# Angle-Resolved Hollow-Core Fiber-Based Curvature Sensing Approach

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

_{0}, t

_{1}, and t

_{2}. The tubes with thicknesses t

_{1}and t

_{2}are placed in orthogonal directions within the fiber microstructure (tubes represented in yellow and red in Figure 1a). Such a configuration of cladding tubes allows for a curvature-dependent fiber transmission spectrum to be obtained and for an angle-resolved curvature sensing approach to be attained, as will be detailed in the following. The curvature radius is defined as R and the curvature angle (i.e., the direction of the curvature with respect to the fiber cross-section) is represented by θ (as illustrated in Figure 1a). In the simulations to be described in the following, the core diameter (D

_{C}) was set as 20 μm (which entails a mode field diameter, MFD, of approximately 15.7 μm, $MFD~\frac{\pi}{4}{D}_{C}$ [39]). The diameter of the cladding tubes, D

_{t}, was chosen to be 9 μm. Under our convention, θ = 0° refers to the situation in which the center of the curvature is located in the +x direction, and θ = 90° stands for a curvature whose center lies in the +y direction. Analogously, θ = 180° refers to a curvature which has its center in the −x direction and θ = 270° means a curvature with center in the −y direction.

_{1}is the refractive index of the core, n

_{2}is the refractive index of the cladding material, and m is the resonance order. Here, we consider n

_{1}= 1 (for air) and n

_{2}= 1.45 (for silica). The value of n

_{2}= 1.45 was used in the simulations for simplicity as, within the wavelength range considered in the simulations (from 650 nm to 850 nm), the refractive index of silica varies from 1.4565 to 1.4525 [42]. The thickness of the cladding tubes determines the resonant wavelengths and, hence, the spectral positions of the high-loss regions in the fiber transmission spectrum. For the fiber proposed herein, as the tubes in the fiber structure display three different thicknesses, several resonances are expected to appear in the fiber transmission spectrum.

_{0}= 450 nm, t

_{1}= 650 nm, and t

_{2}= 800 nm (for a straight fiber). Here, it is worth observing that the choice of the thicknesses of the cladding tubes is constrained to two aspects. The first one considers that the thicknesses of the tubes must provide sufficiently separated resonances in the fiber transmission spectrum (so it does not hinder the identification of the resonances’ shifts when performing the sensing studies). The second aspect considers the fabrication tolerances in state-of-the-art HCPCF (~1% cross-sectional tube thickness variation [43]). The thicknesses t

_{0}, t

_{1}, and t

_{2}used in the simulations reported herein meet these two constraints.

_{1}occurs (tube at +x direction). Analogously, around λ = 835 nm, coupling between the core mode and the modes in the tube with thickness t

_{2}takes place (tube at +y direction). Around λ = 750 nm, the guided mode is highly confined into the fiber core, as, at this wavelength, robust inhibition of the coupling between the core and cladding modes is achieved.

## 3. Results

#### 3.1. The Fiber’s Bend-Dependent Response

_{1}and t

_{2}resonances (around λ = 680 nm and λ = 835 nm, respectively) for different curvature radii and representative curvature angles θ. The spectral shifts presented in Figure 2 are defined as $\mathsf{\Delta}\lambda \equiv {\lambda}_{R,\theta}-{\lambda}_{R=0}$, where is ${\lambda}_{R,\theta}$ stands for the resonant wavelength for a fiber bent at a curvature radius R and angle θ, and ${\lambda}_{R=0}$ denotes the resonant wavelength for the straight fiber. In a hypothetical experimental realization of the sensor, Δλ could be determined by coupling broadband light into the fiber core and by following the resonances’ wavelength shifts in the fiber transmission spectrum.

_{1}and the resonance around 680 nm (t

_{1}resonance) blueshifts for smaller R (Figure 2a). Otherwise, when the center of the curvature rests in the –x direction (θ = 180°), the guided mode intensity distribution is displaced towards the tube with thickness t

_{1}, and the resonance associated with the tube with thickness t

_{1}redshifts for smaller R (Figure 2b).

