take advantage of temperature insensitive. The system was evaluated for strain measurements at range of 600 με, the measurement error is ± 1 με when the loop length of the strain sensor is 4km and it is ± 0.3 με when the loop length is 500 m, including a drift error due to the variation of environment such as temperature. Furthermore, the measurement range of the proposed measuring principle can reach 20000με, because the fiber would exceed the line elastic range and be snapped after more than 2% of the fiber length increase. And a quasi-distributed strain measurement system based on the proposed strain sensor has been designed. Based on these advantages above, the proposed strain sensor may find important applications in structural health monitoring, pipeline security monitoring, aircraft structural health monitoring, ocean-bottom seismic system, and so on.
Funding
National Natural Science Foundation of China (No. 61271030); Fundamental Research Funds for the Central Universities (ZYGX2015J050).
Acknowledgments
This work is supported by the National Natural Science Foundation of China (No. 61271030) and Fundamental Research Funds for the Central Universities (ZYGX2015J050).
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