1. Introduction

Distributed optical fiber sensors have been shown to be an invaluable tool in many applications – such as structural health monitoring, acoustic wave sensing, and in biomedical fields – because of their accuracy, sensitivity, flexibility, and immunity to EM interference [1,2]. One of the most demanding sensing tasks is shape sensing or fiber shape reconstruction, it requires the measurement of strain in multiple fibers or in multiple cores of a single fiber in order to reconstruct a continuously varying shape of a cable or fiber. Optical fiber shape sensing refers to different use cases, from monitoring of large structures such as airplane wings and wind turbines, with single core fibers bonded to the structure [3,4], to the deflection of small objects such as needles, catheters, and other surgical tools using multicore fibers [5–7]. Typically, these sensors use wavelength division multiplexed (WDM) fiber Bragg grating arrays that are interrogated by measuring the optical reflection spectrum of each individual grating. However, WDM Bragg gratings measure strain only at discrete locations and they are typically limited to less than 100 sensing points with spatial resolution greater than 1 mm [8]. Another method of fiber shape sensing employs Brillouin scattering to measure distributed fiber strain in multicore and single core fibers [9–11]. While such schemes can be applied over long lengths, they have spatial resolution on the order of centimeters. Moreover, both the proximal and distal ends of the fiber must be accessible for the sensor to be interrogated with Brillouin scattering. Distributed shape sensing has also been shown using phase-sensitive optical time-domain reflectometry, where only one end of the fiber is needed to be interrogated, but this approach is still limited by a spatial resolution of 10 cm [12].

For more accurate shape reconstructions, measurement of fiber strain with spatial resolution less than 1 mm is required. To date, such high spatial resolution can only be achieved using swept wavelength interferometry (SWI), alternatively known as Optical Frequency Domain Reflectometry (OFDR). In OFDR, spatial resolution is governed by the wavelength scan range, and maximum measurable fiber length is determined by the wavelength step size. The spatial resolution can be less than 100 µm and fibers over tens of meters can be measured. Strain and shape sensing have been shown by using the discrete Bragg gratings or Rayleigh scatter with OFDR [13,14]. Although sensing with Rayleigh scattering is distributed along the fiber and therefore provides high spatial resolution, this backscattered signal is very weak and exhibits a randomly varying phase, which limits interrogator speed and stability. Efforts to improve the back scatter by using UV enhanced fiber has been shown, but this fiber still displays uncontrolled phase [15]. A significant improvement in both signal-to-noise ratio and phase information was demonstrated using nearly continuous arrays of weak uniform period gratings written into multicore fibers [16]. Moreover, the outer cores in these waveguides were spun during draw so that they formed helices around the center core. Such quasi-continuous gratings in twisted multicore fibers have greatly improved the accuracy and speed of fiber shape reconstruction using OFDR. These fibers have enabled OFDR interrogation to be used in applications such as medical shape sensing [17,18] and sensing micron sized deformations along a fiber [19,20].

Compared to other methods, OFDR requires a more complicated interrogator, which uses a precisely tuned narrow linewidth source and phase sensitive measurements. Applying OFDR to four separate channels for shape sensing adds yet more complexity. Typically, this requires an optical switch to measure the different cores in succession, which can degrade the signal quality in dynamic environments. Alternatively, an input/output network can be used to distribute the input signals to the four cores and collect the back reflected signal from each core. To obtain simultaneous measurements, such schemes would require four separate detector modules. This additional complexity brings increased cost, and can compromise accuracy, speed, and stability.

