1 Introduction

Curvature measurement plays more and more important role in practical engineering applications, such as structural health monitoring [1], civil engineering [2], medicine [3] and aerospace [4]. The traditional electromagnetic sensors based on electrical/magneto-resistive effect are susceptible to environmental corrosion and electromagnetic interference [5, 6], which largely limits the demands of compact and in-suit curvature sensing applications. Due to the desirable advantages of high sensitivity, compact structure, lightweight, anti-electromagnetic interference, corrosion resistance, and independence of the environment, the fiber-optical sensor has been extensively investigated in curvature detection [7]. A variety of fiber-optical curvature sensing schemes have been developed. Fiber Bragg grating [8, 9] (FBG) and long-period fiber grating (LPFG) sensors are widely used in curvature sensing, such as sensors engraved in the center core FBG of a seven-core optical fiber [10], with a curvature sensitivity of − 7.27 dB/m−1 in the curvature range of 0 m−1 to 1 m−1. The maximum sensitivity of the curvature sensor of the proposed LPFG and multi-mode fiber (MMF) spliced into the LPFG-MMF-LPFG structure is − 21.08 nm/m−1 (0.246 m−1–0.738 m−1) [11] and the maximum temperature sensitivity of 60 pm/°C. However, the curvature measurement range of the applied encoder is small, limiting its application in a wide range of fields. Specialty fibers are also being actively used for curvature sensing [12]. Tang et al. utilized a 25.7 cm photonic crystal fiber (PCF) with dual cores, sandwiched between two cones, to create a curvature sensor [13]. The sensor achieved a sensitivity of 18.29 nm/m−1, while the maximum temperature sensitivity was − 30.98 pm/°C. Zhao et al. proposed a curvature sensing structure that uses an MMF, a tapered core single-mode fiber (TCSMF), and an MMF cascade [14]. The TCSMF is 12 cm long and the curvature sensor has a curvature sensitivity of − 28.29 nm/m−1 at 2.79 m−1–3.24 m−1. But the application of special optical fibers increases the production cost and the sensor size is large, which is unsuitable for a small range or a certain point of curvature detection [15]. Fiber-optic interferometers are easy to fabricate, have high sensitivity and accuracy, and provide a good platform for curvature measurements. In 2019, Cheng et al. spliced a 5-mm-long HCF and two SMFs, with the joints made into upper cones, to form a Mach–Zehnder interferometer (MZI), obtaining curvature and temperature sensitivities of − 4.28 dB/m−1 and 25.76 pm/°C [16], respectively. In 2020, M. García et al. proposed an MZI structure consisting of MMF–HCF–MMF, where the HCF length is 2.5 mm [17]. The maximum curvature sensitivity is 19.58 dB/m−1 (1.84 m−1 to 2.94 m−1) and the temperature-induced maximum curvature contrast of 0.55 dB over the 35–60 °C range. In the same year, H. Piad et al. presented an MZI by splicing a section of capillary hollow-core fiber between two SMFs, and the curvature sensitivity of the sensor was − 5.62 dB/m−1 (0 m−1–2.68 m−1). The temperature-induced contrast was 0.95 dB between 25 °C and 60 °C [18]. In 2022, Qi et al. made FP interferometric curvature sensors by sandwiching a section of 1197 µm long HCF between two sections of SMF. The maximum curvature sensitivity was 0.2345 nm/m−1 and the highest temperature sensitivity was 0.0172 nm/°C [19], respectively. Recently Zhu et al. used MZI with SMF–HCF–SMF ends fused into the shape of a peanut, with a maximum curvature sensitivity of − 648.17 pm/m−1 and maximum temperature sensitivity of 25.42 pm/°C [20]. Apart from mechanical external forces, temperature changes can also cause curvature changes in fiber-optic sensors. Thus, improving temperature crosstalk has become a crucial area of research for developing these sensors.

In this paper, we present an HMZI sensor based on HCF microbubble structure for curvature monitoring. The curvature sensor consists of SMF fused to HCF. In the curvature measurement, different curvatures are achieved by continuously shortening the distance between the optical fibers under the effect of gravity, and the curvature experiments yielded a curvature sensitivity of − 1.48 dB/m−1 within the range of curvatures from 1.22 m−1 to 3.46 m−1. Temperature experiments are carried out to confirm the impact of temperature changes on the sensor. Temperature-induced contrast does not exceed 0.30 dB in the 35–60 °C, which shows that our novel curvature sensor is insensitive to temperature and ensures the accuracy of curvature sensing. Our proposed HMZI curvature sensor is compact, suitable for a large range of curvatures, practical, inexpensive to produce, and easy to apply in a variety of complex environments. The work provides a good idea for fiber-optic sensing to measure curvature.

