1 Introduction

Optical fiber sensors are widely used in various fields such as biomedical [1], food processing [2], civil engineering [3], and aerospace [4], due to their small size, resistance to electromagnetic wave interference, high sensitivity, and low production cost. These sensors enable the detection of physical parameters like temperature [5], refractive index [6], strain [7], magnetic field [8], and torsion [9]. Over the years, researchers have introduced several types of optical fiber sensors, including Mach–Zehnder interferometer (MZI) [10], Sagnac interferometer (SI) [11], Fabry–Perot interferometer (FPI) [12] fiber Bragg grating (FBG) [13], and long-period fiber grating (LPFG) [14].

Strain is a significant physical parameter used to assess the health of machinery and equipment, bridges, and buildings, making it of utmost importance in daily life and industrial production [15, 16]. Researchers have developed strain sensors based on the Vernier effect to greatly enhance sensor sensitivity [17, 18]. However, many optical fiber strain sensors encounter cross-sensitivity issues with other physical quantities, especially temperature [19], leading to measurement errors during sensor operation [20]. To improve measurement accuracy, researchers have proposed various optical fiber sensors to address cross-sensitivity challenges.

In 2020, Gao et al. introduced an optical fiber sensor using FMF fused to SMF with a core-offset and cascaded FBG [21]. The sensor exhibited distinct sensitivities for different spectrum dips, enabling the differentiation of wavelength shifts caused by strain and temperature, thus facilitating real-time monitoring of strain and temperature with temperature sensitivity of − 34.3 pm/℃ and 10.7 pm/℃, and strain sensitivity of − 2 pm/με and 0.67 pm/με.

In 2022, Xiao et al. developed a Mach–Zehnder interferometer-based optical fiber sensor connecting SMF and commercialized panda-type polarization maintaining fiber (PMF) through peanut-shaped tapers [22]. The different responses of the two fibers to temperature and strain changes were utilized to distinguish between temperature and strain variations, resulting in strain and temperature sensitivities of − 1.441 pm/με and 78.29 pm/℃, respectively.

In 2023, Chang et al. proposed an optical fiber strain and curvature sensor based on a three-layer core few-mode fiber, demonstrating a strain sensitivity of − 2.78 pm/με and a curvature sensitivity of − 6.17 pm/m−1. The sensor was insensitive to temperature changes, effectively eliminating cross-sensitivity issues between temperature and strain [23].

To address the cross-sensitivity between strain and temperature, this paper proposes an optical fiber strain sensor composed of FMF, SMF, and NCF. Core-offset is introduced at the fusion point of NCF and FMF to excite higher-order modes in the fiber and enhance strain sensitivity. Experimental results reveal that the sensor exhibits high strain sensitivity, with the interference spectrum showing a blue-shift with increasing strain. The strain sensitivity is − 19.19 pm/με, and the temperature sensitivity is − 2 pm/℃. The strain-temperature crosstalk is only 0.104 με/℃, effectively mitigating the influence of temperature on the sensor's performance and enhancing its utility value.

2 Working principle

This paper proposes a sensing structure based on SMF-FMF-NCF-FMF-SMF with core offset, as shown in Fig. 1. Two end faces of one NCF section are fused to the FMF using the core-offset method. The other end faces of the two FMFs are coaxially fused to the SMF.

Fig. 1
figure 1

Structural illustration of the sensor

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The input light is coupled from SMF1 into the core of FMF1 and then into the NCF at the first core-offset point, where higher-order modes are excited due to a mode field mismatch. At the second core-offset point, part of the light is coupled into the core of FMF2, while the other part propagates in the cladding of FMF2. The refractive indices of the cladding and the core are different, creating an optical path difference between the two beams, resulting in mode interference when coupled into the core of SMF2, generating interference peaks. The intensity of the interfering light I can be expressed as [24]:

$$I={I}_{0}+\sum_{m=1}^{n}{I}_{m}+2\sum_{m=1}^{n}\sqrt{{I}_{0}{I}_{m}}\mathrm{cos}\left(\Delta {\varphi }_{m}\right)$$
(1)

where \({I}_{0}\) represents the intensity of the core fundamental mode and \({I}_{m}\) is the intensity of higher-order cladding mode. \(\Delta {\varphi }_{m}\) denotes the phase difference between the fundamental mode and the higher-order cladding mode, which can be expressed as:

$$\Delta {\varphi }_{m}=\frac{2\pi\Delta {n}_{eff}^{m}L}{\lambda }$$
(2)

where \(\Delta {n}_{eff}^{m}\) is the effective refractive index difference between the fundamental mode and the higher-order cladding mode. \(L\) is the sensor length and \(\lambda\) is the wavelength of the input light. When the condition \(\Delta {\varphi }_{m}=(2k+1)\pi\) is satisfied (\(k=\mathrm{1,2},3\dots \dots\)), the interference light intensity \(I\) reaches its minimum value, and the resulting interference dip can be expressed as:

$${\lambda }_{m}=\frac{2\Delta {n}_{eff}^{m}L}{2k+1}$$
(3)

where \({\lambda }_{m}\) represents the wavelength at which the interferometric light intensity \(I\) takes its minimum value. From Eq. (3), the free spectral range (FSR) can be expressed as:

$$FSR\approx \frac{{\lambda }_{m}^{2}}{\Delta {n}_{eff}^{m}L}$$
(4)

