Temperature-Insensitive Two-Dimensional Vector Bending Sensor Based on Fabry-Pérot Interferometer Incorporating a Seven-Core Fiber

We proposed and experimentally demonstrated a temperature-insensitive two-dimensional (2D) vector bending sensor on the basis of Fabry-Pérot Interferometer (FPI). The sensor is fabricated by splicing a section of silica capillary optical fiber (COF) between a trench-assisted seven-core fiber (SCF) and a multimode fiber (MMF). For orientation angle from 0° to 360°, the vector bending responses of six outer-core FPIs were studied at a step of 30°. From experimental results, the proposed sensor exhibits strong bending-direction dependence, and the maximum sensitivity is 200.6 pm/m−1. The curvature magnitude and orientation angle can be reconstructed through the dip wavelength shifts of any two off-diagonal outer-core FPIs. Hence, there are 12 various combinations from six outer-core FPIs, and the average value can be taken in multiple reconstructions to develop exact 2D vector bending measurement. For curvature magnitude and orientation angle reconstructed, the average relative error is 6.2% and 3.3%, respectively. In addition, experiments verified the proposed sensor is insensitive to ambient temperature changes.

Among the above curvature sensors, some can be used in two-dimensional (2D) curvature measurement by writing gratings [5]- [12]. Compared with one-dimensional (1D) curvature measurement that can only distinguish positive or negative directions, 2D curvature sensors can tell any curvature direction from 0°to 360°. However, besides high-powered lasers such as UV, femtosecond, and CO 2 lasers, the writing process of gratings is complex, which requires accurate alignment technique. In addition, some of these 2D sensors are created with serial FBGs, and the packaging of these FBGs is also very challenging.
In contrast, fiber optic interferometers have the advantages of simple fabrication, high sensitivity, and versatility [26]. Among various interferometer-based fiber sensors, FPI sensors have been demonstrated in different configurations, for examples, producing an air bubble between two sections of multimode fibers (MMFs) [27] or between single-mode fibers (SMFs) [28], creating the hollow cavities through chemical etching [29], or various laser ablation technologies, containing excimer lasers and femtosecond lasers [26], and utilizing a cold splicing approach based on resin to produce FP cavity [30]- [31]. However, these configurations always depend on challenging manufacturing processes, using dangerous chemicals, or expensive This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see https://creativecommons.org/licenses/by/4.0/ high-powered lasers. And when the FPI cavity is filled with a resin, it will show high sensitivity to environmental temperature variations, and due to the uneven filling of the resin, the compensation process of temperature effect becomes complicated. Recently, the formation of FPI using a silica capillary optical fiber (COF) has been studied extensively [32]- [34]. They can be used to monitor a variety of parameters, such as, refractive index, strain, and magnetic field. Due to the low thermal-optical coefficient of air in the cavity, these sensors are insensitive to temperature. Moreover, in the process of manufacturing, only cutting and splicing are required, so they have the superiorities of high repeatability, low cost, and simple fabrication. However, the current FPI curvature sensor based on COF can only be used for 1D curvature measurement [35].
In this work, we present and experimentally confirm a temperature-insensitive 2D vector bending sensor based on FPI. The proposed sensor is made through splicing a section of COF between the MMF and trench-assisted seven-core fiber (SCF). The SCF-COF and COF-MMF interfaces act as two FPIs reflectors. The reflection spectra of six outer-core-based FPIs are monitored to measure curvature from different fiber orientation angles. The maximum sensitivity of 200.6 pm/m −1 is obtained, which is 3 times higher than that of a FBG vector bending sensor [8]. The orientation angle and curvature magnitude can be reconstructed through the dip wavelength shifts of any two off-diagonal outer-core FPIs. In addition, since the low thermal-optical coefficient of air, the proposed sensor is nearly insensitive to temperature, which has also been verified by temperature experiments.

