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Abstract
This study proposes a distributed large-curvature sensor based on ring-core few-mode fiber (RC-FMF) and differential pulse-pair Brillouin optical time-domain analysis (DPP-BOTDA). The RC-FMF is adhered to a thin steel substrate and an asymmetric hump shape is reconstructed using the Frenet-Serret algorithm. The proposed curvature sensor demonstrates a larger curvature-sensing range, excellent tolerance to bending-induced optical loss, and increased Brillouin gain coefficient. The proposed sensor also demonstrates longer sensing distance and continuous absolute measurement compared to other sensors. The proposed model can be applied to the end tracking of soft robotics and structural health monitoring of civil infrastructures.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In the past three decades, Brillouin optical time-domain analysis (BOTDA) has been widely studied due to its advantages of high sensing accuracy and continuous absolute measuring ability of measurands [1,2]. Novel techniques such as differential pulse pair [3,4], Raman/Brillouin distributed amplification [5,6], pulse coding [7,8], frequency agility or chirped-pulse modulation [9,10], and optical chirp chain [11] have been developed to improve the sensor’s performance in terms of spatial resolution, sensing distance, and measurement time. Meanwhile, special optical fibers have intrinsic advantageous properties over traditional single-mode fibers (SMF), thus have been proposed to achieve various distributed sensing applications. These include gold-coated fibers for high-temperature strain applications [12–14], polymer optical fibers for large-scale strain sensing [15,16], few-mode or multimode fibers for bending-insensitive sensing [17], and polarization-maintaining fibers for multi-parameter sensing [18]. These novel sensing techniques and special optical fibers boost BOTDA-based distributed sensors in practical applications, such as monitoring the structural health of civil infrastructures [19] or power transmission lines [20].
Compared with these applications, distributed curvature sensing has gained increasing attention because it shows potential for shape reconstruction, which can be used for continuous end tracking of motional objects such as needles [21] or soft robotics [22]. In general, the curvature information of a shape can be reconstructed by utilizing the bending-induced strain of an off-center sensor. Wylie et al. first reported distributed shape sensing with BOTDA by measuring the bend-induced strain, where the sensing fiber was attached to both sides of a steel tape to obtain the difference in the Brillouin frequency shift (BFS) [23]. Ba et al. proposed a temperature-self-compensated distributed shape sensor by adhering the polarization maintaining fiber (PMF) on both sides of a steel strip, based on differential pulse pair BOTDA (DPP-BOTDA) with a spatial resolution of 10 cm [24]. Various shape profiles were patterned in the PMF and the shapes were reconstructed using the Frenet-Serret algorithm. A compact scheme demonstrated by Zhao et al. used multi-core fiber to identify the curvature and curvature direction, wherein three outer cores of the integrating seven-core fiber were used to construct a shape sensor array to identify bending [25]. However, a common issue is that the curvature range of these shape sensors is limited, which is due to the use of SMF. In this case, there is a bending-induced optical loss for a bending radius smaller than 1.25 cm. Moreover, for a two-end-access sensing setup, such as BOTDA, the macro-bending-induced loss deteriorates the distributed sensing signal. Furthermore, extreme conditions can lead to the breakdown of the entire sensing signal along the sensing fiber, such as a sharp bend [17,26,27].
Bending-loss-insensitive distributed sensing is necessary to solve this problem, and bending sensitivity is highly desired. Several works have been proposed to achieve bending-loss-insensitive distributed sensing, but they could not quantitatively identify bending [17,26]. In 2016, Wu et al. demonstrated a few-mode fiber (FMF)-based distributed curvature sensor using a quasi-single-mode BFS for the first time, to the best of our knowledge [28]. The FMF showed curvature sensitivity and tolerance to bending-induced optical loss, in which the maximum BFS change was 14 MHz at a bending radius of 0.9 cm. However, the measured curvature range was low due to the macro loss of the fiber during bending. In 2019, Shen et al. reported a distributed curvature sensor based on BOTDA with a spatial resolution of 1 m [29]. Moreover, the bending radius was 0.5 to 1.5 cm and the corresponding BFS change was from 32.9 to 7.81 MHz. In 2021, our group proposed a distributed curvature sensor with higher sensitivity, corresponding to BFS variations of 91.7 to 9 MHz for a bending radius of 0.3 to 2.02 cm [30]. However, the sensing length of this fiber was short because of its high background absorption loss, which hindered long-distance sensing applications.