2 Sensor structure and principle analysis

Figure 1 gives the diagrams of the proposed SMF-HCF-SMF sensing structure, including the detailed parameters of SMF, HCF, and the fusion structure of SMF-HCF-SMF. The lead-in and lead-out splicing points were formed at the length of 30 µm. When fusing SMF and HCF, it is important to note that direct fusion splicing between these two types of fibers can cause the hollow core of the HCF to collapse. Therefore, extra care should be taken when performing fusion splicing operations between SMF and HCF. When directly fusing the HCF, it is important to keep in mind that the HCF may be exposed to temperature changes. This causes the hollow part of the fiber to expand, which leads to the deformation of the fiber, thus affecting the experiment. Qi et al. [19] prevents the hollow part from collapsing, weakly discharged, and manually mode several times during the fusion splicing process [21]. But this operation requires many manual adjustments and a lot of manual actions that are more accidental. To avoid such a problem, the paper cleverly uses the temperature effect on the hollow part of the HCF. The ends of the HCF are first fused using a fusion splicer. The HCF is then collapsed, forming a structure similar to a centerless fiber. Finally, the processed HCF is fused to SMF, as shown in Fig. 1d. It can be a good solution to the problem of hollow core collapse, while also effectively avoiding the expansion and deformation of the hollow part due to temperature changes during the fusion welding process. As shown in Fig. 1c, the fiber optic curvature sensor can be obtained successfully according to this process of sequence fusion bonding.

Fig. 1
figure 1

a and b The geometric parameters of the core and cladding of SMF and HCF, repectively. c Fabrication process diagram of SMF-HCF-SMF fusion. d Schematic diagram of sensor structure

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The beam travels to the first fusion point of the SMF and HCF and splits into two beams. One beam propagates as a fundamental mode in the core of the HCF, while another beam propagates as a cladding mode in the cladding. When the two parts of the light reach the second fusion point of the output SMF and HCF, the beams are coupled. Since the propagation distances of the two beams are different, a phase difference naturally occurs when they propagate to the output SMF, resulting in interference. The corresponding optical path is shown in Fig. 2.

Fig. 2
figure 2

Optical paths in SMF-HCF-SMF sensing structures

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Beam propagation method is used to simulate the transmission light power evolution along z-direction. Figure 3a shows the detailed model with the cladding of the SMF and HCF in red and the core in blue. The empty core part of HCF is depicted in green, and the conical tip formed by the heating and collapsing of the HCF is shown in purple. The grey part can be considered as a coreless fiber with a width equal to the outer diameter of the SMF. Figure 3b gives the simulated optical field distribution. As the incident beam passes through the first fusion splice point, the light splits and propagates in the core and the cladding, respectively [22]. Finally, the beams combine and interference at the second fusion point [23]. When the sensing head undergoes a bending change, the output light would change in both the intensity and filed distribution.

Fig. 3
figure 3

a Schematic diagram for the building simulation model. b Simulation of beam propagation

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In a hollow microbubble MZI curvature, the interference intensity could be described as [24]:

$$I={I}_{1}+{I}_{2}+2\sqrt{{I}_{1}{I}_{2}}\mathit{cos}\left(\Delta \phi \right)$$
(1)

where \({I}_{1}\) and \({I}_{2}\) denote the light intensity [25] within the cladding and hollow microbubbles, respectively. \(\Delta \phi\) is the phase difference between the two beams that can be calculated by:

$$\Delta \phi =2\pi \left({n}_{cladding}-{n}_{air}\right){L}_{HCF}/\lambda =\frac{2\pi \Delta {n}_{eff}{L}_{HCF}}{\lambda }$$
(2)

\({n}_{cladding}\) and \({n}_{air}\) are the refractive indices of the cladding and the hollow core in the HCF, respectively. \(\Delta {n}_{eff}\) represents the difference between the two. \({L}_{HCF}\) represents the length of the HCF and \(\lambda\) is the wavelength of the light wave.

According to the interference cancellation condition, the resonant dip for the m-level interference with a wavelength of λm, can be expressed [26] as:

$${\lambda }_{m}=2\Delta {n}_{eff}{L}_{HCF}/(2m+1)$$
(3)

Based on the travelling distance of the removable platform and the initial distance, the curvature C can be approximated [27] as:

$$C=1/R=\sqrt{\frac{24x}{{L}^{3}}}$$
(4)

where x represents the travel distance between the two removable platforms and L is the initial distance between the two clamping tables.

The normalized transmittance [28] for multimode interference can be expressed as:

$${T}_{MZI}={B}_{m}{cos}^{2}\left(\frac{\pi L_{HCF}}{\lambda }\bullet \Delta {n}_{m}\right)$$
(5)

where \({B}_{m}\) is the spectral intensity coefficient, \(m\) is the number of cladding mode orders, \({\Delta }_{m}\) is the effective refractive index difference between the core and cladding modes of order m. It can be seen that the spectral transmission intensity could be tuned by the refractive index of the HCF core and cladding.