In this work, we take a different approach to shape sensor interrogation that requires only a single high spatial resolution OFDR measurement while still maintaining a compact distal fiber end compatible for use in medical catheters and other compact applications. We use a 498 µm long graded index (GRIN) fiber lens spliced to the distal end of a twisted multicore fiber to reflect light from a given outer core to the second opposite core 180 degrees away. In this way, both the input light and the distributed backscattered light used in the OFDR interrogator follow a path forward and backward through the multicore fiber. At the proximal fanout, we splice the single core fiber output of the GRIN lens reflected signal to another single core fiber that connects to the third outer core. The light entering this core is then reflected by the same GRIN lens back into a fourth core. Critically, we show that the back reflected light from these four cores can be used effectively in bend and shape sensing with 80 µm spatial resolution. We compare the performance of our single measurement micro-turnaround shape sensor with a 1 × 4 optical switch system that interrogates each core sequentially and found similar performance for a range of curvatures from 2.2 m^-1 to 52 m^-1. We also performed shape reconstruction with our shape sensor fiber wrapped around two posts that were rotated with respect to each other. The reconstructed fiber shapes showed good agreement with experimental results. We expect that our micro-turnaround shape sensor will find use in applications that require a compact distal fiber end and where demands of cost and/or performance require the use of a single channel OFDR system.

2. Sensor and GRIN design

Figure 1(a) depicts the experimental setup. The fiber under test (FUT) is a twisted multicore fiber inscribed with quasi-continuous weak uniform period fiber Bragg gratings (See Fig. 1(b) and described in detail in [16]). The fiber has one center core and six outer cores, where the end face image is shown in Fig. 1(d). The outer cores are 35 µm from the center core and twist along the length of the fiber with an average period of 2 cm (or 50 twists per meter).

 

Fig. 1. (a) Experimental setup; OFDR: optical frequency domain reflectometer, TFB: taper fiber bundle, FUT: fiber under test; (b) Diagram of the twisted multicore fiber with continuous gratings; (c) Side image of GRIN fiber spliced to FUT before metal coating was applied; (d) End face image of FUT; (e) Ray tracing diagram of the FUT and turnaround device. For clarity, only the first four cores are shown and the twist of the FUT is not shown.

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The use of graded index multimode fibers as lenses to refocus a point source of light is well known [21–23]. Such a short length of graded index multimode fiber is often referred to as a graded index or GRIN lens. In this work, we exploit this effect to allow the OFDR to measure backscatter signal from all outer cores of the multicore shape sensing fiber. To interrogate the outer cores of the FUT serially, the proximal end of the fiber is spliced to a tapered fiber bundle (TFB) multicore to single core fanout that has seven single core fiber inputs and one multicore fiber output. The distal end is spliced to the GRIN turnaround device. A sideview image of the FUT spliced to the GRIN fiber is shown in Fig. 1(c). The GRIN fiber has a length of 498 µm, core diameter of $a = 115.3\; \mu m\; $, cladding diameter of 127 µm, and core index profile of $n(r )= {n_0}\left( {1 - \frac{{{g^2}}}{2}{r^2}} \right)$, where ${n_0} = 1.4687$ and $g = \frac{{NA}}{{a{n_0}}} = 0.003153\; \left( {\frac{1}{{\mu m}}} \right)$ and $NA = 0.2663$. In such a fiber, a focused spot will refocus after a length $P = \pi /g$ and therefore if the GRIN has a length of $\frac{\pi }{{2g}} = 498\; \mu m$ the light will expand, reflect off the end of the fiber and refocus into an opposite core. After splicing to the shape sensor fiber, the GRIN lens and shape sensor fiber were dipped in a reflective silver coating, up to about 15 mm from the end of the GRIN lens. As shown schematically in Fig. 1(e), the GRIN fiber can focus light from one of the offset cores to another offset core that is at the same radius but rotated 180° from the first core. To access offset cores that are less than 180° degrees from Core 1, the fanout single core output from Core 2 was spliced to Core 3, which is 120° from Core 2. Note that Core 3 is arbitrarily chosen and can be the core 60° from Core 2 as well. Core 4 was also spliced to Core 5 and 6, but these cores were left out of our analysis for reasons listed below.