This study proposes a distributed large-curvature sensor up to 350 m−1 based on a ring-core few-mode fiber (RC-FMF) and DPP-BOTDA. A high spatial resolution of 5 cm was used to characterize the curvature sensitivity of the RC-FMF, which showed a BFS variation of 48 MHz at a curvature of 350 m−1. The Brillouin gain coefficient of the proposed sensor increased with the curvature by a factor of 2 for a curvature of 350 m−1, which showed a high sensing accuracy compared to those of circular-core-based curvature sensors. The RC-FMF showed excellent tolerance to bending-induced loss and intrinsically low loss, indicating its potential application in long-distance applications. The RC-FMF was adhered to a thin steel substrate so that the curvature could be characterized by the bending-induced strain. The bending direction was distinguished by identifying the tensile or compressive strain-induced positive or negative variation in the BFS. Using the Frenet-Serret-frames algorithm, an asymmetric hump shape with the largest curvature of 30 m−1 was successfully reconstructed based on the proposed curvature sensor. The proposed distributed curvature sensor showed potential for shape reconstruction, which demonstrated its possible application in the end tracking of soft robotics and structural health monitoring of civil infrastructures.
2. Experimental setup
The RC-FMF was fabricated using the traditional vapor deposition method, in which the refractive indices of the core and cladding were 1.4728 and 1.4597, respectively. The refractive index profile of RC-FMF is shown in ref. 29. The inner and outer diameters were 7.38 and 15.14 µm, respectively. This setup supported 4 modes named LP01, LP11, LP21, and LP31, and their attenuations at the C band were 0.29, 0.29, 0.31, and 0.32 dB/km, respectively [31]. Compared to conventional SMF, the RC-FMF can maintain most of its optical field in a bend region owing to its unique ring-core-structure; compared to circular-core-based FMFs, the RC-FMF can enhance its Brillouin gain in the bend region thanks to the reduced effective mode area. These two main advantages brought by RC-FMF make it suitable for Brillouin-scattering-based curvature sensors.
The experimental setup of the DPP-BOTDA model is shown in Fig. 1, which coincides with that of the standard BOTDA. The light from a narrow linewidth laser of 100 Hz (NKT photonics, X15) was divided into two beams, called a pump and probe. Subsequently, the pump beam was modulated into a pulsed light via an electro-optic modulator MZM1, which was driven by an arbitrary function generator that could generate an electrical pulse. Then, the pump light was polarization-scrambled, amplified by an erbium-doped fiber amplifier, and sent into the RC-FMF via a circulator. In the lower branch, the probe beam was modulated into a carrier-suppressed double sideband, which was generated via an MZM2 that was driven by a radio frequency (RF) synthesizer with a frequency shift of the Brillouin center frequency. An isolator was used to block the pump light, and the powers of the pump and probe were 100 and 2 mW, respectively. The probe was amplified using the pump light through the SBS effect in the RC-FMF, filtered out using a fiber Bragg grating with a 3 dB bandwidth of 8 GHz, and detected with a photodetector (PD). The data of the distributed Brillouin signal was recorded with an oscilloscope by tuning the frequency shift from 10 to 10.6 GHz with steps of 4 MHz using an RF synthesizer. Two pump pulses were sent with a 0.5 ns pulse-width difference that equaled 5 cm spatial resolution. This is because a high spatial resolution is extremely necessary for a simplified curvature measurement. The setups for the temperature, strain, and curvature measurements are shown in the inset of Fig. 1. RC-FMF-based distributed large-curvature sensor. The RC-FMF was spliced into SMF pigtails using tapered splicing with repeated arc discharge, and the splicing loss was approximately 1.5 dB.
Fig. 1. Experimental setup of the DPP-BOTDA model. Inset: setups for temperature, strain and curvature measurement
3. RC-FMF-based distributed large-curvature sensor
First, the optical loss of the RC-FMF with respect to the bending radius was tested. The bending losses of the RC-FMF and SMF were experimentally tested with the standard ‘winding the fiber’ measurement method. The loss of the SMF fiber began to exponentially increase at the bending radius of 1.25 cm, and continued up to 55 dB at 0.7 cm, as shown in Fig. 2(a). In contrast, the loss of RC-FMF at all the bending radii was less than 0.02 dB, demonstrating that it was bending-loss resistant, as shown in the small black squares in Fig. 2(a). The optical field distribution of the RC-FMF was obtained with numerical simulations under a bending radius of 0.28 cm using finite element method (FEM) software. The simulation results are shown in the inset of Fig. 2(a). The optical field of the RC-FMF shifted to one side of the cross-section of the ring core, where most of its energy was maintained even at the smallest bending radius of 0.28 cm. This is mainly due to the ring fiber-core structure, which exhibited excellent bending resistance in the simulation results.
