1. Introduction

Fiber Bragg grating (FBG) sensors have been widely used to measure mechanical properties such as strain, tilt, or curvature since the FBG sensors offer a variety of advantages over conventional sensors [1]. For example, the FBG sensors are ultra-thin and ultra-light so that they can be easily applied to objects to be measured. In addition, the FBG sensors are immune to electromagnetic interference and also allow rapid measurement. Specifically, sensing curvatures using the FBG sensors has gained great attention in various applications from small scale to large scale: e.g., biomedical applications and structural health monitoring. To measure the curvatures of structures, the FBG sensors primarily detect the change of Bragg wavelength of the FBGs, induced by mechanical deformation (either elongation or compress) of optical fibers incorporating the FBGs. The curvatures are then estimated by a relationship between a Bragg wavelength shift and curvature [2,3]. However, a considerable level of deformation of the structures may lead to the permanent damage or breakage of the sensors due to the rigidity of the optical fibers [2].

To overcome the limitation of the FBG sensors entailing axial deformation of the optical fibers, different structures of FBG sensors have been widely investigated. For example, Baek et al. proposed a bending type of sensors incorporating tilted fiber Bragg gratings (TFBGs) [4]. The TFBGs lead to the change of the intensity of the transmitted light through the cladding of an optical fiber according to bending, while preserving the light reflected back through the core. Since the optical fiber with the TFBGs itself acts as a bending sensor, the sensor can measure the high curvature of bending up to 20 m⁻¹ with high durability. However, use of the sensor in various applications is still limited due to its principle and measurement configuration. The transmitted light through the cladding should be measured at one end of the fiber, while the other end is used for the entrance of the source light. Furthermore, the asymmetric structure of the TFBGs results in different sensitivities even for the same magnitude of curvatures created in different directions. Hence, the direction of bending should be specified, and an appropriate calibration for the specific direction of bending is required prior to use of the sensor.

These issues with cladding-mode Bragg gratings can be addressed by placing a short length of a multi-mode fiber between two single-mode fibers via splicing. It allows the light passing the cladding to be reflected back to the same end of the fiber used for the entrance of the source light [5,6]. However, the direction of bending could not be differentiated from each other due to the cylindrically symmetric structure of the sensors. In addition, the measurable curvature is limited below 2.5 m⁻¹ as the intensity of the light passing cladding-mode Bragg gratings becomes very low at the high curvature of bending [5].

The direction of bending can be differentiated by attaching an optical fiber with FBGs either on the top or the bottom of a bending plate such as a metal plate or a carbon-fiber-reinforced plastic [7–9]. The FBGs is thus subjected to either increase or decrease of the Bragg wavelength according to the tensile or compressive surface strain induced on the structure. In addition, these sensors have little power loss in detecting the light reflected from the FBGs even at high curvature, and also take into account the wavelength shift rather than directly detecting the intensity change of light. However, these types of sensors may not be suitable for measuring the high curvature of bending, since the sensitivities of the sensors are primarily dominated by the material properties and shapes of base plates used rather than optical fibers. For instance, the Young’s moduli of the base plates are typically higher than the moduli of optical fibers, and the base plates are thicker than the diameters of the optical fibers. The neutral axis of bending is thus nearly located at the center of the base plate, which leads to a large offset distance between the FBGs and the neutral axis of bending; the neutral axis is described as an axis in the cross section of a beam, where no longitudinal stresses or strains are applied. As a result, even small deflection of bending may lead to relatively high bending strain on the FBGs due to the large offset. It may either cause the unstable measurement of the wavelength spectrum or the breakage of the fiber.

These issues could be mitigated by embedding an optical fiber with FBGs in silicone rubber sheets [10,11]. Accordingly, use of soft base materials enable to measure high curvature up to 80 m⁻¹. However, such soft base materials should have a certain level of thickness to yield a satisfactory level of sensitivity. Otherwise, the neutral axis of bending would be created near the center of the optical fiber, which leads to little change in the Bragg wavelength. Consequently, such a thick sensor would still limit to maximum allowable curvatures.