3. OFDR and data processing

We measure the distributed strain along the fiber sensor using OFDR. This technique is well known and has been used to measure discrete attenuations, strain and temperature in optical fibers [13,24,25]. Briefly, the OFDR technique sends a narrow linewidth signal into the fiber to be measured. The weak, distributed back reflection from the entire fiber length is interfered with a reference beam from the same laser. As the laser is scanned over a given range of frequencies, a spectral interferogram $R(\omega )$ is recorded in which every position on the fiber is represented by a spectral oscillation frequency. A Fourier transform of this spectral interference results in a time domain quantity [24]:

Here, the phase derivative is related to a vacuum Bragg wavelength that varies along the optical fiber. This local effective Bragg wavelength depends on the local strain and temperature of the fiber in the same way that a discrete fiber Bragg grating would. In our fiber, the presence of a weak quasi-uniform periodic modulation along the fiber results in a large value of ${R_{OFDR}}(t )$ and a very well-defined value of ${\phi _{OFDR}}(t )$ and hence, ${\lambda _{Bragg}}$. The weak gratings thus greatly simplifying the measurements compared to those that rely on Rayleigh backscattering.

A trace of ${R_{OFDR}}(z )$ for our novel fiber assembly, using a commercially available OFDR (Luna OBR 4600), is shown in Fig. 2(a). The y-axis units are power in dB/mm ($20{\log _{10}}{R_{OFDR}})$ normalized to a 1 mm length. The OFDR trace shows back reflection from six transits through the six outer cores of the fiber, alternating between forward and backward direction of light propagation. The output of the OFDR is spliced into the TFB fanout fiber corresponding to Core 1 of the FUT (Splice 1). A slight increase in backscattered signal is observed at Splice 1 because of the difference in Rayleigh scattering between the output fiber of the OFDR (SMF-28) and the TFB fanout fiber. The large spike in the OFDR trace after the input splice is caused by the TFB itself. The enhanced backscattered signal of the FUT from the continuous gratings is clearly seen (Core 1). The GRIN turnaround device, spliced to the distal end of the FUT, appears as a spike, followed by a drop in signal due to attenuation and reflection loss. We note that there was another FUT-to-FUT splice near the GRIN lens splice. The signal is then spatially reversed with respect to the signal from Core 1 and propagates toward the fanout at the proximal end of the FUT (Core 2). The output of the Core 2 fanout is spliced to the Core 3 fiber of the fanout. Signal in Core 3 and Core 4 will also propagate in a similar fashion as Core 1 and Core 2, although with lower amplitude due to losses in the TFB and multicore splices. Core 5 and Core 6 were also spliced into the signal path and their OFDR signals can be seen as well. However, they were not used in this set of experiments because the signal degraded too much due to losses in the system. Most of this attenuation is due to the TFB and two unoptimized multicore fiber splices rather than the GRIN lens turnaround. Each GRIN reflection, including the splice to the multicore fiber, has a roundtrip loss of ∼2 dB. The other losses are estimated as follows: each one-way pass between the TFB multicore output-to-FUT splice has a loss ranging from 1.6 to 3 dB, the TFB itself has a 3 dB roundtrip loss, and FUT-to-multicore splice near the GRIN lens has a splice loss ranging from 0.3 to 1.4 dB. Also, the grating reflectivity was ∼2-3 dB lower in Core 3 and Core 4.

 

Fig. 2. (a) OFDR trace of the shape sensing system; (b) Cropped and flipped amplitude of the four outer cores; (c) Local changes in Bragg wavelength of the four outer cores [box of Fig. 2(b)]; (d) Geometry of fiber cross section. (e) Data processing steps for shape reconstruction.