_{2}. When the center of curvature lies in the +y direction (θ = 90°), the guided mode intensity distribution is displaced away from the tube with thickness t

_{2}and its correspondent resonance blueshifts when R decreases (Figure 2c). Otherwise, when the center of curvature rests in the −y direction (θ = 270°), the guided mode intensity distribution is displaced towards the tube with thickness t

_{2}and the resonance associated with the tube with thickness t

_{2}redshifts for reduced R (Figure 2d). Remarkably, one observes a weak impact on the spectral positions of the resonances when the curvature direction is orthogonal to the tubes’ location within the fiber microstructure (i.e., t

_{1}resonance mildly shifts for θ = 90° and θ = 270°, and t

_{2}resonance slightly shifts for θ = 0° and θ = 180°).

_{1}and t

_{2}as a function of the curvature radius for representative curvature angles. By observing the data in Figure 3a (corresponding to the wavelength shifts associated with the resonance of the cladding tube with thickness t

_{1}) at the curvature radius of 3 cm, for example, one sees that Δλ values vary from −3.64 nm when θ = 0° to −0.47 nm when θ = 90° and to 3.60 nm when θ = 180°. In turn, data in Figure 3b (corresponding to the wavelength shifts associated with the resonance of the cladding tube with thickness t

_{2}) at the curvature radius of 3 cm show that Δλ values change from –0.23 nm when θ = 0° to −3.47 nm when θ = 90° and to −0.46 nm when θ = 180°. The results in Figure 3 also show that the variation of Δλ values due to changes in θ is smaller for larger curvature radii.

#### 3.2. Calibration of the Fiber Response

_{1}= 680 nm and λ

_{2}= 835 nm (identified as Δλ

_{1}and Δλ

_{2}, respectively) for θ between −180° and 180° and R = 3 cm, 5 cm, 7 cm, 10 cm, and 15 cm. The data in Figure 4 allow for the calibration of the fiber optical response by conveniently fitting Δλ for different R and θ. Here, we empirically chose a sinusoidal fitting function, as shown in Equation (1), to account for Δλ. The choice of the function in Equation (1) was driven by the observation of the sinusoidal trend of the data points in Figure 4. In Equation (1), Δλ

_{0}, Λ, θ

_{0}, and δ are fitting parameters. The fitted functions are presented as dashed and dotted lines in Figure 4. The coefficients of determination (R

^{2}) associated with the fits ranged from 0.987 to 0.999 for t

_{1}resonance and from 0.972 to 0.977 for t

_{2}resonance.

_{0}, Λ, θ

_{0}, and δ can be plotted as a function of R and fitted by using empirical functions. Figure 5 exhibits graphs of Δλ

_{0}, Λ, θ

_{0}, and δ as a function of R, together with the fitting curves. Table 1 shows the fitting functions which have been used to adjust Δλ

_{0}, Λ, θ

_{0}, and δ trends as a function of R. It is worth observing that Λ is related to the amplitudes of the wavelength shifts and δ to the conversion between radians and degrees. In turn, θ

_{0}accounts for the cosine-like and sine-like behaviors of Δλ

_{1}and Δλ

_{2}, respectively.

_{0}, Λ, θ

_{0}, and δ with R was obtained by the fitting curves in Figure 5, one can assume Δλ

_{0}, Λ, θ

_{0}, and δ as functions of R and, thus, rewrite Equation (1) by considering Δλ as a function of R and θ. Therefore, we obtain Equation (2), where Δλ

_{0}(R), Λ(R), θ

_{0}(R), and δ(R) are the fitted functions.

_{1}= 680 nm and λ

_{2}= 835 nm − $\mathsf{\Delta}{\lambda}_{1}\left(R,\theta \right)$ and $\mathsf{\Delta}{\lambda}_{2}\left(R,\theta \right)$, respectively, which have been calculated by considering Equation (2) and the adjusted functions. Indeed, these plots are the basis of the sensing operation principle reported herein, which relies on considering the pair (Δλ

_{1}, Δλ

_{2}) and associating it with the bending radius and angle univocally. The analytical functions shown in Figure 6 are, thus, the calibration curves of the sensor response. Indeed, the plots in Figure 6 are strongly correlated to the ones shown in Figure 4, from which the fitting parameters have been determined.