3 Results and discussion

The schematic diagram of the curvature sensing system based on the HCF assisted microbubble MZI is shown in Fig. 4a. The sensing structure is represented by the red line segments, which consist of SMF–HCF–SMF. The broadband source (BBS) emits light from 1250 to 1660 nm. Through HCF-MZI sensing head, it reaches the spectral analyzer (OSA) and finally generates the spectral data. The ends of the sensor are placed on removable platforms. As the distance between the two removable platforms changes, the sensing structure bends under gravity, causing a change in curvature and a shift in the transmission spectrum of sensors. In conducting the experiments, one of the removable platforms is fixed and the other is moved axially along the fiber to change the curvature of the sensing structure.

Fig. 4
figure 4

The experimental setup of the proposed sensor in terms of (a) curvature sensing, and (b) temperature sensing

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The initial state without bending (C = 0 m−1) is defined at L = 19 cm, which denotes the distance between two fiber tables. As shown in Fig. 5, the corresponding transmission spectrum is measured, there is a large attenuation in 1300–1350 nm, which is not favorable for subsequent observation. For subsequent experiments, the band at 1452 nm is chosen as the sensing frequency (marked in red circle in Fig. 5).

Fig. 5
figure 5

Transmission spectrum of the sensor when it is not bent

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Experiments to measure curvature are conducted in a room temperature environment at approximately 25 °C. By moving removable platforms and changing the distance by 0.5 cm each time, different curvature degrees of the sensor are reached so that the curvature varies from 1.22 m−1 to 3.46 m−1. The sensing performance of different curvatures could be obtained with transmission spectra in Fig. 6a.

Fig. 6
figure 6

a The measured transmission spectra of different curvatures. b The curvatures dependent transmission with a linear fitting

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It can be easily seen that at 1452 nm, the transmitted intensity is decreasing and the depth is increasing with the increase of curvature. It could be due to gravity causing the fiber-optic sensor to bend downward, and the bending naturally causes one side of the fiber-optic sensor to bulge outward due to stretching. At the same time, the other side is concave inward due to compression. Under different bending scenarios, the optical path and optical travel of the MZI are different, resulting in different modes. Meanwhile, the symmetry propagation of the core and cladding modes within the HCF in which the bending occurs is broken [28], which leads to signal leakage. The greater the curvature, the more serious the signal leakage is, and the transmission intensity naturally decreases. It is evident that the dip wavelength shift is less than 0.2 nm and can be considered linear, which means that wavelength fluctuations can be ignored when testing the curvature. The curve of transmittance versus curvature is obtained by data fitting. As can be seen in Fig. 6b, the transmittance intensity shows a good linear relationship with the curvature in the range from 1.22 to 3.46 m−1. The degree of linearity is 0.989. It can be noticed that as the curvature gradually increases, the transmittance intensity decreases, and the bending sensitivity of the intensity response can be obtained as − 1.48 dB/m−1.

Next, curvature sensing performance considering the external temperature variations are conducted to investigate the effect of temperature on the HMZI curvature sensor, as shown in Fig. 4b. The fiber-optic curvature sensor is placed on a temperature control platform that allows for temperature control during the temperature-sensing process. The beam passes through the curvature sensors in different temperature environments and reaches the OSA. The sensor ends are fastened to a movable platform in a non-bending state with x = 0 cm and the sensing section is smoothly placed on the heated platform. There should be a warning that before the formal experiment, a simple test is conducted to check whether the heating platform can ensure the sensor part is heated uniformly as a way to reduce the experimental error. After all the preparatory work is completed, the transmitted spectra is experimentally measured in a tunable temperature range from 35 to 60 °C with a step of 5 °C.

In the temperature experiments, transmission intensities at different temperatures are selected and analyzed. The fluctuation range of the transmission intensity due to temperature change can be obtained. As shown in Fig. 7, the maximum instability is only 0.30 dB between 35 °C and 60 °C, which indicates that the HMZI sensor is less affected by temperature in the common temperature range from 35 to 60 °C. After the above experiments, our proposed HMZI curvature sensor can be applied to various complex situations.

Fig. 7
figure 7

The measured temperature (35–60℃) dependent transmissions

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4 Conclusion

In summary, we have proposed and experimentally demonstrated a temperature-insensitive HMZI curvature sensor with a large curvature range. The sensor consists of an SMF–HCF–SMF fused structure to excite the optical interference between the multi-mode in the core and the cladding modes. The measured spectral transmission at the wavelength of 1452 nm shows a linear response to the applied curvature, which exhibits a sensitivity of − 1.48 dB/m−1 in a curvature range from 1.22 m−1 to 3.46 m−1. Our experimental results also have a good performance in temperature resistance, and temperature-induced perturbation to the curvature sensing is below 0.30 dB in a temperature range from 35 to 60 °C. Compared with recently reported fiber-optic curvature sensing schemes, the HMZI is compact, large-range, and temperature-insensitive for related curvature monitoring applications.