## Abstract

We propose and experimentally demonstrate a distributed directional torsion sensor based on an optical frequency domain reflectometer (OFDR) using a helical multicore fiber (MCF). A theoretical model is first established to reveal that the ability of the torsion direction discrimination stems from the fiber design of the central-offset cores with helical structure and the shorter helical pitch holds higher sensitivity. Such a distributed torsion sensor is then experimentally realized by using an OFDR system with an adjacent sensing distance of 9.4 mm. Comparative experiments with three different MCFs fully prove the theoretical predication. Finally, a distributed directional torsion sensor is realized with a linear sensitivity of 1.9 pm/(rad/m) by using the helical MCF with a helical pitch of 6 mm. Such a torsion sensing system would find potential applications in the fields of bionic robotics, 3-D shape sensing, oil drilling and so on.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Distributed torsion sensing is of great interest in various industrial applications including the bionic robot [1,2], 3-D shape sensing [3,4], and structural health monitoring [5,6]. For instance, oil drilling requires the distributed torsion measurement in the bit and in the pipe to improve drilling efficiency and avoid drill pipe twist-off [7]. Medical image guidance requires distributed torsion information for finally reconstructing the 3-D shape of the medical catheters and surgical instruments in minimally invasive surgery [8]. Fiber-optic sensors have the advantages of strong electromagnetic interference resistance, small size, lightweight, and corrosion resistance, and thus measurements of twist/torsion have been topics of intense research within the fiber-optic sensor community. Up to now, various optical fiber torsion sensors have been investigated, for example, specially designed gratings [9–11] and in-line fiber interferometers [12–14]. Optical fibers with helical structure were usually employed to identify the torsion direction. For instance, helical Bragg grating waveguides can be directly written in the fiber by using the femtosecond-laser processing technology [10]. With the help the CO_{2} laser scanning, helical structures in photonic crystal fiber also has been realized for torsion direction discrimination [11]. The helical structure also can be fabricated into the multi-core fiber (MCF) by pre-twisting and then heating with a CO_{2} laser splicing system [13]. The interference between the lights guided in the multi-cores were employed for monitoring the torsion. However, these sensors based on gratings and interfeometers are usually developed for torsion sensing at a single portion, few works were reported to realize the distributed or quasi-distributed torsion sensing [15,16].

In this work, our idea is to develop a distributed torsion sensor with the capability of torsion direction discrimination. Optical frequency domain reflectometry (OFDR) is promising distributed sensing technology with a unique advantage of high spatial resolution. The measurement principle of the OFDR is based on measuring the wavelength shift of the Rayleigh scattering spectrum. Rayleigh backscattering lights at multiple locations along the optical fiber will generate an interference spectrum. The distributed Rayleigh scattering in the fiber is regarded as random weak gratings. The wavelength shift induced by strain/temperature can be obtained by implementing the cross-correlation of the reference and measurement signals in spectrum domain. OFDR has been developed for distributed strain sensing [17], temperature sensing [18], shape sensing [19,20], and polarization analysis [21]. In this work, we present a distributed directional torsion sensing system by using the OFDR technology and the helical multicore fiber (MCF). Theoretical and experimental studies are successively implemented for investigating the capability of the torsion direction discrimination and the dependence of the torsion sensitivity.

## 2. Working principle

To understand the working principle of our proposed torsion sensor, we build up a theoretical model of the helical fiber under external torsion. In our design, the fiber needs at least one eccentric core with helical structure, and the helical MCF used in out experiment is just an example to demonstrate our idea. The helical core is flattened into a triangle on a plane as shown in Fig. 1. The width equals the circumference at radius *r*, where *r* represents the radial offset from the helical core to the fiber center and the height equals to the helical pitch *h*.