Fig. 2. (a) Bending-induced optical loss of RC-FMF compared with SMF. Inset: optical field of RC-FMF at the bending radius of 0.28 cm. (b) Demonstration of spatial resolution of 5 cm.
Next, a spatial resolution of 5 cm was verified using the DPP-BOTDA model shown in Fig. 1. The distributed BFS of the RC-FMF showed that a strain was imposed in the middle and the 10% to 90% rising region corresponded to the 5 cm spatial resolution, as shown Fig. 2(b). This was the result of using a differential pulse-pair of 0.5 ns. The distributed strain and temperature characterization of a 50 m RC-FMF were conducted and the results are shown in Fig. 3. The strain coefficient of the RC-FMF was 0.0413 MHz/µɛ and the temperature coefficient was 0.968 MHz/°C. This was obtained by applying a linear fitting in the strain range of less than 9000 µɛ and the temperature range of less than 80°C, respectively. These parameters coincide with those of the FMFs.
Subsequently, the large-curvature sensing ability of the RC-FMF was analyzed. The bending-insensitive RC-FMF was continuously winding on cylinders with different radii. The three-dimensional distributed Brillouin gain spectrum (BGS), that is, the intensity of the BGS vs. frequency and fiber length, at bending radii of 3.2, 2.6, 2, 1.76, 1.5, 1.25, 1, 0.6, and 0.285 cm are shown in Fig. 4(a). The region under different bending radii showed a gradual elevation of BFS and enhancement of BGS intensity, which can be observed in the middle of Fig. 4(a). The BFS variation with different bending radii were plotted to quantitatively characterize the curvature with BFS, as shown in Fig. 4(b). In contrast, the theoretical BFS of the RC-FMF with bending radii was calculated with a growing bending radius of 0.5 mm, which was determined using the same methodology and definition as those used in Ref. [30]. The power-averaged central shift of the mode field to the neutral axis was defined as the overall shift distance under the bending condition. This was because there was significant distortion in the mode field of the sensing fiber as well as the lateral shift. The theoretically calculated values could then be determined using the strain coefficient and overall shift distance of the mode field divided by the bending radius. Following the above method, the calculated results are shown in Fig. 4(b) as a blue curve. The experimental measurement results aligned well with the theoretical calculations, which justified the calculation and experimental method.
Fig. 4. (a) BGS of the bending radii at 3.2, 2.6, 2, 1.76, 1.5, 1.25, 1, 0.6, and 0.285 cm. The black dotted curve represents the BFS of the RC-FMF under these curvatures. (b) Measured BFS vs. the bending radii. The blue curve represents the calculated values and red squares represent the plotted BFS variations with different bending radii. (c) BGSs of the RC-FMF at different bending radii. (d) Effective mode area vs. bending radius.
The BGSs of the straight fiber and those at the bending radii of 2, 1, and 0.285 cm are shown in Fig. 4(c). The BGSs were increasingly enhanced in terms of the intensity and central frequency (BFS) with decreasing bending radii values. The intensity increased by a factor of 2.13 and the BFS showed an increase of 48.1 MHz compared to the straight fiber at a bending radius of 0.285 cm (350.8 m−1). The linewidth was approximately 66 MHz for all bending radii without obvious widening. As discussed above, bending induces strain, and strain induces a decrease in the intensity of the BGS in SMF [32], which broadens the linewidth of the BGS accordingly. However, this tendency was not observed in this experiment. Conversely, the intensity of the BGS increased with bending-induced strain, and the linewidth of the BGS remained the same. FEM software was used to conduct a simulation to identify the optical field variation in order to quantitatively explain why the SBS gain increased with curvature. The effective mode area was calculated for a growing bending radius of 0.5 mm and the calculation results are shown in Fig. 4(d). The normalized effective mode area gradually decreased with a decrease in the bending radii values, which enhanced the Brillouin gain coefficient, that is, the intensity of the BGS. More specifically, the effective mode area decreased from 148.4 to 62.7 µm2 at bending radii of 3.5 to 0.28 cm with a factor of 2.36, as shown in Fig. 4(d). In contrast, the maximum intensity of BGS increased from 0.297 to 0.635 at the same bending radii, with a factor of 2.13, as shown in Fig. 4(c). The small margin was likely due to the strain and bending-induced loss, which lowered the intensity of the BGS and subsequently decreased the factor. The linewidth of the BGSs maintained constant values for all the bending radii, which was likely due to the increase in the Brillouin gain coefficient-induced narrowing effect that counteracted the linewidth broadening.