In this paper, we present a novel FBG curvature sensor that can measure high curvature up to 200 m⁻¹. To measure such high curvature, the sensor is designed to have an ultra-thin structure with the similar modulus of an optical fiber used, while precisely managing the location of the FBG fiber from the neutral axis of the sensor. We investigate the sensitivity and maximum measurable curvature for different sensor designs that lead to different offset distances of the FBG fiber from the neutral axis of each sensor. Furthermore, the curvature sensor with multiple FBGs in a single fiber allows the accurate shape reconstruction of multi-bend structures. Finally, the shape reconstruction of various objects is successfully demonstrated using the proposed FBG sensor.

2. Sensor principle and design

2.1 Fiber Bragg grating (FBG) principle

A fiber Bragg grating (FBG) reflects the specific wavelength of light depending on the periodic grating pitch as in Eq. (1).

The stain induced on an FBG is defined by a beam bending theory, in which the strain is proportional to the curvature of bending and also to the distance offset of the optical fiber from the neutral axis:

2.2 Sensor design and fabrication

To design an ultra-high curvature sensor with an FBG, it is important to manage the appropriate minimal offset distance of the FBG, $d$. For instance, smaller offset distance the sensor has, larger high curvature it can detect for the same amount of the Bragg wavelength shift. Meanwhile, the offset should be greater than a certain level of distance to be sensitive enough to strain induced on the FBG node as in Eq. (3). In addition, the sensor should be flexible enough to be bent at high curvature without breakage.

Hence, we propose a curvature sensor embedded in a material of which the Young’s modulus is similar to an optical fiber used as shown in Fig. 1. This design would be suitable for measuring high curvature by minimizing the overall thickness of the sensor for ensuring flexibility, and also by precisely managing the offset distance from the neutral axis. Epotek Epoxy 301 (Epoxy Technology Inc. Billerica, MA) is used for a molding material, which offers the similar Young’s modulus to the optical fiber used.

 

Fig. 1 Illustration of the FBG-embedded curvature sensor. Inset on the left shows the cross-section of the curvature sensor.

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We define the minimal thickness of the sensor, $t_{min}$, to provide a satisfactory level of sensitivity with high structural flexibility in curvature sensing as in Eq. (5).

We design three types of sensors to investigate the effects of the offset distance on the sensitivity and the maximum allowable curvature. The offset distance of the sensors varies from 20 µm to 100 µm, while the width is set to be 1.8 mm for all sensors. Such a high aspect ratio in the cross-section of the sensors is designed to allow only in-plane bending, while minimizing twist of the sensors. The specifications of the sensors are summarized in Table 1.

Table 1. Specification of fabricated curvature sensors.

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The proposed sensors are fabricated using jig fixtures illustrated in Fig. 2. The jig fixtures assist in aligning the optical fiber in the middle of the sensor, and also adjusting the overall thickness of the sensors. The fabrication procedure follows four steps: (1) align an optical fiber with FBGs on the main jig fixture, (2) fill the epoxy into a gap that determines the thickness of the sensor, and (3-4) cure and detach the sensor from the jig fixtures. Fiber holders and aligners are used to position the fiber in the middle of the main jig fixture. The holders are located above the bottom of the sensor by 15 µm as described as the offset $d_o$, in order to float the optical fiber from the bottom. To maintain the consistent offset distance of the FBG nodes along the sensor, a pre-strain of about 0.87 mε is applied to the optical fiber during the fabrication. After assembly of the jigs, the epoxy is filled into the gap using a vacuum chamber, and then cured in an oven at 65°C for 2 hours. As the optical fiber is fully enclosed with the epoxy, it could prevent the fiber from sliding or dislocation of the fiber.

 

Fig. 2 Fabrication procedure of the FBG curvature sensor.

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The FBGs (FBGS, Geel, Belgium) are inscribed in the optical fibers via a draw tower grating (DTG) technology. Unlike common FBG fabrication processes, the DTG technology offers the uniform diameter of an FBG fiber, since it does not necessitate recoating the fiber after writing Bragg gratings. As a result, it allows maintaining the consistent offset distance along the sensor during fabrication. A single curvature sensor has three fiber Bragg gratings 4 mm long, which is regularly spaced at an interval of 20 mm in an optical fiber of 130-µm diameter with ORMOCER coating; the diameter of cladding is 80μm. To detect the wavelength spectrum of the reflected light from the FBGs, we use an optical interrogator system (SmartScan SBI, Smartfibres, Bracknell, UK) composed of a tunable laser, a circulator and a photodiode as shown in Fig. 3(a).

