# Double Antiresonance Fiber Sensor for the Simultaneous Measurement of Curvature and Temperature

^{1}

^{2}

^{*}

## Abstract

**:**

^{−1}and −0.90 nm/m

^{−1}, in a curvature range of 0 m

^{−1}to 1.87 m

^{−1}, and temperature sensitivities of 21.7 pm/°C and 16.6 pm/°C, in a temperature range of 50 °C to 500 °C, regarding the external resonance and internal resonance, respectively. The proposed sensor is promising for the implementation of several applications where simultaneous measurement of curvature and temperature are required.

## 1. Introduction

^{−1}within a curvature range up to 2.14 m

^{−1}. Moreover, a hybrid sensor for the simultaneous measurement of temperature and curvature has been proposed by Cheng et al. [37], with the ARHCF being spliced between two SMFs. The acquired sensitivities for the temperature and curvature were 25.76 pm/°C and −4.28 dB/m

^{−1}, respectively. Further sensors have already been developed to measure these parameters [38,39].

## 2. Fiber Geometry

## 3. Principle of Operation

_{11}) as well as its intrinsic properties. Considering the perturbation theory model, encountered in [44], the effective refractive index of a propagating mode is described by the following expansion:

_{m,n}, $\varphi $ is the accumulated phase between two consecutive reflections of a light ray in the silica glass structure, and the parameter $\u03f5={({n}_{\mathrm{Si}}/{n}_{\mathrm{air}})}^{2}$. Developing the set of equalities from Equations (2)–(8), one concludes that the effective refractive index can be given by:

_{11}mode, attained by using Equation (9). It is also presented the simulated profile of the real part of the effective refractive index of the HSCF in the spectral range of the visible and infrared, resorting to the COMSOL Multiphysics. The COMSOL Multiphysics simulation (version 5.6) was carried out over a wavelength range between 600 nm and 1600 nm, in steps of 1 nm near the resonance wavelengths and steps of 10 nm in the regions away from these. The Sellmeier equation was used to estimate the refractive index of silica for each wavelength.

_{11}mode, one has to analyze the first cladding mode, that is, the hybrid mode HE

_{12}. Furthermore, the parameters r and w, which were considered to be the core radius and the silica strands thickness, will change since in the external resonance the fiber is comparable to a capillary with a core radius of d and thickness equal to the difference between the fiber radius and the core radius. With the conjecture of a complex effective refractive index, where the imaginary part is associated with the losses subjected to the mode propagation, it is possible to attain the profile loss associated with the HSCF, by considering that the major factor that induces losses in the HSCF and the significant diminishing of the optical power is the confinement loss (CL) of the propagating mode. Since in the HC-PCFs, the light is guided within air, loss factors such as absorption and the Rayleigh scattering are too small [2,45]; therefore, they were disregarded. The confinement loss is as follows [46]:

## 4. Results and Discussion

#### 4.1. Sensor Design and Experimental Setup

#### 4.2. Spectral Characteristics

^{−1}low pass filter was applied. The experimental proceedings were performed by applying a Gaussian fit to the entire depression band (IR dip) and by monitoring the ER dip (λ

_{62}).

#### 4.3. Sensor Characterization

^{−1}. The decrease of height was done in steps of 2 mm. In Figure 10, the experimental results attained for the IR and ER are presented. Both components presented a shift towards smaller wavelengths (blue shift). From the results, one can infere that both resonance responses presented a linear tendency, leading to curvature sensitivities of (−0.22 ± 0.02) nm/m

^{−1}(r

^{2}= 0.95064) and (−0.90 ± 0.02) nm/m

^{−1}(r

^{2}= 0.99575), for the ER and IR, respectively. The difference between the correlation coefficients can be attributed to the low pass filter applied in the data processing. Further studies, which are not within the scope of this work, should be performed regarding the best filter to be applied in the context of an application.

^{2}= 0.99809) and (16.6 ± 0.7) pm/°C (r

^{2}= 0.98517) for ER and IR, respectively. The magnitude of these values is in agreement with what was established in the literature [50], where it is expected to achieve a higher sensitivity for the thickest resonant structure, that is, for the ER.

^{ext}) and IR response (Δλ

^{int}) to variations on the curvature (ΔC) and temperature (ΔT) can be described by the following expressions:

^{−1}, °C, and nm, respectively. This outcome potentially enhances the use of this sensor in the simultaneous measurement of these parameters, making it a good candidate for several applications.

## 5. Conclusions

^{−1}and −0.22 nm/m

^{−1}, in a curvature range of 0 nm/m

^{−1}to 1.87 nm/m

^{−1}, were attained for the IR and ER, respectively. On the other hand, temperature sensitivities of 21.7 pm/°C and 16.6 pm/°C were respectively achieved for the ER and IR. The proposed inline sensor is innovative due to its reduced dimensions, robustness, and capability on measuring more than one parameter without needing a complex design configuration or using several sensing heads, but instead merely resorting to the ARROW guidance properties.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Microscopic (400$\times $) picture of the HSCF cross-section. (

**b**) Geometrical scheme of the HSCF.

**Figure 2.**Scheme of the optical paths of light in the internal AR guiding process, which occurs in the core area, and the external AR, that occurs in the outer cladding section.

**Figure 3.**(

**a**) Imaginary effective refractive index with wavelength obtained from Equation (9) (Numerical). (

**b**) Real component of the effective refractive index with a wavelength attained by the COMSOL Multiphysics (Simulation) and by Equation (9) (Numerical). The expected frequency of the IR is represented by dash lines, according to Equation (1).

**Figure 4.**(

**a**) Numerical solution of the external AR spectrum. (

**b**) Numerical solution of the internal AR spectra. (

**c**) Numerical solution of the transmission spectrum of the HSCF.

**Figure 5.**(

**a**) Schematic representation of the sensor fabrication, wherein the HSCF is spliced between two SMFs and (

**b**,

**c**) longitudinal view of the splicing area for different arc discharge times.

**Figure 7.**Transmission spectra of a HSCF sensor with lengths of (

**a**) 7.50 mm and (

**b**) 9.98 mm. The dash lines represent the central wavelength of each transmission band.

**Figure 8.**(

**a**) Representation of the simulated and experimental transmission spectrum of the 7.50 mm HSCF. (

**b**) Amplification of the transmission spectra in the range of 1200 nm−1600 nm, where the external AR modulation is observable. (

**c**) Values of the ER dips attained from the numerical and experimental spectrum, and the theoretical values from Equation (1).

**Figure 9.**Transmission spectrum of the 7.20 mm long sensor and the respective curve attained by applying a 0.11 nm

^{−1}low pass filter on the spectral range of interest. The ER wavelength (λ

_{62}) analyzed is also indicated in the figure.

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**MDPI and ACS Style**

Pereira, D.; Bierlich, J.; Kobelke, J.; Ferreira, M.S.
Double Antiresonance Fiber Sensor for the Simultaneous Measurement of Curvature and Temperature. *Sensors* **2021**, *21*, 7778.
https://doi.org/10.3390/s21237778

**AMA Style**

Pereira D, Bierlich J, Kobelke J, Ferreira MS.
Double Antiresonance Fiber Sensor for the Simultaneous Measurement of Curvature and Temperature. *Sensors*. 2021; 21(23):7778.
https://doi.org/10.3390/s21237778

**Chicago/Turabian Style**

Pereira, Diana, Jörg Bierlich, Jens Kobelke, and Marta S. Ferreira.
2021. "Double Antiresonance Fiber Sensor for the Simultaneous Measurement of Curvature and Temperature" *Sensors* 21, no. 23: 7778.
https://doi.org/10.3390/s21237778