4. RC-FMF-based 2D shape reconstruction
A distributed 2D shape reconstruction process was analyzed for curvature sensing that complied with the advantages of RC-FMF. A strain was produced if the RC-FMF endured a curvature when it was fixed on a steel tape. The strain-induced BFS variation is determined as follows:
where $d$ is the distance between the centers of the RC-FMF and steel tape, $\kappa = 1/R$ is the curvature, R is the bending radius, and ${C_\varepsilon }$ and ${C_\kappa }$ are the strain and curvature coefficients, respectively. The ${C_\kappa }$ of the sensor was determined if the temperature was kept constant, resulting in a linear response to the curvature. The RC-FMF experienced compressive or tensile strain when bending downward or upward, which corresponded to negative or positive changes, as shown in Figs. 5(a) and 5(b), respectively. Therefore, the bending direction could be identified based on the signs of the BFS change.Fig. 5. Principle of distinguishing the bending direction of the proposed distributed shape sensor based on: (a) compressive strain and (b) tensile strain. The inset is a cross-sectional photo of RC-FMF. (c) Theory of 2D shape reconstruction based on the Frenet-Serret algorithm.
A 2D shape can be described as a vector $r(s)$, where s is the arc length of the curve, as shown in Fig. 5(c). $r(s)$ is the integral of the unit tangent vector ${\textbf T}(s)$ and is expressed as:
where $r(0)$ is the initial value at the location $s = 0$. ${\textbf T}(s)$, indicating the direction of the curve at each location, can be calculated according to the formulas:The RC-FMF was adhered to a 100 µm thick steel strip using epoxy resin. The fiber then needed to fully dry and solidify the sensor to a stable condition. The measuring limitation of the homemade sensor was explored by first characterizing its curvature, ranging from 18 to 31 m−1, which was larger than that of the sensors in Refs. [23–25]. The resulting curvature coefficients after using the linear fitting were 11 and 10.6 MHz/m−1 for up- and down-bending, respectively, as shown in Fig. 6. The curvature direction could be identified by distinguishing between the positive and negative BFS variations.
Shape reconstruction was then conducted using the above method. The RC-FMF was glued in the middle of the steel strip, as shown in Fig. 7(a). An irregular hump shape that was 20 × 6.7 in height and width, and the length is 47 cm, respectively, was imposed on the proposed sensor and fixed with 3 M tape. The frequency-position mapping of the distributed BGS of the RC-FMF under the shape from Fig. 7(a) is shown in Fig. 7(b). The hump shape produced a positive BFS change in the middle and a negative BFS change on both sides of its slopes. This variation can be observed at the positions of 15.42 (positive), 15.38, and 15.56 m (negative) in Fig. 7(b). It should be noted that the bending induced a large strain in the position between 15.3 and 15.6 m. Conversely, the BGS before and after this region remained the same. This is different from SMF-based shape sensors, which show a strain-induced optical loss that can lower the BGS intensity. The RC-FMF-based shape sensor was resistant to the optical loss induced by bending and increased the intensity of the BGS, indicating the advantages of high sensing accuracy.
Fig. 7. (a) Photo of the actual shape, (b) position-frequency mapping of the RC-FMF of the imposed shape, (c) BFS distribution of the actual shape, and (d) comparison of the reconstructed and actual shapes via the Frenet-Serret algorithm.
