Fiber-optic curvature sensing has great potential for a number of medical and industrial applications because alternative solutions are practically nonexistent, at least for similar small sizes. FBGS has shown the feasibility of performing curvature sensing using a newly developed multicore fiber (MCF) inscribed with draw-tower gratings (DTG^{®}).

The resulting MCF-DTG^{®} exhibits the advantageous properties of a standard single-core DTG, including the option for system designers and integrators to take advantage of the densely configured arrays of DTGs. The emergence of the MCF-DTG is destined to present a paradigm shift in curvature, shape, and deflection sensing and will for many applications be the preferred solution over conventional methods.

#### The DTG Approach

The DTG approach vertically integrates the drawing process of the optical fiber with the inscription and coating steps in a single setup as seen in Figure 1.^{1} The fully automated process can inscribe a high-density quasi-distributed array of DTGs along the length of the optical fiber. This array can be configured of different Bragg wavelengths or with a uniform wavelength profile.

Based on the dynamic nature of the draw-tower inscription process, the typical length of an inscribed DTG can range from 1 to 10 mm, while their centers’ separation typically starts from 10 mm. Shorter spacing down to ~1.5 mm has also been achieved with a minor modification to the setup. Individual reflectivity can range from 0.01 up to 25 percent. A few advantages of the DTG process are as follows:

- Fabricated sensors exhibit ultimate robustness with a uniform and smooth surface.
- Dense and long arrays of DTGs can be inscribed and can be configured in different cross-sectional structures.
- The process is uniquely positioned for both small- and large-volume production (up to several tens of thousands DTGs per day), and costs are reduced.

#### Multicore Fiber DTGs

MCFs are specially configured optical fibers with multiple single-mode cores sharing the same cladding. The cores can all be addressed individually via a customized fiber optic fan-out element.

A uniquely produced cross-sectional profile of a seven-core (one central, six outer symmetrically spaced) MCF-DTG sample is shown in Figure 2. Its cladding diameter is 150 μm with an average core diameter around 5 μm and an outer diameter of a unique coating (Ormocer^{®}) of about 215 μm.^{2} A significant parameter to be considered is the distance between its center and outer cores, which is about 50 μm.

The DTG fabrication process has successfully been adopted for writing fiber Bragg gratings (FBGs) into the MCFs.^{3} Hence, one can simultaneously produce DTGs of specific configurations in all seven cores at the same exact axial location and with the same wavelength. This precision in the inscription of these high-density DTGs in the MCF represents a major milestone.

#### Curvature Sensing with MCF-DTGs

A curvature sensing system comprises a sensor, an interrogator, and the software that manages all the algorithms for data readout and processing. The key constituent for this type of sensing is based on simultaneous and real-time monitoring of the induced strain in a minimum of three outer cores of the MCF-DTG.

Depending on the curvature orientation of the MCF, some of the DTGs on the outer cores will experience a relative longitudinal tension or compression with respect to the central core and, therefore, will register positive- or negative-induced strain changes, respectively. To calculate the local curvature (or bending radius), the relative strains are measured, processed, and analyzed. To reconstruct the curvature profile of an MCF, the gathered data can then be plotted as function of the DTG position along the optical fiber.

Algorithms for calculating curvature have been well-presented in the literature.^{4–6} The induced strain in each core is proportional to the curvature and the distance from the outer core to the neutral axis (NA) of the structure. In principle, the NA is an axis that passes through the center of the fiber's cross-section as shown in Figure 3. It represents the points of zero-bending strain, which can be oriented in any direction. To parametrize this degree of freedom, use the angles α_{n} (*n*=1, 2 or 3) that represent the angular orientation of the n^{th} outer core with respect to the NA. Basically, it is sufficient to know only one angle since the other angles are fixed by design. The distance of each outer core to the NA can be found by making a perpendicular projection of the core position to the NA, which can be expressed in terms of the angle α_{n} and the distance *d* between the outer core and the central core.

The induced strain *ε _{n}* of the n

^{th}outer core can then be expressed as:

*ε _{n}* = (

*d*/

*R*)

*.sin*

*α*

_{n}where 1/R represents the curvature *C* expressed in terms of the bending radius *R.* This leads to a set of three equations from which *R* and *α _{n}* can be calculated irrespective of the longitudinal strain, temperature, and twist of the fiber, as these parameters can all be considered as common-mode effects. At the same time — similar to a single-core DTG — the center core can still be used to monitor the temperature or strain.

For a fixed value of *d*, the value of the induced strain in the outer core increases linearly with the curvature. Consequently, the curvature sensitivity is constant over the full bending range of the MCF-DTG. The minimum detectable curvature changes for a given *d* is defined by the resolution of the interrogator, whereas the overall accuracy is not only defined by the interrogator's accuracy but also by the uncertainty and variation on the parameter *d* over the length of the MCF.

To increase the MCF's curvature sensitivity, the parameter *d* can be increased. Increasing the cladding diameter will, for example, allow scaling of the parameter *d* accordingly. Nevertheless, this tuning range is limited by the maximum cladding diameter the drawing process can withstand.

In case the requested sensitivity cannot be achieved using the MCF approach (e.g., industrial applications where very large bending diameters of +100 m need to be monitored), an alternative solution based on a multi-fiber-bundle (MFB) can be considered. The MFB consists of several fibers, which are consolidated in a cable-like structure to achieve a similar measurement concept. In an MFB design, it is easier to enlarge the parameter *d.*

However, the major drawback of the MFB approach is the error induced by the co-alignment of the different grating positions in the different optical fibers, which reduces the curvature accuracy. In addition, the mechanical stability over time and the operating temperature of the MFB is also more difficult to control but nevertheless is crucial to ensure a consistent strain transfer to the different fiber strands in the MFB. Finally, the overall cost of large volume fabrication, testing, and deployment of the MFB is inherently far more expensive than the MCF-DTG approach because the MFB is composed of at least four individual fibers.