II. SENSOR FABRICATION AND MEASUREMENT PRINCIPLE
The proposed 2D vector bending sensor is created via splicing a section of COF between the SCF and MMF, as illustrated in Fig. 1. The SCF-COF and COF-MMF interfaces act as two FPIs reflectors. As a result, in the configuration, seven separate FPIs can be produced. Fig. 2(a) reveals a micrograph of the cross section of SCF (YOFC, MC1010-A, China), with a hexagonal arrangement of six outer cores and one central core. The diameter of SCF is 150 μm and between neighboring cores, the distance is 42 μm. As a separate channel, each core possesses three claddings: outer, the secondary, and the inner-field cladding,  with diameters of 30, 19, and 9 μm, respectively. The micrograph of COF (TSP100170) cross section is shown in Fig. 2(b), where the diameters of cladding and air hole respectively are 140 and 100 μm. For the MMF, its core diameter and cladding diameter are 105 and 125 μm, respectively, which is just as one reflector of the FPIs. It is noted that when the fusion splicer (ATOMOWAVE SFS-A60+) is utilized to splice three fibers, manual mode and multiple weak discharges were employed for preventing the air hole collapse in COF [23]. The fusion mode was set as photonic crystal fiber (PCF)-SMF, the discharge power was lower than the standard value of 100 bit. Discharge stop time, discharge time, as well as duration time was 500 ms, 300 ms, and 2 seconds, respectively, that is, a total of three discharges, as shown in Table I. The microscope image of this proposed structure is shown in Fig. 2(c). The COF length is about 225 μm.
On the basis of two-beam interference model, for the output interference light intensity, the expression can be displayed as [36]: where the I 1,i and I 2,i represent the light intensities reflected through two interfaces, i denotes the core number of the SCF shown in Fig. 1(a), and ϕ denotes the difference of optical phase between two beams, which can be expressed with [36]: in which L i refers to the i th FPI cavity length, n 0 represents the air refractive index, and λ is the wavelength of light in vacuum. When conforming ϕ = (2m i + 1)π,(m i = 0, 1, 2 . . .), for the reflection spectrum on the basis of FPI, the dip wavelength can be described as [36]: where m i is constant depending on the length of FP cavity, indicating the m i th resonant wavelength. According to (3), the distance between two adjacent interference wavelengths, that is, the free spectral range (FSR), can be expressed as [33]: From (3), we can clearly see that in reflection spectrum, the dip wavelength is determined by the cavity length L i and refractive index n 0 , thus, any change in the above two parameters will cause the shifts of dip wavelength. When strain occurs, the corresponding dip wavelength shifts in the FPI reflection spectra can be written as [30]: where ε i = ΔL i /L i is the locally-induced strain in i th core. Fig. 3 reveals the distribution of seven cores. If the presented sensor bends within a specific fiber orientation angle, various strains will be generated in each fiber core. Although some cores are likely to undergo the local compressive strain, others are probably to be elongated locally, determining by the relative angular position between the local bending axis and a specific core. Especially, the strain induced by bending in the core i along the SCF at any point can be displayed as [37]: in which d is the distance between outer cores and center core, C is the curvature magnitude, θ ν can be defined as the angular offset between 0°(optional as reference) and the bending direction of fiber and θ i denotes the angle between 0°and i th core, as reflected in Fig. 2. Replacing (6) into (5), the (5) can be described as: Due to the low elasticity-optical coefficient of air, Δn 0 /n 0 can be neglected. Depending on the position of the core i on the fiber and the fiber rotation angle, the dip wavelength will be red-or blue-shifted. In addition, the absolute value of (7) is maximum when θ i and θ ν are equal or the difference is π, that is, the direction applied by the curvature is exactly the same as or opposite to the core position. The value of (7) is zero when the core position is orthogonal to the bending direction, that is, the difference between θ i and θ ν is π/2 or 3π/2. However, the center core (core 1) will not be affected by the curvature [8]. The temperature dependence can be expressed by the following formula [30]: among which, in parentheses, first term represents the normalized thermal-optical coefficient (β), and the second term refers to the normalized thermal-expansion coefficient (α). Since the thermal-optical coefficient in air is low, the presented sensor should have low sensitivity of temperature.