A detailed asymmetrical BFS distribution was obtained with a larger slope on the left side than that on the right side, as shown in Fig. 7(c). The hump shape was reconstructed using Eqs. (1) to (3) and the results are shown in Fig. 7(d). In contrast, the actual shape was drawn using the front view of the photo and point-to-point description method. The actual and reconstructed shapes aligned well with each other except for the inflection point on both sides, which was due to the spatial resolution that could not identify the large variation of curvature in such a small region, as shown in Fig. 7(d). The slope of the left side was higher than that of the right side, which was the same as shown in Figs. 7(a) and (c). Thus, the results shown in Fig. 7(d) verify the effectiveness of the proposed sensor.
The accumulated error is a critical issue in the shape reconstruction process that determines the deviation of the reconstructed shape from the actual shape. This accumulated error is inevitable because it originates from each single-point calculation, and is an accumulated amplification process. Therefore, a large strain variation was introduced for a large-curvature sensing process and the Brillouin-scattering-based distributed sensing technique was selected as it can continuously measure the absolute variation. This sensing ability is intrinsically greater in BOTDA than in Optical Frequency Domain Reflectometry (OFDR) [22], because OFDR measures the relative variation of strain that is not suitable for large-curvature sensing. In addition, a 3 dB enhancement of the SNR of the BGS was observed at a curvature of 350 m−1 due to the unique structure of the proposed distributed curvature sensor. This signal enhancement phenomenon cannot be found in OFDR-based shape sensing, indicating the advantages of BOTDA for large-curvature reconstruction. Nevertheless, insufficient spatial resolution led to a large deviation of the inflection point on both sides of the hump shape, as shown in Fig. 7(d). A higher spatial resolution at the millimeter level using Brillouin optical correlation domain analysis is preferred if the reconstruction length is not the main factor considered. In this study, a reconstruction process from both ends of the shape was used to suppress the accumulated error induced by the large curvature in order to demonstrate the large-curvature sensing ability. Following this direction, distributed large-curvature shape sensing was demonstrated using the proposed sensor.
5. Conclusion
This study proposed a distributed shape reconstruction with an RC-FMF and DPP-BOTDA with a high spatial resolution of 5 cm. The proposed distributed curvature sensor exhibited bending-induced loss as low as 0.02 dB for all bending radii down to 0.285 cm, where the respective Brillouin gain coefficient increased by a factor of 2. This was because the ring-core structure of the RC-FMF maintained its maximum optical energy at a reduced optical field distribution during bending. Using these advantages, the RC-FMF was adhered to a thin steel substrate so that the curvature could be characterized by the bending-induced strain, and the curvature direction could be distinguished by identifying the compressive or tensile strain-induced positive or negative variation of the Brillouin frequency shift of the proposed sensor. A 2D shape with a large curvature of up to 30 m−1 was successfully reconstructed. Although this curvature is higher than that of most reported works [23,24,34], it is still much lower than the curvature measurement up-limit of 350 m−1, indicating a large space to improve. This requires a very high-spatial-resolution distributed curvature sensor that is bending-loss-resistant, and at the same time, the accumulated error during the shape reconstruction process should be suppressed as well as possible, which is considered to be conducted in future work.
For 3D shape sensing, axial rotations-induced reconstruction error should also be considered, as it will generate a deviation from the real information of curvatures and curvature’s direction to the calculated values. For short-distance like several meters, this axial rotation effect can be solved by carefully aligning in the layout process [25]. However, the axial rotation induced error cannot be avoided for long-distance shape sensing. In this case, calibration of the rotation must be conducted, and a preset shape using the sensing fiber or inside the fiber cladding such as using helical multicore fiber [35] could be a solution. This concern should be addressed in the following work to boost the performance of shape sensing.
Nevertheless, the proposed RC-FMF-based sensor exhibits a potential for large-curvature distributed shape sensing, which could be used in applications such as the state feedback of soft robotics and structural health monitoring of civil infrastructures.
Funding
Program for Guangdong Introducing Innovative and Enterpreneurial Teams (2019ZT08X340); National Natural Science Foundation of China (62005055, U2001601); Guangdong Provincial Key Laboratory of Photonics Information Technology (2020B121201011); Program of Marine Economy Development Special Fund (Six Marine Industries) under Department of Natural Resources of Guangdong Province (GDNRC [2021]33); Basic and Applied Basic Research Foundation of Guangdong Province (2021A1515011891).
Acknowledgments
The authors thank the experimental discussion and support from RealPhotonics Corporation Ltd.
Disclosures
The authors declare that there are no conflicts of interest related to this study.
Data availability
The data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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