## 4. Discussion

_{1}and Δλ

_{2}. The first studied case (Test #1, Figure 7a) considers that, in a hypothetical measurement, Δλ

_{1}and Δλ

_{2}have been measured as Δλ

_{1}= (2.48 ± 0.05) nm and Δλ

_{2}= (−1.45 ± 0.05) nm. By intersecting the latter Δλ

_{1}and Δλ

_{2}values with system calibration curves (as shown in Figure 6), we obtained the 2D color plots shown in Figure 7a, which maps the R and θ that could correspond to the considered Δλ

_{1}and Δλ

_{2}. The intersecting region between the Δλ

_{1}and Δλ

_{2}plots readily communicates the R and θ values expected from the calibration. For the Δλ

_{1}and Δλ

_{2}values used in this example, the calibration allowed for the determination of R = (4.1 ± 0.2) cm and θ = (30 ± 2)°. Indeed, the BPM simulations using R = 4.0 cm and θ = 30° yielded Δλ

_{1}= −2.48 nm and Δλ

_{2}= −1.45 nm. Therefore, the method reported herein allows for R and θ to be obtained via the knowledge of Δλ

_{1}and Δλ

_{2}.

_{1}= (−1.72 ± 0.05) nm and Δλ

_{2}= (−1.00 ± 0.05) nm (Test #2) to be determined. Similarly, Figure 7b presents color maps on the considered Δλ

_{1}and Δλ

_{2}values as reported from the calibration. In this case, our methods allow R = (6.2 ± 0.3) cm and θ = (30 ± 2)° to be determined, which are, once again, in good agreement with the BPM simulations’ results (which yields Δλ

_{1}= −1.72 nm and Δλ

_{2}= −1.00 nm for R = 6.0 cm and θ = 30°). The two other examples’ results (Test #3 and Test #4) are summarized in Table 2 (together with the data from Test #1 and Test #2). Remarkably, the sensing approach reported herein allows the curvature radius and angle to be determined if Δλ

_{1}and Δλ

_{2}are known. All the R and θ values retrieved in the test cases are consistent with the estimated error bars, and the maximum percentual error between expected and retrieved values has been found to be 5.8% for R values and 6.6% for θ values. A new angle-resolved curvature sensing approach based on HCPCF technology has thus been developed.

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**(

**a**) HCPCF cross-section diagram (t

_{0}, t

_{1}, and t

_{2}: thicknesses of the cladding tubes; θ: curvature angle). (

**b**) Typical transmission spectrum for a straight fiber and mode profiles at selected wavelengths.

**Figure 2.**Simulated wavelength shifts (Δλ) associated with t

_{1}and t

_{2}resonances as a function of the curvature radius for (

**a**) θ = 0°, (

**b**) θ = 90°, (

**c**) θ = 180°, and (

**d**) θ = 270°.

**Figure 3.**Wavelength shifts (Δλ) associated with the tubes with thicknesses (

**a**) t

_{1}and (

**b**) t

_{2}as a function of the curvature radius tube for representative curvature angles, θ.

**Figure 4.**Wavelength shifts as a function of the curvature angle (θ) for different curvature radii (R) for the resonances associated with the tubes with thicknesses (

**a**) t

_{1}(Δλ

_{1}) and (

**b**) t

_{2}(Δλ

_{2}).

**Figure 5.**Parameters Δλ

_{0}, θ

_{0}, δ, and Λ as a function of the curvature radius (R), corresponding to the resonances associated to the tubes with thicknesses (

**a**) t

_{1}and (

**b**) t

_{2}.

**Figure 6.**Calibration curves: (

**a**) $\mathsf{\Delta}{\lambda}_{1}\left(R,\theta \right)$ and (

**b**) $\mathsf{\Delta}{\lambda}_{2}\left(R,\theta \right)$.

**Figure 7.**Color plots from calibration for selected Δλ

_{1}and Δλ

_{2}values. The intersecting region between Δλ

_{1}and Δλ

_{2}plots allows the curvature radius and angle expected from the calibration to be obtained. (

**a**) Test #1, (

**b**) Test #2, (

**c**) Test #3, and (

**d**) Test #4.

**Table 1.**Fitting functions used to adjust Δλ

_{0}, θ

_{0}, δ, and Λ trends as a function of R (a, b, c, and d are representative fitting constants).