The length of the helical core can be calculated as:

When the helical fiber experiences an external torsion with a twist rate of *γ*, the helical core has a shear strain *ɛ _{s} = rγ*, and the width of the triangle will become (2π

*r*+

*ɛ*). Here, the sign of external torsion (±) represents the clockwise (CW) and counterclockwise (CCW) direction, respectively. After applying the external torsion, the length of the strained core changes to be

_{s}h*l*

_{ɛ}+

*l*). Considering that the

*l*

_{ɛ}≈

*l*in the denominator term, the upper equation can be approximately written as

*h*is inversely proportional to the initial bias torsion γ

_{0}, i.e.

*h*= 2π/γ

_{0.}Substituting it into the equation above, the torsion induced strain in the fiber core can be finally expressed as

*P*e = 0.1994 is the effective elastic-optic coefficient of the optical fiber, and λc = 1550 nm is the central wavelength of the wavelength sweeping range in the OFDR system. Generally, the theoretical sensitivity before approximation can be expressed as

For the first case of the fiber without a helical structure, i.e. γ_{0 }= 0, only the quadratic term in Eq. (5) is retained, and the wavelength shift is simplified to be

For the second case of the fiber core with a relatively large twist bias, it is reasonable to neglect the quadratic term in Eq. (5), and the wavelength shift can be linearized with respect to the external torsion. Since the helical redial is in the order of micrometer, the term (*rγ _{0}*)

^{2}is minuteness to be neglected. After the above approximation, Eq. (6) can be simplified as:

We conducted a theoretical calculation according to Eq. (6) to discuss the torsion sensitivity, as shown in Fig. 2. If the helical pitch of *h* is sufficiently large, i.e. larger than approximately 9 mm, the quadratic term of the shear strain in Eq. (5) cannot be neglected, and hence the wavelength shift shows a nonlinear relationship with the twist rate (Fig. 2(b)), and the theoretical sensitivity is lower than 1.06 pm/(rad/m) when the helical pitch is larger than 9 mm. The nonlinear phenomenon is not obvious, but can be observed by comparing the wavelength shift at ±100 rad/m. For example, the wavelength shift for helical pitch of 9 mm are 100 and 125 pm, respectively. This nonlinear phenomenon was verified in our later experiment. If the helical pitch of *h* is small enough, it makes sense to retain only the linear term of the shear strain. Therefore, linear response between the wavelength shift and the twist rate is obtained when *h* is smaller than 8 mm (Fig. 2(a)), and the sensitivity is expected to be 4.8 pm/(rad/m) with the helical pitch of 2 mm. In consideration of the theoretical sensitivity discussion above, it is expected to design a distributed torsion sensor with high sensitivity and linear response in both CW and CCW direction by employing a fiber with shorter helical pitch.

## 3. Experimental setup

Figure 3 illustrates the OFDR system established in our lab for distributed torsion sensing. A tunable laser (Santec, TSL-550) with a linewidth of 400 kHz is linearly swept, and the output light is divided into two parts by a 1:99 coupler. The small fraction of the 1% light is then further split by a 50/50 coupler for constructing a Mach-Zehnder interferometer with a 10 m delay in one arm. This interferometer functions as an auxiliary interferometer for obtaining the instantaneous frequency of the sweep light source and then compensating the nonlinearity of the sweep process. Because of the frequency tuning nonlinearity of the tunable laser, no linear relationship exists between the beat frequency and the time. An effective interpolation algorithm is usually used for compensating the nonlinearity of the sweep process [18]. First, we obtained the instantaneous frequency of the sweep light source thought the auxiliary interferometer and then resampled the beat signal obtained from the main interferometer. The 99% light from C1 is launched to a main interferometer with a local arm and a measurement arm. The measurement arm is used to collect the Rayleigh backscattering light in the FUT through a circulator. In order to suppress the polarization fading effect in optical fibers, a polarization diversity receiver is employed to receive the Rayleigh backscattering light in two orthogonal states of polarization (SOP), i.e. s and p states. The light from the local arm is split evenly between two orthogonal SOPs by adjusting the polarization controller in front of the polarization beam splitter (PBS). All data from both the auxiliary and main interferometers are detected by the photoelectric detectors (Thorlabs, PDB470C), and then acquired by a four-channel data acquisition card (ATS9440) at a sampling rate of 20 MS/s.