 

Fig. 3 (a) The measurement setup for the FBG curvature sensor. (b) The spectrum data during three kinds of deformation, upper and down bending and straight state.

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Since the optical fiber is located below the neutral axis of the sensor, positive bending (upper bending) yields extension of the fiber. Accordingly, it lengthens the Bragg grating pitch of the FBG, which shifts the peak wavelength of the FBG toward longer wavelength as depicted in Fig. 3(b). On the other hand, negative bending (down bending) leads to compression of the fiber. It thus shortens the grating pitch, as resulting in the lower peak wavelength compared to the nominal peak wavelength. In addition, the amount of the peak shift varies upon the magnitude of curvature. Therefore, we can measure both the magnitude and the direction of curvature by detecting the shift of the peak wavelength of the FBG using the interrogator.

2.3 Multi-bend curvature sensor

We also present a multi-bend curvature sensor that can measure its entire shape in a 2-D space. The sensor incorporates seven fiber Bragg gratings in a single optical fiber with an interval of 20 mm. The overall dimension of the sensor is 280 µm thin and 120 mm long. The seven FBGs can be simultaneously detected, as each node has a different nominal Bragg wavelength.

To reconstruct the shape of the sensor from curvatures detected at the multiple FBG nodes, we adopt the Frenet-Serret formulas, which describes a particle moving along a continuous curve with three orthonormal bases in ${ℝ}^3$ [12]. The formulas are as follows:

The shape of the sensor is however accurately reconstructed only when correct curvatures are given for the curve. Otherwise, position error is accumulated while integrating the unit tangent vectors over the total length of the sensor, which results in erroneous reconstruction. Therefore, it is essential to find the correct offset distance of each FBG node described in Eq. (4) for the accurate reconstruction. The offset distance $d_i$ at the ith node is thus found prior to the 2-D reconstruction of the multi-bend sensor via a calibration procedure introduced as below. For calibration, we first bend the FBG sensor upward and downward with any known curvatures ${\kappa }_s^{FBG}$. For a specific curvature ${\kappa }_s^{FBGs}$, deflections to the up and the down result in the strains on the FBG node, ${\epsilon }_i^{up}$ and ${\epsilon }_i^{down}$, respectively.

3. Results and discussion

3.1 Evaluation of curvature sensor

First, we evaluated the structural precision of the three types of the sensors listed in Table 1 by taking the cross-sectional photomicrographs of the sensors as shown in Fig. 4(a). For the evaluation, we fabricated the total three sensors for each design and measured the thickness and offset distance at various locations of each sensor; the locations were regularly spaced by 10 mm for a length of 80 mm. The values obtained at each location were averaged for each sensor design, and the overall mean values were also calculated regardless of the locations.

 

Fig. 4 (a) Cross-sectional photomicrographs of the curvature sensors of which thickness. (b) Average thicknesses at the various locations of the sensors with error bars (n = 3); Dotted lines indicate the target values—200, 280 and 360 µm. (c) Average offset distances at the various locations with error bars; the design values—20, 60, 100 are depicted as dotted lines.

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The overall thicknesses were measured to be 197.4 ± 4.1, 276.3 ± 3.9, and 363.9 ± 4.7 µm for the sensors designed for thicknesses of 200, 280, and 360 µm, respectively in Fig. 4(b). The deviations from the target thicknesses at the locations varied within a range of 10 µm for all three sensors, which results in the maximum 4.2% error in thickness for the sensor of the 200-µm thickness. We obtained the overall offset distances as 18.8 ± 4.1, 59.2 ± 5.0, and 99.2 ± 5.0 µm for the distances targeted to 20, 60, and 100 µm, respectively in Fig. 4(c). We observed the similar patterns of deviation for the offset distances; the distances also varied within a range of 10 µm. However, it leads to relatively large error in the offset distance because the offset distances are much smaller than the overall thicknesses. Accordingly, the maximum error was found to be 38% in the sensor of the 200-µm thickness. These variations could have occurred, because the optical fiber was not tightly pulled while fixing it, in order to minimize the application of initial tensile strain on the fiber during fabrication. Besides, it could also have been made due to the thermal expansion of the fiber while curing the epoxy.