III. TWO-DIMENSIONAL VECTOR BENDING EXPERIMENTS
Before conducting the curvature measurement experiment, the reflection spectra of seven FPIs are measured first as shown in Fig. 4. According to Fig. 4, the loss values of FPIs are different. This may be caused by using ordinary cutting knife that will lead to unevenness end face of SCF, and slight offset will also be introduced in the process of manually alignment in splicing the SCF and COF. Moreover, the measured values of the FSR of the seven-core FPIs near 1550 nm are 5.32, 5.53, 5.49, 5.39, 5.39, 5.63, and 5.53 nm, respectively, which agree well with the theoretical calculated value 5.34 nm. It can be observed that between the FSRs in seven FPIs, there exist a small difference. The differences between measured and theoretical FSR values are mainly owing to the COF air hole was slightly squeezed in splicing the COF and SCF, and can be discovered from Fig. 2(c).
The 2D bending response for the presented sensor was studied with the experimental device reflected in Fig. 5. An amplified spontaneous emission with a range of 1508-1583 nm (ASE, MAX-RAY ASE-CL-20-B-T) as a light source into the port 1 of the optical circulator (OC). The port 2 of OC connects with a fanin/out device, which was spliced with the sensor head mounted on two translation stages by two rotary fiber holders. Finally, an optical spectrum analyzer (OSA, YOKOGAWA AQ6370D) was utilized to monitor reflection spectra with 0.02 nm resolution and linked with port 3 of OC. The stage on the left is fixed, and the other on the right can move controllably inward to raise curvature value. The curvature expression on the basis of the experimental device can be expressed as [17]: in which R is the radius of bending fiber, x is the displacement of stage on the right, and l = 17 cm is the original distance between two stages. The bending characteristics are achieved through moving the stage on the right in steps of 0.8 mm at a time, allowing the application of curvature, which reaches 5.93 m −1 . For each of the curvature step, the reflection spectra were gathered. Via rotating two rotary optical fiber holders at the same time, the bending characterization of each optical fiber orientation angle was performed in steps of 30°between 0°a nd 360°. In order to prevent the sensor head twisting during the experimental measurements, we put the sensor head into a segment of capillary to increase its weight. Experiments have verified the direction dependence of the presented sensor. As shown in Fig. 6(a), θ is defined as the fiber orientation angle, and θ = 0°is the initial orientation angle of the sensor head. Keeping the fiber orientation angle, the dip wavelength shifts of six outer cores were recorded with applied curvatures, which are shown in Fig. 7(a). We can see from Fig. 7(a) that the FPIs corresponding to cores 2, 3, and 7 are in the compression, while which of cores 4, 5, and 6 are in tension, respectively displaying negative or positive dip wavelength shifts. In the consideration of SCF is completely symmetrical and cores 2 and 5 are 42 μm away from central core, their sensitivities should be equal. The values acquired in the experiment respectively are −157.3 and 155.8 pm/m −1 for cores 2 and 5, with similar magnitudes. For the cores along other diagonal lines, including cores 3 and 6 as well as cores 4 and 7, the sensitivities were −126.6, 132.8, 16.4, and −29.9 pm/m −1 , respectively. In summary, the cores along the same diagonal lines have the similar sensitivities in magnitude, which are lower than that obtained for cores 2 and 5. According to the (7), the core with larger θ i will have smaller wavelength shifts, proving that the experimental responses are in agreement with theoretical analysis above. Then rotating the sensor head by 30°(θ) using the rotary fiber holders, the SCF cross section reflected in Fig. 6(b) can be obtained. The dip wavelength shifts of six outer cores under the same curvature were recorded and shown in Fig. 7(b). The results in Fig. 7(b) show that when θ, the fiber orientation angle varies, for each core, the response also changes with the strain of radial position of each core (reflected in (7)). When θ = 60°shown in Fig. 7(c), we can see that the