To demonstrate the proposed sensor, we employed three types of multicore fibers (MCFs) for the torsion test. Both the first and second are commercially available seven-core fibers from FIBERCORE. The first one has a helical structure with a pitch of approximately 15.4* *mm according to the data sheet. This fiber has a central core, and 6 outer cores arranged in a hexagonal array, as shown in Fig. 3. The core-core separation is approximately 35 µm, and the NA of each core is approximately 0.2. In addition to the non-existence of helical structure, the second fiber has similar parameter equivalent to the first one. The third one is homemade seven-core helical fiber. This fiber was fabricated by means of twisting the second fiber during hydrogen-oxygen flame heating [23]. This post-processing method can be used to fabricate the fiber with a designed helical pitch. In order to balance sensing performance and the additional insertion loss induced by the post-processing, the third fiber was designed and fabricated with a helical pitch of 6* *mm, as shown in inset of Fig. 3. In order to sequentially interrogating each core of the helical MCF, the three types of fibers are spliced to a fan-in spatial Mux coupler, and a 1×7 optical switch is connected to the seven pigtails of the fan-in coupler. In order to reduce the reflection at the end of the fiber, a knot was tied at the end of the fiber during the experiment.

In our experiment, the tunable laser is swept from 1545 to 1555* *nm at a rate of 16* *nm/s for 0.2048 s acquisition time, which gives a determined wavelength range of 3.278* *nm. In order to ensure the enough wavelength range, the sweep rang is usually larger than the really used wavelength range. The determined wavelength range of 3.278* *nm gives a theoretical spatial resolution of 235 µm in the distance domain. Two hundred data samples are selected for the cross correlation, giving a gauge spatial resolution of 4.7* *cm. In order to obtain sufficient sensing points and mitigate spectral leakage when implementing the short-time fast Fourier transformation, we selected the hamming window with 80% overlapping between adjacent windows. Hence, the distance between adjacent sensing points is 9.4* *mm.

Figure 4 illustrates the Rayleigh backscattering signal as a function of the fiber length. Five distinct strong reflection points are detected along the sensing fiber link. The first four reflection points are enlarged in the inset. The first one is located at approximately 1 m which indicates the APC-APC connector between the pigtail of the second port of the optical circulator and the input pigtail of the optical switch. The second one represents the internal reflection of optical switch. The third one is the APC-APC connector between one of seven output pigtails of the optical switch and the input pigtail of the Fan-in coupler. The fourth one indicates the strong reflection inside the Fan-in coupler. The last one is located around 60 m indicating the reflection at the end of the fiber.

Our signal processing procedure is the same as the commonly used OFDR system [24]. Firstly, both reference and measurement signals before and after torsion are converted to the spatial domain via fast Fourier transform. Secondly, a certain location signal is selected via a hamming window and converted to the spectrum domain via the short-time fast Fourier transform. Finally, cross-correlation is performed between the reference and measurement signals in spectrum domain to obtain the spectral shift. The cross-correlation peak is spectral shift at the selected location, which reflects the torsion variation. It is worth noting that the local spatial signal after applying the window function is zero-padded to increase the frequency resolution of the cross-correlation and hence improve the torsion measurement accuracy. The total time taken to make a complete measurement in the six out cores of the MCF is around 30 s which including the date acquisition and processing in each core, and this time can be shorten by using the parallel processing algorithm.

Figure 5(a) illustrates the normalized Rayleigh backscattering spectrum (RBS) before and after torsion. Here, the measurement spectrum was collected with the torsion of 15.7 rad/m. After implementing the cross-correlation of the measurement and reference Rayleigh backscattering spectrum, and a clear local spectral shift of 10.6 pm can be found, as shown in Fig. 5(b).