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Precise shape reconstruction requires the measurement of local curvature and twist along the fiber, typically with spatial resolution of less than 1 mm (∼80 µm in this work). These parameters are derived from the strain field over the fiber cross section. To measure the local strain, we extract data from the first four cores of the OFDR trace in Fig. 2(a). The trace for Core 1 is taken from the measurement in Fig. 2(a) by cropping before Splice 1 and after the first turnaround at 8.5 m. The trace for Core 2 is obtained by cropping off the data before the turnaround and after Splice 2 and then inverting the trace. The traces for Core 3 and Core 4 are generated in a similar fashion. Core 3 is cropped between Splice 2 and the second turnaround at 20 m. Core 4 is cropped between the second turnaround and Splice 3 then spatially reversed. The four traces are then translated so that they are all aligned at the location of the turnaround, and splice 1 is at just after z = 0. The various components of the system can be seen aligned in Fig. 2(b). The turnaround point is aligned at ∼5.6 m, splices into the multicore fiber are aligned near 2.6 m, and the tapered region of the TFB is aligned at ∼2 m. Note that the splices among the single core fanout outputs do not occur exactly at the halfway point between the micro-turnaround points. This is because the single core pigtails in the fanout are not exactly the same length. Therefore, unlike the turnaround and the multicore splice to the fanout, the reflections from these points are not at the same location for each trace. These reflections may combine with the other reflection spikes to generate spurious peaks in the traces. Such reflection artifacts may also arise from stray reflections at the GRIN micro-turnaround.

OFDR processing, as described above, will convert the phase data for these traces into a local Bragg wavelength using Eq. (2). Figure 2(c) shows the difference in local Bragg wavelength, $\mathrm{\Delta }{\lambda _{Bragg}}(z )$, for a straight fiber and a fiber experiencing constant curvature. When a fiber is bent, one side of the fiber is under tension, while the other side is under compression. With the outer cores rotating along the fiber, $\mathrm{\Delta }{\lambda _{Bragg}}(z )$ will give a sine wave profile, since each outer core will alternate between tension and compression through one twist period of ∼2 cm. Note that $\mathrm{\Delta }{\lambda _{Bragg}}(z )$ for Core 1 and Core 2 are anti-phase from each other as the cores are 180° apart, which also holds true for Core 3 and Core 4. Moreover, the oscillations in Core 1 (and Core 2) are 60 degrees out of phase with respect to Core 3 (and Core 4) since these cores are rotated by 60 degrees with respect to each other. By knowing the local changes in Bragg wavelength, the strain of the outer cores can be then computed with the equation $\varepsilon (z )= \; \frac{1}{\eta }\frac{{\mathrm{\Delta }{\lambda _{Bragg}}(z )}}{{{\lambda _{Bragg}}}}$, where ɛ is the strain experienced by the core and $\eta \sim 0.69$ accounts for the strain-optic effect. Unlike other shape sensing fiber core configurations [16], our fiber plus the turnaround device provides no signal from the center core. The center core provides a reference since it does not undergo strain when the fiber is bent. In our analysis, we obtained a reference value of ${\lambda _{Bragg}}$ in a calibration step in which the fiber is measured while being held completely straight with no added axial strain.

We perform the fiber shape reconstruction by integrating the Frenet-Serret equations:

To calculate $\kappa (s )\; $ and $\tau (s )$, the vectors pointing to the helically twisting cores, ${\hat{\rho }_u}(s )$ must be accurately determined along the fiber [Fig. 2(d)]. Although the fiber has an average twist rate of 50 twists per meter, the local twist rate can vary due to fiber manufacture. Thus, another calibration step is performed, similar to [28]. The fiber is carefully placed on a flat plane surface to not introduce external twist while it is continuously bent. By measuring the local variation of $\mathrm{\Delta }{\lambda _{Bragg}}(s )$, the vectors pointing to the cores locally, ${\hat{\rho }_u}(s )$, can be determined. This set of values of ${\hat{\rho }_u}(s )$ are then taken as the local reference coordinates, $\hat{\rho }_u^{ref}(s )$, to compute the curvature when the fiber is in an arbitrary shape. Using the distributed strains ${\varepsilon _u}(s )$ of the outer cores, a local curvature vector, $\vec{\kappa }(s )$, of the FUT can be calculated from these reference vectors using the relation: $\vec{\kappa }(s )= \textrm{}\frac{1}{r}\mathop \sum \limits_{u = 1}^N \hat{\rho }_u^{ref}(s ){\varepsilon _u}(s )$, where r is the distance between the center core and outer cores, $N = 4$ is the number of cores, and u is the label of the core [27]. After $\vec{\kappa }(s )$ is calculated, the bend angle, $\theta (s )$, along the fiber may be determined as well. Since this fiber is under non-zero curvature, this bend angle can be calculated by the difference in angle of $\hat{\rho }_u^{ref}(s )$ and ${\hat{\rho }_u}(s )$. The local torsion can then be determined with the following: $\tau (s )= d\{{\theta (s )} \}/ds$. These values are then used in the Frenet-Serret Equations [Eq. (3)] to compute ${\mathbf T}(s )$ and the fiber shape. We assume the initial conditions of $\kappa (0 )= 0$ and $\tau (0 )= 0$. Also, ${\mathbf T}(0 )$, ${\mathbf N}(0 )$, and ${B}(0 )$ are three arbitrarily chosen orthonormal unit vectors. In our case, we chose ${\mathbf T}(0 )= [{0\; 0\; 1} ]$ for the initial fiber to point in the z-direction. The data processing steps are summarized in the flow chart of Fig. 2(e).