In our experiments, the stretching method is used to generate axial strain, the magnitude of which can be expressed as [25]:

$$\Delta \varepsilon =\frac{\Delta L}{L}$$
(5)

where \(\Delta L\) is the change in length of the fiber and \(\Delta \varepsilon\) is the strain generated when stretching the fiber. As the fiber is stretched, the refractive index of the fiber changes due to the elasto-optical effect [24]. The refractive index change rates of the fiber cladding and core are not uniform, resulting in a change in \(\Delta {n}_{eff}^{m}\) as well. The combined effect of the change in fiber length and the change in \(\Delta {n}_{eff}^{m}\) will result in a shift of \({\lambda }_{m}\). Additionally, due to the thermo-optic effect [26], temperature changes in the sensor will also lead to refractive index changes in the fiber, resulting in a wavelength shift of \({\lambda }_{m}\). The corresponding wavelength shift \(\Delta \lambda\) can be expressed as [22]:

$$\frac{\Delta \lambda }{{\lambda }_{m}}= \left[1+\frac{1}{\Delta {n}_{eff}^{m}}\bullet \frac{\partial (\Delta {n}_{eff}^{m})}{\partial \varepsilon }\right]\Delta \varepsilon + \left[\frac{1}{\mathrm{L}}\bullet \frac{\partial L}{\partial T}+\frac{1}{\Delta {n}_{eff}^{m}}\bullet \frac{\partial (\Delta {n}_{eff}^{m})}{\partial T}\right]\Delta T$$
(6)

where \(\Delta T\) is the varying ambient temperature. From Eq. (6), it can be observed that \(\Delta \lambda\) is positively correlated with \(\Delta \varepsilon\) when the sensor temperature remains stable. Therefore, the strain generated by the optical fiber can be measured by detecting the wavelength shift \(\Delta \lambda\) for the purpose of strain sensing.

3 Device fabrication of sensing structures

The device fabrication process of the proposed optical fiber strain sensor is detailed in this section. The sensor components include SMF, FMF, and NCF. The SMF-28e (Corning) single-mode fiber with cladding and core diameters of 125 μm and 8.2 μm, respectively. The FMF (YOFC) used has cladding and core diameters of 125 μm and 18.5 μm, with refractive indices of 1.444 and 1.4499 for the cladding and core, respectively. This FMF can supporting four modes (LP01, LP11, LP21 and LP02 mode). The NCF (YOFC) has a diameter of 125 μm and a refractive index of 1.444. A commercial fusion splicer, FITEL S179A (Furukawa, Japan), was used for the fabrication process.

The sensor fabrication process is illustrated in Fig. 2:

  1. 1.

    The SMF and FMF are separately cleaved using an optical fiber cleaver to obtain flat end faces.

  2. 2.

    The SMF and FMF are placed on the V-slot of the fusion splicer and fused together. The FMF is then cut to the appropriate length.

  3. 3.

    During the fusion operation of NCF and FMF, the fibers are aligned in the Y direction, and the offset of the fiber in the X-axis direction is adjusted. The fusion splice is then completed, and the NCF is cut using a fiber optic cleaver.

  4. 4.

    The above steps are repeated to complete the fusion on the other side of the NCF.

Fig. 2
figure 2

a Fabrication process, b Micrograph of optical fiber sensor

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Compared to coaxial fusion, the core-offset fusion method is employed to generate a mode field mismatch and excite higher-order modes in the fiber. Despite the incorporation of multiple fiber sections within the aforementioned fiber structure, the fusion process between SMF and FMF can be executed using an automated fusion splicer program. The only requirement is to adjust the offset of optical fibers and ensure the fiber end face remains flat during the fusion process. Consequently, the fabrication procedure remains uncomplicated.

Previous research has demonstrated the efficacy of the core-offset structure in significantly enhancing the sensitivity of fiber sensors [27, 28]. To achieve a higher contrast interference spectrum, an appropriate core-offset value needs to be selected. Therefore, four optical fiber sensors with different core-offset values (0 μm, 20 μm, 40 μm, and 62.5 μm) were fabricated. The FMF length \({L}_{F}\) was 2 cm and the NCF length \({L}_{N}\) was 5 mm, respectively. The interference spectra were measured, and the results are shown in Fig. 3.

Fig. 3
figure 3

Interference spectra with different core-offset a 0 μm, b 20 μm, c 40 μm, d 62.5 μm

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For core-offset values of 0 μm, 20 μm, 40 μm, and 62.5 μm, the insertion loss of the fiber sensor increases with increasing core-offset, from 8.57 dB for coaxial splicing to 19.53 dB, 21.72 dB, and 24.16 dB, respectively. Additionally, the contrast of the measured interference spectrum is 7.45 dB, 19.58 dB, 12.09 dB, and 13.24 dB, respectively.