Parameter | Fitting Function (t _{1} Resonance) | Fitting Function (t _{2} Resonance) |
---|---|---|

Λ | $\mathsf{\Lambda}=a/{R}^{b}$ | $\mathsf{\Lambda}=a/{R}^{b}$ |

δ | $\delta =a+bR+c{R}^{2}$ | $\delta =a+bR$ |

θ_{0} | ${\theta}_{0}=a+bR+c{R}^{2}$ | ${\theta}_{0}=a+bR$ |

Δλ_{0} | $\mathsf{\Delta}{\lambda}_{0}=a+bR+c{R}^{2}$ | $\mathsf{\Delta}{\lambda}_{0}=a+bR$ |

Test | Results from the System Calibration | Results from BPM Simulations |
---|---|---|

#1 (Figure 7a) | $\mathsf{\Delta}{\lambda}_{1}=\left(-2.48\pm 0.05\right)\mathrm{nm}$ $\mathsf{\Delta}{\lambda}_{2}=\left(-1.45\pm 0.05\right)\mathrm{nm}$ $\iff $ $R=\left(4.1\pm 0.2\right)\mathrm{cm}$ $\theta =\left(30\pm 2\right)\xb0$ | $\mathsf{\Delta}{\lambda}_{1}=-2.48\mathrm{nm}$ $\mathsf{\Delta}{\lambda}_{2}=-1.45\mathrm{nm}$ $\iff $ $R=4.0\mathrm{cm}$ $\theta =30\xb0$ |

#2 (Figure 7b) | $\mathsf{\Delta}{\lambda}_{1}=\left(-1.72\pm 0.05\right)\mathrm{nm}\text{}$ $\mathsf{\Delta}{\lambda}_{2}=\left(-1.00\pm 0.05\right)\mathrm{nm}$ $\iff $ $R=\left(6.2\pm 0.3\right)\mathrm{cm}$ $\theta =\left(30\pm 3\right)\xb0$ | $\mathsf{\Delta}{\lambda}_{1}=-1.72\mathrm{nm}$ $\mathsf{\Delta}{\lambda}_{2}=-1.00\mathrm{nm}$ $\iff $ $R=6.0\mathrm{cm}$ $\theta =30\xb0$ |

#3 (Figure 7c) | $\mathsf{\Delta}{\lambda}_{1}=\left(-1.77\pm 0.05\right)\mathrm{nm}$ $\mathsf{\Delta}{\lambda}_{2}=\left(0.70\pm 0.05\right)\mathrm{nm}$ $\iff $ $R=\left(5.8\pm 0.2\right)\mathrm{cm}$ $\theta =\left(-28\pm 3\right)\xb0$ | $\mathsf{\Delta}{\lambda}_{1}=-1.77\mathrm{nm}$ $\mathsf{\Delta}{\lambda}_{2}=0.70\mathrm{nm}$ $\iff $ $R=6.0\mathrm{cm}$ $\theta =-30\xb0$ |

#4 (Figure 7d) | $\mathsf{\Delta}{\lambda}_{1}=\left(-0.84\pm 0.05\right)\mathrm{nm}$ $\mathsf{\Delta}{\lambda}_{2}=\left(-0.71\pm 0.05\right)\mathrm{nm}$ $\iff $ $R=\left(12.7\pm 0.9\right)\mathrm{cm}$ $\theta =\left(45\pm 5\right)\xb0$ | $\mathsf{\Delta}{\lambda}_{1}=-0.84\mathrm{nm}$ $\mathsf{\Delta}{\lambda}_{2}=-0.71\mathrm{nm}$ $\iff $ $R=12.0\mathrm{cm}$ $\theta =45\xb0$ |

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**MDPI and ACS Style**

Guimarães, W.M.; Cordeiro, C.M.B.; Franco, M.A.R.; Osório, J.H.
Angle-Resolved Hollow-Core Fiber-Based Curvature Sensing Approach. *Fibers* **2021**, *9*, 72.
https://doi.org/10.3390/fib9110072

**AMA Style**

Guimarães WM, Cordeiro CMB, Franco MAR, Osório JH.
Angle-Resolved Hollow-Core Fiber-Based Curvature Sensing Approach. *Fibers*. 2021; 9(11):72.
https://doi.org/10.3390/fib9110072

**Chicago/Turabian Style**

Guimarães, William M., Cristiano M. B. Cordeiro, Marcos A. R. Franco, and Jonas H. Osório.
2021. "Angle-Resolved Hollow-Core Fiber-Based Curvature Sensing Approach" *Fibers* 9, no. 11: 72.
https://doi.org/10.3390/fib9110072