4. Results and discussion

Figure 3(a) shows the fiber bend reconstruction using our shape sensing system for several curvatures. The FUT was wrapped around spools of various diameters with minimal twist with curvatures ranging from 52 m^-1 to 2.2 m^-1. Our shape sensing system is also compared against one where the fiber does not have a GRIN turnaround device and each core is interrogated individually with the use of an optical switch [Fig. 3(b)] [16,20]. As shown in Fig. 3(c), the curvature sensing capability of the current system is accurate and comparable to past systems as they are in good agreement with each other and with the theoretical value.

 

Fig. 3. Reconstruction of FUT under various curvatures w/ GRIN turnaround device (a) and w/o the device (b); (c) Comparison of curvature results for current and previous shape sensing systems.

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Finally, we demonstrate the capability of the current system in 3D shape sensing over 0.55 m. The fiber was wrapped helically around two optical posts. In this way, the fiber followed the position of the two posts while always having non-zero curvature. These two posts were then bent at 5 different angles (-90°, -45°, 0°, 45°, and 90°) and interrogated with a commercially available OFDR (Luna OBR 4600). The FUT was scanned with a sweep rate of 10 nm/s and a scan range of 10.32 nm, which gives a spatial resolution of 77.7 µm. The FUT length was about 3 m long, but only 0.55 m of the FUT is shown in the experiment because that was the length needed to wrap the FUT around two posts. The shape reconstructions are shown in Fig. 4. We added cylinders to the shape reconstruction to show that the reconstructed shape was indeed wrapping around a straight cylinder. Since the shape reconstruction gives an arbitrary frame, the bottom post was used as a reference. The shape reconstruction in each case was rotated so that the axis around which the bottom spiral rotated was always in the same orientation, and the azimuthal orientation of the top post was always the same. Qualitatively speaking, the reconstructions are well matched to the pictures of the fiber. All the reconstructions are wound neatly around both posts, where the curvatures are not significantly larger or smaller than the half inch posts. Also, each reconstruction shows six fiber revolutions around the posts, the same as shown in the pictures. For a more quantitative result, a top post was created virtually and fitted against the reconstructed shape to give an estimate of the angle between the posts, which gives angles of -85°, -44°, 0°, 52°, and 91°. We note that a precise wrapping of the posts is not necessary, and the fiber was not bonded with the posts during the entire experiment.

 

Fig. 4. Photo and shape reconstruction of fiber wrapped around two posts at different angles: -90°, -45°, 0°, 45°, and 90°.

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In summary, we have presented a novel single channel multicore fiber assembly for use in curvature and shape sensing. By using a GRIN micro-turnaround device that is spliced to the distal end of a multicore fiber, we interrogated multiple cores in a single fiber simultaneously without the need of an optical switch or multiple detectors. We showed that the curvature measurements are accurate and comparable to the traditional multi-channel method. We also demonstrated the capability of our sensing system in shape reconstructions of a fiber wrapped around two posts. Our approach promises improved acquisition speed, stability, cost, and performance in shape sensing applications that requires a compact distal fiber endpoint.