Next, we investigated the performance of the curvature sensors in terms of sensitivity and maximum measurable curvature. The peak shifts of the Bragg wavelengths of the FBG nodes were measured for various curvatures from 10 m⁻¹ to 200 m⁻¹: for the total eight radii of curvatures–100, 80, 50, 30, 20, 10, 7, and 5 mm. In addition, the sensors bent both upward and downward directions, which resulted in the positive and the negative wavelength shift, respectively. The measurement was repeated and averaged for three trials for each curvature.

The maximum measurable curvature of a curvature sensor was determined by the minimum allowable radius of curvature at which the peak shift of the Bragg wavelength could be reliably detected by the optical interrogator, as the radius of curvature decreased. As a result, the sensors with thicknesses of 200 and 280 µm could measure curvature up to 200 m⁻¹, whereas the sensor with a thickness of 360 µm only up to 140 m⁻¹. It is noted that the material properties of the optical fiber and the epoxy and/or optical loss may limit the maximum measurable curvature. For example, the optical fiber itself can tolerate curvature greater than 1000 m⁻¹ without breakage. Such a high curvature, however, may lead to a considerable level of optical loss in the fiber due to the significant drop in light intensity to be detected. Besides, the maximum curvatures may also be determined by the elastic limit of the epoxy used as the based material of the sensors. Given the elastic as 0.03ε, the maximum allowable curvatures are thus derived as 300, 210, and 170 m⁻¹ for the sensors with the thickness—200, 280, and 360 µm, respectively: calculated by substituting the half of each sensor thickness and the elastic limit into Eq. (3), since the greatest strains are subjected to occur at the surfaces of the sensors. Therefore, the limiting factor to measurable curvature would be primarily due to optical loss in the fiber for a highly thin sensor, while it would be due to the elastic limit of the epoxy for thicker sensors.

The sensitivity of a curvature sensor is defined as the relative change of the peak wavelength with respect to curvature, which also corresponds to a slope in Fig. 5(a) indicated as a dotted line. We found that the peak wavelength linearly increases as the magnitude of curvature increases up to its maximum limit as shown in Fig. 5(a). The sensitivities were then measured to be 21.3, 51.9, and 91.9 pm/m⁻¹ for the sensors of thicknesses of 200, 280, and 360 µm, respectively. Accordingly, higher sensitivity is obtained for a thicker sensor as expected in Eq. (4). The linearity of the sensors however become degraded at high curvature, as approaching the detection limits of the sensors. Difficulties in testing the sensors on the small radii of curvatures might have led to inaccurate measurement; the sensors were supposed to be firmly wrapped around corresponding sizes of cylinders, but their own stiffness hindered to keep the sensors on the top or the bottom of smaller cylinders without slip. It is also found that a less sensitive sensor shows a larger error in curvature sensing. The larger error may be due to that the less sensitive sensor is prone to be more susceptible to external conditions, such as temperature. Furthermore, the less sensitive sensor would be easily affected by the noise fluctuation of the optical interrogator, because the sensor undergoes a lower signal change in the interrogator.

 

Fig. 5 (a) The reflected wavelength peak shift of three types of curvature sensor during bending with various curvature. (b) Comparison of the measured curvature error.

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3.2 Multi-bend curvature sensor

The curvature sensor can be used for sensing the 2-D shape of a multi-bend structure by arranging multiple FBGs in a single fiber. Given the multiple curvatures at the FBG nodes from Eq. (4), we reconstruct the entire shape of the multi-bend curvature sensor using Eq. (6). To obtain accurate curvatures, the calibration is required for finding the individual offset distances of the FBG nodes as in Eq. (10). Hence, we calibrated the multi-bend curvature sensor with known curvatures, 10 and 20 m⁻¹ prior to the reconstruction.