SCF core-distribution is the same as in Fig. 6(a), but the location of the original core is different. As a result, cores 7 and 4 should possess similar curvature sensitivity amplitudes. In accordance with the symmetrical arrangement of the outer core, their sensitivities shall be similar to that of cores 2 and 5 (θ = 0°), respectively. And the sensitivities of other cores should be lower in magnitude than cores 4 and 7. From the experimental results, as shown in Fig. 7(c), cores 4 and 7 were 145.3 and −178.1 pm/m −1 , respectively, which were slightly different between the two cores and from cores 5 and 2 (θ = 0°). This small difference may be caused by irregularly cutting fiber ends and slight deformation of COF air hole when splicing COF and SCF during sensor manufacturing. In addition, the sensitivities of cores 2, 3, 5, and 6 shown in Fig. 6(c) were  −132.7, 1.3, 135.5, and −4.3 pm/m −1 , respectively, which had a lower magnitude in comparison with that in cores 4 and 7. Therefore, the experimental results were consistent with the above analysis. Similarly, when θ = 90°, as shown in Fig. 7(d), the SCF position was the same with Fig. 7(b), but the positions of i th core were different. Therefore, the curvature sensitivity of each core in Fig. 7(d) should be the same as the curvature sensitivity of the corresponding core located in the same angular position in Fig. 7(b). And the data represented in the Fig. 7(b) and 7(d) showed that our analysis were consistent with the experimental results.
Accordingly, in order to clearly see the curvature characteristics of each core at different fiber rotation angles, the outcomes of two FPIs along the identical diagonal (cores 2 and 5) are displayed in the Fig. 8(a) and 8(b). Linear fittings are applied to the dip wavelength shifts at given fiber orientation angles θ, and the slope denotes the curvature sensitivity. The outcomes suggest that for the FPI based on two outer cores along an identical diagonal line, curvature sensitivities possess an opposite dip wavelength shifts and similar amplitude at any orientation angle θ. For a fixed core, with different orientation angles θ, the dip wavelength shifts are different (red-or blue-shifted), this is mainly due to the change of the core position (θ i ), which will lead to the change of dip wavelength shifts in (7). Fig. 8(c) and 8(d) reveal that the dip wavelength of cores 2 and 5 changes as the orientation angles θ increases. All the changes of dip wavelength under a specific curvature exhibit good consistency with the sinusoidal behavior [8]. Fig. 9 plots the data of cores 2, 3, and 4 in polar coordinates, and clearly reflects the symmetrical "8"-shaped distributions, in which it is easy to see the angle difference of 60°. The directions of the maximum curvature sensitivities are 85°, 145°, and 205°for cores 4, 3, and 2, respectively. Therefore, the original 0°can be acquired through the calculation on the basis of the determined responses [39].
With an aim of understanding the maximum curvature sensitivity S i , the sinusoidal curves were fitted into the data of experiment in Fig. 10 to obtain the following values: 173.8, 167.7, −172.8, −173.0, −200.6, and −181.7 pm/m −1 , for cores 2, 3, 4, 5, 6, and 7, respectively. It is 3 times higher than the one reported (59.47 pm/m −1 ) for FBGs written on a similar SCF [8], and 2 times higher than the one reported (81.5 pm/m −1 ) for 1D bending vector sensor on the basis of the similar air cavity FPI [35]. The curvature sensitivities of each outer-core FPI are slightly different, due to errors produced during fiber cleaving and splicing. Additionally, we can clearly see that between cores 2 and 4, the change of angle is 120°, which is in accordance with the angle separation of 120°in SCF. And the angle change for cores 2 and 3 is in accordance with the angle separation of 60°i n the SCF.
In addition, increasing and decreasing curvatures experiments were performed to verify the proposed sensor is repeatable. Fig. 11 shows that linear fit curves of the dip wavelength shifts of the six outer cores with increasing and decreasing curvatures at a fixed orientation angle. Experimental results show that the proposed sensor has good curvature repeatability.
