# Negative Curvature Hollow Core Fiber Based All-Fiber Interferometer and Its Sensing Applications to Temperature and Strain

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## Abstract

**:**

_{11}and HE

_{12}. A small portion of light guiding by means of Anti-resonant reflecting optical waveguide (ARROW) mechanism is also observed. The transmission dips, resulting from multimode interferences (MMI) and ARROW effect have a big difference in sensitivities to strain and temperature, thus making it possible to monitor these two parameters with a single sensor head by using a characteristic matrix approach. In addition, the proposed sensor structure is experimentally proven to have a good reproducibility.

## 1. Introduction

## 2. Experimental Setup

_{out}) of 125 μm, an inner diameter (D

_{in}) of 76 μm, each thin silica capillary tube has an inner diameter (D

_{tube}) of 14 μm, a silica strut thickness (t) of 1.4 μm and a core diameter (D

_{core}) of approximately 45.2 μm.

## 3. Theoretical Analysis

_{eff}) with a finite-element method (FEM). Throughout the simulation, the input light wavelength is set to 1300 nm, the refractive indices (RIs) of the core and cladding of the SMF are 1.4519 and 1.4469 and the RIs of the silica cladding of NCHCF and air are 1.4469 and 1.0, respectively. The RI of the silica cladding is calculated from the Sellmeier equation [20]. Figure 2a shows examples of the simulation results of different modes in an NCHCF, including the fundamental air core mode, high order air core modes, supermodes (hybrid modes between air core modes and capillary air cladding modes) and silica cladding modes. It is found that the air cladding modes propagating within the capillary tubes have slightly lower n

_{eff}than that of the fundamental air core mode (HE

_{11}), thus high order air core modes are easier to couple into the capillary air cladding, and those modes propagating in the air core and capillary air cladding interfere with each other, producing the supermodes. In the NCHCF, most of the light is confined in the air core and capillary tubes, and only a very small portion of light is coupled into the outer silica cladding.

_{1}and I

_{2}are light intensities of the two modes involved in the interference, φ is the phase difference between the two modes. Assuming the incident light has a wavelength of λ, ∆n

_{eff}is the difference between the effective refractive indices of the two interfering modes and L is the length of the sensor head, then:

_{dip}) shown in the transmission spectrum are thus can be derived as:

_{1}and λ

_{2}are the central wavelengths of the two adjacent transmission dips. As can be seen from Equations (4) and (5), the positions of the transmission dips and their corresponding FSRs are dependent on the difference of effective RIs of the interfering modes and the length of the NCHCF. Since the RIs of the fiber core, cladding and fiber length are dependent on the surrounding temperature and strain applied to the fiber, the proposed NCHCF-based fiber interferometer can be easily employed as a temperature (T) and stain (ε) sensor by measuring the wavelength shifts of the transmission dips. The temperature and strain sensitivities can be obtained from Equation (4), as in [22]:

_{eff}, L and their corresponding changes towards strain and temperature variations, and the selected λ

_{dip}.

## 4. Results and Discussion

#### 4.1. Spectral Response in Air

_{1}, A

_{2}, A

_{3}, A

_{4}, B

_{1}, B

_{2}, B

_{3}, B

_{4}, C

_{1}, C

_{2}, C

_{3}, C

_{4}, D

_{1}, D

_{2}, D

_{3}, D

_{4}, respectively. The relationship between the spatial frequencies (corresponding to the above frequency peaks) and the sensor length is plotted in Figure 3c. In the figure, the straight lines are the linear fittings for the data obtained from Figure 3b. As one can see from the figure, the spatial frequency increases linearly with the sensor length, indicating that interferences in samples with different sensor lengths are created by the same modes. The result agrees well with the theoretical prediction, where the spatial frequency (ξ) can be expressed as [26]:

_{eff}is the differential modal group index. The slope of the fitted curve in Figure 3c is determined by the difference in the effective group index for the interfering modes.

#### 4.2. Temperature and Strain Measurement

_{4}, B

_{4}, C

_{4}, D

_{4}) as shown in Figure 5. It is evident that the spectrum corresponding to frequency A

_{4}is the main contribution to dips A and B. It is known that ∆n

_{eff}can be calculated with the following equation [31]:

_{4}is 31.73 nm at 1300 nm, then ∆n

_{eff}is calculated to be 1.13 × 10

^{−3}, which is very close to the simulated ∆n

_{eff}(1.03 × 10

^{−3}) between HE

_{11}and HE

_{12}modes at 1300 nm, it is thus concluded that the modes coupling between HE

_{11}and HE

_{12}are the dominant modal interferences, and produce the large extinction ratios dips in transmission.