As shown in Fig. 4, a fast Fourier transform (FFT) was performed on the interference spectrum. It can be observed that the fiber structures with core-offset values of 20 μm, 40 μm, and 62.5 μm possess a dominant peak with a frequency around 0.020 nm−1 and some weak peaks. Furthermore, there is a significant increase in amplitude compared to the fiber structure without core-offset, indicating that higher-order modes are effectively excited in the core-offset structures. As a result, the fringe contrast of the interference spectrum is greatly improved compared to the coaxial fused structure. Among the core-offset structures, the one with 20 μm core-offset exhibits a higher FFT peak and interference spectrum contrast, as well as a lower insertion loss, making it the optimal choice for subsequent measurements.

Fig. 4
figure 4

FFT of the spectrum with different core-offset

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4 Experimental results and analysis

The experimental setup for strain measurement is shown in Fig. 5. A 1250 nm ~ 1650 nm broadband light source (BBS, Golight) served as the light source, and an optical spectrum analyzer (OSA, Yokogawa, AQ6370) was utilized to measure the output interference spectrum. The sensor was placed between two three-dimensional stages, and the distance between the two fixed points of the optical fiber was increased by turning the knob. The initial distance between the stages was 25 cm, and each knob turn allowed for a distance increase of 0.03 mm. Based on Eq. (5), the strain in the optical fiber was calculated to increase by 120 με after each stage movement. This process was repeated five times, incrementally increasing the strain from 0 με to 600 με. The change in the interference spectrum was recorded using the OSA, and the results are presented in Fig. 6. It is evident that as the strain increases from 0 με to 600 με, the interference peaks dip1, dip2, and dip3 shift towards the shortwave direction (blue shift). Specifically, the interference peaks dip1, dip2, and dip3 shifted by − 11.7 nm, − 9.4 nm, and − 6 nm, respectively. As shown in Fig. 7, the strain sensitivities of dip1, dip2, and dip3 were calculated to be -19.19 pm/με, − 15.26 pm/με, and − 9.55 pm/με, respectively. The corresponding linearities were determined to be 0.9908, 0.95225, and 0.93909, respectively. All three peaks exhibited excellent strain sensitivity and linearity, with dip1 displaying the highest sensitivity and linearity.

Fig. 5
figure 5

Experimental setup for strain sensing

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Fig. 6
figure 6

Interference spectrum changes as strain increases from 0με to 600με

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Fig. 7
figure 7

Fitting results for each interference peak with strain increase

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Fiber sensors can be influenced by various external parameters, especially temperature. Therefore, a temperature sensing experiment was conducted, and the experimental setup is depicted in Fig. 8. The sensor was affixed to a heated platform, and the temperature of the platform was incrementally increased from 30℃ to 70℃, with 10℃ increments. The interference spectrum was recorded by the OSA during the temperature changes, and the results are presented in Fig. 9. It can be observed that the transmission of the three interference peaks increased as the temperature rose. Figure 10 displays the outcomes of the linear fits for the wavelength shifts of dip1, dip2, and dip3. The temperature sensitivities were found to be − 2 pm/℃, 19.2 pm/℃, and 8.4 pm/℃, respectively, with corresponding linearities of 0.85714, 0.99769, and 0.92308. The crosstalk of temperature at the strain sensor for the different interference peaks was calculated to be 0.104 με/℃, − 1.258 με/℃, and − 0.880 με/℃, respectively. These findings indicate that the strain sensor exhibits lower sensitivity to temperature, and under normal operating conditions, the cross-sensitivity effects of temperature variations can be disregarded.

Fig.8
figure 8

Experimental setup for temperature sensing

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Fig. 9
figure 9

Change in interference spectrum as temperature increases from 30℃ to 70℃

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Fig. 10
figure 10

Fitting results for each interference peak with increasing temperature

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A comparison between the proposed optical fiber strain sensor and other existing optical fiber strain sensors is presented in Table 1. It can be observed that the proposed sensor exhibits higher strain sensitivity compared to other sensors, while also exhibiting low temperature sensitivity. Notably, the crosstalk originating from temperature variations in strain measurements stands at a mere 0.104 με/℃, representing the lowest value among comparable strain sensors. This characteristic significantly diminishes the potential influence of temperature on strain measurement accuracy during routine operational conditions. Furthermore, our sensor’s structural configuration boasts advantages of simplicity and cost-effectiveness upon its fabrication process.

Table 1 Sensitivity of different optical fiber strain sensors
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5 Conclusion

In conclusion, a strain fiber sensor based on SMF-FMF-NCF-FMF-SMF with core offset was proposed in this paper. Within a strain range of 0–600 με, the sensitivities of the different interference peaks of the sensor reached − 19.19 pm/με, − 15.26 pm/με, and − 9.55 pm/με, respectively, all exhibiting excellent linearity. Furthermore, the crosstalk of temperature to strain sensors was 0.104 με/℃, − 1.258 με/℃, and − 0.880 με/℃, respectively, effectively avoiding measurement errors caused by temperature variations in the working environment. The proposed strain sensor features a novel design, easy fabrication, low cost, and high sensitivity. Therefore, the sensor shows great potential in strain sensing fields, including structural health monitoring, biomedical, food processing, civil engineering, aerospace, and other areas.