Besides above experiments, we have also measured the temperature effect of the seven independent FPIs, ranging from 30°C to 100°C. The dip wavelength variations of seven FPIs with increasing temperatures are displayed in Fig. 12. For seven FPIs, their temperature sensitivities are 0.40, 0.67, 0.39, 0.57, 0.29, 0.52, and 0.20 pm/°C, respectively. It can be seen that there are some differences between the temperature sensitivities of the seven FPIs. This is due to the uneven cutting end face of the SCF during the manufacturing process, resulting in some slight differences between the cavity lengths of the seven FPIs. However, this does not affect the measurement of 2D vector bending. In comparison with 200.6 pm/m −1 of the maximum    curvature sensitivity, the cross sensitivity between temperature and curvature is calculated as temperature sensitivity divided by curvature sensitivity [40], which is only 0.0009 m −1 /°C.

IV. CURVATURE MAGNITUDE AND ORIENTATION ANGLE RECONSTRUCTION
The correlation between orientation angle and curvature sensitivity for each outer-core FPI can be fitted with the sinusoidal function, as displayed in Fig. 10. The fitting equation is described as [6]: where S i represents the curvature sensitivity in direction θ, and y i , A i , and θ i are constants which acquired from Fig. 10. Any two functions of off-diagonal outer cores from Fig. 10, such as cores 2 and 4, can form a matrix as follows [6]: where Δλ 2 and Δλ 4 are the dip wavelength shifts of cores 2 and 4 FPIs, respectively. C represents the curvature magnitude of presented sensor in direction θ. Δλ 2 and Δλ 4 can be obtained by curvature measuring. Therefore, the orientation angle θ and curvature magnitude C can be acquired with (11). A set of sampling data was applied to detect the reconstruction property for seven-core FPI. For a specific set of conditions (θ = 30°a nd C = 4.8 m −1 ), for six outer-core FPIs, their detected dip wavelength shifts (Δλ i ) are −0.72, −0.32, 0.43, 0.71, 0.44 and −0.58 nm, respectively. The curvature magnitude C and orientation angle θ can be reconstructed by (11). There are 12 different combinations in total, as shown in Table II [38], revealing excellent reliability and high accuracy of the method for reconstructing the orientation angle and curvature magnitude.
In summary, structures similar to the proposed sensor have not been studied before. Although its sensing principle is similar to [30], due to the low thermo-optical coefficient of air, the problem of temperature compensation can be ignored. At the same time, only cutting and welding techniques are required in manufacturing, so the cost is low. Compared with the grating writing techniques [8], [15], expensive lasers and precise alignment techniques are not required. In addition, compared with the fused polymer optical fiber [14], the used optical fiber can be easily obtained and does not need to be damaged by the fiber fusing. Curvature sensitivity of the proposed sensor is high and the relative errors of curvature magnitude and orientation angle reconstruction are small. It has important research prospects in 2D vector bending measurement.

V. CONCLUSION
We have presented and experimentally confirmed a temperature-insensitive 2D vector bending sensor in a SCF. In all 360°directions, for the six outer-core FPIs, their bending responses were studied at a step of 30°. The curvature sensitivities of six outer-core FPIs have strong angular dependences, and the maximum sensitivity is 200.6 pm/m −1 . It is about 3 times higher than the one (59.47 pm/m −1 ) based on FBGs written on a similar SCF [8], and 2 times higher than the one (81.5 pm/m −1 ) reported for 1D bending vector sensor on the basis of similar air cavity FPI [35]. Moreover, the temperature effect influence on the proposed sensor can be nearly ignored. The curvature magnitude and orientation angle could be reconstructed utilizing the obtained outer-core FPI dip wavelength shifts and fitted sine functions of curvature sensitivity against orientation angle. There are 12 different combinations which can be used for reconstructing the curvature magnitude C and orientation angle θ. For a given set of orientation angle and curvature magnitude (30°, 4.8 m −1 ), the reconstructed average value of 12 various combinations (29°, 4.4 m −1 ) has a relative error of 6.2% for curvature magnitude C and 3.3% for the orientation angle θ. The temperature-insensitive 2D vector bending sensor has significant potential in fiber shape sensing applications.