## 4. Experimental results

Firstly, the multi-core fiber with a helical pitch of 15.4 mm is employed to implement the torsion measurement. A segment of 1.2 m long helical MCF is fixed between two rotators, as shown in Fig. 6(a). Prior to the fiber torsion, the reference signals of the 7 cores are collected and stored. In the torsion experiment, one of the rotators is gradually rotated to 18 turns with a step of 3 turns in both CW and CCW direction, thereby yielding a torsion range of 94.2 rad/m with a step of 15.7 rad/m, and the measurement signals are collected after each rotation.

Figure 7(a) shows the wavelength shift results when the torsion changed from −94.2 to 94.2 rad/m in an interval of 15.7 rad/m in the 1.2 m twist segment between 58.0 and 59.2 m along the helical MCF. When the helical fiber is additionally twisted in the same direction as the initial twist bias of the helical fiber, i.e. CW in our experiment, the additionally applied torsion will elongate the outer helical cores; the additional torsion in the opposite sense compresses these outer cores. In a nutshell, CW and CCW torsions in our experiment will equivalently stretch and compress the outer helical cores, respectably. Finally, the peak wavelengths of the cross-correlation exhibit a redshift and blue shift, as shown in Fig. 7(a). Such opposite phenomenon of wavelength shift can be used for distinguishing the torsion direction. The capability of directional torsion sensing is the result of the initial twist bias of the helical fiber. Figure 7(b) illustrates the wavelength shift as a function of the torsion. It is evident that torsion sensitivity is different between CCW (∼0.604 pm/(rad/m)) and CW (∼1.096 pm/(rad/m)) directions. Such nonlinear trend is proved by the theoretical simulation according the Eq. (7).

Figure 6(b) illustrates the mechanical configuration for implementing the distributed torsions measurement. A 1.2 m long helical MCF is equally divided into three segments by four rotators, namely A, B, C, and D. To apply the distributed torsions, Rotator A and C were fixed and did not rotate. Only Rotator B and D were rotated to different angles in both CW and CCW directions, and the experimentally applied torsions of all segments are listed in Table 1. In case 1, only Rotator B is rotated by 720 degrees in CCW direction, and hence the 1^{st} segment suffered a torsion of −31.4 rad/m in CCW direction, while the 2^{nd} segment was exposed to the oppositely signed rotations, i.e. 2^{nd} segment in a CW direction of 31.4 rad/m. In cases of 2, 3, and 4, the rotator B was rotated by 1080 degrees in CCW direction, and hence the 1^{st} and 2^{nd} segments were twisted by a torsion of −47.1 rad/m and 47.1 rad/m in CCW and CW directions, respectively. In later 3 cases, rotator D was sequentially rotated by 720, 1080, and 1800 degrees in a CW direction, giving a torsion of 31.4 rad/m, 47.1 rad/m and 78.5 rad/m in 3^{rd} segment, respectively.

Figure 8 illustrates the measured wavelength shifts along the 1.2 m long helical MCF using the OFDR system. Firstly, it is clear to identify the torsion direction of each sensing segment according to the sign of wavelength shift. The positive wavelength shift presents a torsion in CW direction, and vice versa. Secondly, we obtained the measured torsion by inserting the wavelength shift into Eq. (9), and the sensitivities of *S* are 1.096 and 0.604 pm/(rad/m) corresponding to CW and CCW directions. Finally, the calculated torsions are also listed in Table 1 for comparison with the prior imposed torsion values, which proves that the measured torsions are in a good agreement with the mechanically preset values, and the maximum fluctuation is 3.51 rad/m. Additionally, it is interesting to find that there is always a wavelength dip approaching to zero at the location of Rotator C. This phenomenon is in accordance with the fact that a section fiber around 1 cm long was fixed by glues at all rotators and did not produce any torsion. Therefore, the wavelength approaches to zero at the location of Rotator C. The discrimination of the short-untwisted section fiber at the Rotator C benefits from the high spatial resolution of the OFDR system, i.e. 9.4 mm separation between adjacent sensing points. It is expected to obtain a higher spatial resolution by increasing the wavelength sweeping range.