_{11}and HE

_{12}are well confined inside the central hollow core and they are isolated from the outer environment.

_{m}and λ

_{m+}

_{1}are the central wavelengths of adjacent resonant dips, n

_{1}and n

_{2}are the refractive indices of silica cladding and air, d is the outer silica cladding thickness. The calculated FSR is 32.8 nm which matches well with the experimental measured FSR of 30.4 nm between dips C and D. Silica material has a relatively high thermo-optical coefficient, and hence a higher temperature sensitivity. However, variations of the central wavelengths of the dips produced by the ARROW effect are independent of the change of sensor head length [25], thus they have lower strain sensitivities than those of dips produced by MMI. It is also noted that much higher temperature sensitivities could be potentially achieved by the functionalization of the NCHCF with liquids or other materials, which is under investigation.

_{A}and ∆λ

_{C}are wavelength shifts of dips A and C, and ∆T and ∆ε are temperature and strain variations.

## 5. Reproducibility

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Schematic diagram and (

**b**) SEM image of the negative curvature hollow core fiber (NCHCF); (

**c**) schematic diagram of the experimental setup for strain and temperature measurements. Inset microscope image shows the connecting point between singlemode fibers (SMF) and NCHCF after one arc splice.

**Figure 2.**(

**a**) Examples of different mode profiles (normalized) within the NCHCF, and their corresponding effective refractive indices, simulated by an finite-element method (FEM).; (

**b**) Energy distributions in the XZ plane along the SMF-NCHCF-SMF length (lengths of the input and output SMF, and NCHCF are 200 μm, 500 μm and 3000 μm, respectively) and the mode profile evolution at the cross-section at different fiber structure lengths, simulated by a beam propagation method (BPM).

**Figure 3.**(

**a**) Measured transmission spectra and (

**b**) fast Fourier transform (FFT) spatial frequency spectra corresponding to those in (

**a**), and (

**c**) spatial frequency variations versus the NCHCF length.

**Figure 4.**Measured spectral responses of S-48 versus (

**a**) strain in the range from 0 to 1200 με and (

**b**) temperature in the range from 19 °C to 108 °C, and the corresponding wavelength shifts of dips A, B, and C in relation to (

**c**) strain and (

**d**) temperature. The straight lines are linear fits for the measured data.

**Figure 5.**Measured spectral responses of S-48 and fast Fourier transform (FFT) band pass filter recovered transmission spectra corresponding to different frequencies of A

_{4}, B

_{4}, C

_{4}, D

_{4}.

**Figure 6.**(

**a**) Measured transmission spectra for S-48 in air and water (RI = 1.333) and (

**b**) their corresponding FFT spatial frequency spectra. (

**c**) Measured transmission spectra for S-48 in liquids with different refractive indices (RI).

**Figure 7.**(

**a**) Measured transmission spectra of five reproduced sensor samples with a NCHCF length of 39 mm; (

**b**) Measured transmission spectra with different number of arcs performed on the splice point between SMF and NCHCF.

**Figure 8.**Measured transmission spectra of five sensor samples with an MMF length of ~39 mm for (

**a**) MMF 50/125; and (

**b**) 105/125.

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**MDPI and ACS Style**

Liu, D.; Li, W.; Wu, Q.; Zhao, H.; Ling, F.; Tian, K.; Shen, C.; Wei, F.; Han, W.; Farrell, G.;
et al. Negative Curvature Hollow Core Fiber Based All-Fiber Interferometer and Its Sensing Applications to Temperature and Strain. *Sensors* **2020**, *20*, 4763.
https://doi.org/10.3390/s20174763

**AMA Style**

Liu D, Li W, Wu Q, Zhao H, Ling F, Tian K, Shen C, Wei F, Han W, Farrell G,
et al. Negative Curvature Hollow Core Fiber Based All-Fiber Interferometer and Its Sensing Applications to Temperature and Strain. *Sensors*. 2020; 20(17):4763.
https://doi.org/10.3390/s20174763

**Chicago/Turabian Style**

Liu, Dejun, Wei Li, Qiang Wu, Haoyu Zhao, Fengzi Ling, Ke Tian, Changyu Shen, Fangfang Wei, Wei Han, Gerald Farrell,
and et al. 2020. "Negative Curvature Hollow Core Fiber Based All-Fiber Interferometer and Its Sensing Applications to Temperature and Strain" *Sensors* 20, no. 17: 4763.
https://doi.org/10.3390/s20174763