The sensor was first tested on an Archimedean spiral designed to have continuously varying curvatures from 60 to 180 m⁻¹ along the trajectory of the spiral as shown in Fig. 6(a); the curvatures correspond to the radii of curvatures from 16.4 down to 5.5 mm. We compared the reconstructed shape of the calibrated sensor with the shape of the uncalibrated sensor in terms of curvature and position error. For the uncalibrated sensor, offset distance was explicitly given as 80 µm for all of the FBG nodes as designed (Table 2).

 

Fig. 6 2-D Reconstruction of an Archimedean spiral using the multi-bend curvature sensors. (a) A test pattern fabricated by a laser engraving machine; the red star symbols indicate both ends of the sensor in the slot of the pattern. (b) and (c) Reconstructed trajectories with the uncalibrated and the calibrated sensors, respectively.

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Table 2. Comparison of reconstruction results for the uncalibrated and the calibrated sensors—curvature and position error at each FBG node.

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As a result, the calibrated sensor shows the highly accurate reconstruction of the Archimedean spiral as presented in Fig. 6(c). The position error was only 1.15 mm at the end of the trajectory. On the other hand, the reconstruction result of the uncalibrated sensor significantly differs from the ground truth, because error in curvature sensing consequently yielded large position error in the estimation of the entire shape. Although curvature error was not significant at each node, position error accumulated over the length of the sensor, leading to a large error of 8.63 mm at the end of the trajectory. Hence, it is found that any small variation of the offset distance that might occur during fabrication could not be neglected in reconstruction of multi-bend structures. Furthermore, the proposed calibration procedure allows accurately reconstructing multi-bend structures by providing the correct offset distances of the multiple FBG nodes.

Finally, we applied the multi-bend curvature sensor to a variety of 2-D patterns, and attained accurately reconstructed trajectories for the patterns as shown in Fig. 7.

 

Fig. 7 Reconstruction results of curvature sensor with various shape. The red star symbols indicate both ends of the multi-bend curvature sensor.

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4. Conclusion

We present the ultra-high curvature sensor that can accurately measure high curvature up to 200 m⁻¹. To measure such high curvature, we propose the optimal design for the sensor in terms of the thickness of the sensor and the offset distance of the FBG from the neutral axis of bending. The fabrication method is also developed to have the ultra-thin structure with the similar modulus of the optical fiber used, while precisely managing those design parameters in the submillimeter range. Our sensor design thus overcomes the limitation of current fiber-optic curvature sensors in sensing high curvature, while taking into account the characteristics of both the optical and the physical properties of the sensor. In addition, we introduce a new calibration scheme to simultaneously estimate all of curvatures of multiple FBGs on a single fiber. Given the calibration, the curvature sensor can accurately reconstruct any 2D shapes of multi-bend structures.

We designed three types of the sensors with the different thickness and offset distance for each sensor. The sensitives and the maximum measurable curvatures of these three sensors were investigated by varying curvature from 10 to 200 m⁻¹. As a result, the properly designed curvature sensors could measure curvature up to 200 m⁻¹. To the best of our knowledge, it is the highest curvature that optical fiber sensors can measure in the literature.

Compared to other fiber-optic curvature sensors, the proposed sensor has a great deal of advantages. First, the sensor can accurately estimate the small radius of curvature down to several millimeters, while most of curvature sensors have been designed for primarily detecting large curvature. In addition, the sensor can differentiate the direction of bending with an asymmetric design in the ultra-thin structure. Lastly, the multi-bend curvature sensor can simultaneously measure multiple curvatures along a structure by running the calibration proposed. Consequently, it allows reconstructing the corresponding 2D shape of the structure in real time.

The performance of the sensor can be further improved by mitigating the variation of the offset distance occurred during the fabrication process by applying appropriate tension to the fiber while curing the epoxy. Besides, the calibration method could also be improved by taking more than two known curvatures during the procedure. These would thus bring the more precise structure and calibration of the sensor, which leads to a higher curvature sensor with better accuracy.

In conclusion, the proposed high curvature sensor would be suitable for various applications, specifically for monitoring curvature of small structures. Such applications may include goniometers to detect small anatomical structures, e.g., finger joints, wearable sensors, and robotics [13–16]. Furthermore, use of multi-bend curvature sensors and resulting shape reconstruction can allow tracking entire human body motion in real time, which would be a promising solution to patient monitoring for rehabilitation, biomechanics, and virtual reality.