## 5. Discussion

As presented above, we have demonstrated a promising, fully distributed torsion sensing configuration with twist direction discrimination, even though the experimental torsion sensitivity is relatively low and nonlinear in the CW and CCW directions. In this section, we will discuss the sensitivity and the capability of recognizing the torsion direction. Firstly, the torsion sensitivity is directly related to the pitch of the helical MCF, according to Eq. (10). We tested the torsion sensing performance by using our post-processed fiber with helical pitch of 6 mm. Like the previous experimental results of the fiber with a helical pitch of 15.4 mm, we can identify the torsion direction through the direction of the wavelength shift. The important difference is that the wavelength shift of the fiber with a pitch of 6 mm exhibits a symmetrical shift between CW and CCW torsion directions, as shown in Fig. 9(a). Furthermore, the wavelength shift has a linear response in the whole torsion range from −36 to 36 rad/m. The sensitivity is around 1.9 pm/(rad/m), as shown in Fig. 9(b), agreeing well with the theoretical calculation of 1.6 pm/(rad/m) in Eq. (10).

To discuss the capability of recognizing the torsion direction, we implemented a comparative experiment by using the fiber with straight fiber cores. The experimental results are shown in Fig. 10. It is found that a red wavelength shift is always observed whatever the torsion direction is. More importantly, the wavelength shift is parabolic with the torsion, which also proved our theoretical predict in Eq. (8). In short, the capability of recognizing the torsion direction stems from the fiber design with central-offset helical fiber cores. Furthermore, the shorter the helical pitch, the higher the sensitivity.

In our experiment, we used the multi-core fiber to experimentally demonstrate the proposed sensor scheme, but it is worth noting that the key idea of distributed torsion sensor with direction recognition ability is the optical fiber with eccentric spiral structure. Here, the use of multi-core fiber not only verifies our proposed sensors successfully but also provides a basic study and fiber design guidance for full 3-D optical fiber shape sensing. One problem for 3-D shape sensing is that the tether fiber is likely to experience not only bending, but also varying amounts of torsion along its length. A potential method to separate the bending and torsion is the use of helical multi-core fiber [25]. Our theoretical torsion model and the sensitivity study should be useful to design helical MCF for 3-D shape sensing.

In this paper, we just used the wavelength shift of the Rayleigh scattering spectrum to demonstrate the torsion sensing. Besides the wavelength shift induced by the shear strain, the state of the polarization undoubtedly changes and hence induces polarization fading effect to the Rayleigh scattering spectrum, which will influence the detection of the wavelength shift. In order to suppress this effect and ensure the measurement stability, we have used the polarization diversity receiver in the system as shown in Fig. 3. On the other hand, we also can measure the polarization variation by slightly changing our system to stablish a polarization-sensitive OFDR system [21].

## 6. Conclusion

In summary, we proposed and implemented a distributed directional torsion sensor system by using the OFDR technology and the helical MCF. Both theoretical and experimental results enforce that the central-offset fiber cores with a helical structure in the MCF endows the torsion sensor with the direction recognition ability. Furthermore, the shorter pitch gives higher sensitivity. By employing the OFDR technology, a distributed directional torsion sensor is experimentally demonstrated with a linear sensitivity of 1.9 pm/(rad/m) and an adjacent sensing points separation of 9.4 mm.

## Funding

National Natural Science Foundation of China (61975022, 61905030); Fundamental Research Funds for the Central Universities (2019CDQYGD038).

## Disclosures

The authors declare that there are no conflicts of interest related to this article.

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