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Abstract
Low extinction ratio (ER) and high temperature cross-sensitivity are serious but common problems for most strain sensors based on Vernier effect. In this study, a hybrid cascade strain sensor of a Mach–Zehnder interferometer (MZI) and a Fabry–Perot interferometer (FPI) with high sensitivity and high ER based on Vernier effect is proposed. The two interferometers are separated by a long single-mode fiber (SMF). The MZI is used as the reference arm, which can be flexibly embedded in the SMF. The FPI is used as the sensing arm and the hollow-core fiber (HCF) as the FP cavity to reduce optical loss. Simulation and experiments have proven that this method can significantly increase ER. At the same time, the second reflective face of the FP cavity is indirectly spliced to increase the active length to improve the strain sensitivity. Through the amplification of Vernier effect, the maximum strain sensitivity is –649.18${\mathrm{pm}/\mathrm{\mu} \mathrm{\varepsilon}}$, and the temperature sensitivity is only 5.76$\textrm{pm/}^\circ \textrm{C}$. The magnetic field was measured by combining the sensor with a Terfenol–D (magneto–strictive material) slab to verify the strain performance, and the magnetic field sensitivity is –7.53$\textrm{nm/mT}$. The sensor has many advantages and has potential applications in the field of strain sensing.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Fiber optic strain sensors have been widely investigated for applications in many fields, such as structural health monitoring and civil engineering [1,2]. Such sensors mainly include two types: fiber gratings and interferometers. Although fiber gratings have simple structure and strong robustness, their sensitivity is extremely low at only a few [3–5]. Fiber optic strain sensors based on interferometer are superior to fiber gratings in terms of performance. Many methods and structures of optical fiber strain sensors based on interferometer have been proposed, such as Mach–Zehnder interferometers (MZI) [6,7], Sagnac interferometers [8,9], and Fabry–Perot interferometers (FPI) [10,11].
Among them, air cavity has attracted wide attention because of its small size, high sensitivity, and low temperature cross sensitivity. At present, several methods for making air cavity are reported. For example, a femtosecond laser [12] or CO2 laser [13] is used to fabricate a microstructured air cavity on fiber. However, this method is expensive, and the cavity surface is uneven. Arc discharge technology is used to make elliptical air cavity [14–16]. This method requires many manual arc discharges to adjust the size of bubbles, thereby causing poor repeatability of the sensor. In addition, a hollow core fiber (HCF) [17] or hollow core photonic crystal fiber [18] is directly spliced between two single-mode fibers (SMFs), but their sensitivity is low. In recent years, the Vernier effect has widely received attention from researchers because it can significantly increase the sensitivity of fiber sensors [19,20], it is applied to the measurement of temperature [21], humidity [22], gas pressure [23] and so on. Many researchers have used the Vernier effect for strain sensing [24–27]. Although the above works use the vernier effect to increase the strain sensitivity of the sensor, however, the extinction ratio (ER) is poor ($< $5$\textrm{dB}$). The Vernier effect can be generated by the two cascaded interferometers with small difference in free spectral range (FSR). Therefore, most reports focus more on FSR to generate the Vernier effect, but the ER is often neglected. Low ER affects the sensor accuracy and causes measurement errors because the sensor based on Vernier effect is measured by tracking the shift of Vernier envelope. For FP–based sensors, reducing the cavity length can effectively increase ER. However, this condition leads to a larger FSR of interference fringe, which is not conducive to produce a significant Vernier effect. Another method is to plate metal film on the end face to improve the reflectivity. However, the surface of the metal film is easy to oxidize, and the thickness is difficult to control, thereby also increasing the manufacturing cost [28]. Temperature cross-sensitivity is a non-negligible problem for strain measurement, especially for strain sensors based on Vernier effect. When strain sensitivity is amplified, temperature cross-sensitivity is further increased. Therefore, solving the ER and temperature cross-sensitivity in practical application is particularly important.
In this study, a hybrid cascade MZI and FPI strain sensor with Vernier effect is proposed. MZI is composed of SMF–no-core fiber (NCF) –HCF–NCF–SMF. It has the advantages of small size, good spectral characteristics, and difficult to interfere by the external environment. FPI is connected by SMF–HCF–hollow silica capillary (HSC) –thinner-diameter SMF (TD-SMF). The strain sensitivity can be double amplified due to the long active length and the amplification effect of the Vernier effect. Moreover, because the inner diameter of the HCF is only 40 ${\mathrm{\mu} \mathrm{m}}$, it has a good effect of restraining the light beam. Compared with the common air-FP cavity structure, HCF-FP cavity can reduce the light loss. Thus, more light can return to the lead-in SMF after being reflected by the second end face. Therefore, when the cavity length of HCF-FP cavity exceeds 300 ${\mathrm{\mu} \mathrm{m}}$, the ER can still reach more than 12$\textrm{dB}$. In addition, HCF, HSC and TD-SMF are silica materials, their thermal expansion coefficients are close to each other. When their lengths satisfy the matching conditions, the sensor exhibits extremely low temperature sensitivity [29]. The sensor has the advantages of high strain sensitivity, high ER, and temperature cross-sensitivity; it has potential applications in large temperature changes and high strain sensitivity detection.
2. Fabrication and principle
The structural diagram and the microscope image of the MZI is shown in Fig. 1(a); it is composed of SMF–NCF–HCF–NCF–SMF. Figure 2(a) illustrate the fabrication process of the MZI. The instruments used are fiber cutter (Fujikura CT-104, Japan), fiber fusion splicer (Fujikura FSM-100P, Japan), and optical microscope system. Among them, the length of NCF at both ends is 750${\mathrm{\mu} \mathrm{m}}$. The length and the inner diameter of HCF are 1570 and 40${\mathrm{\mu} \mathrm{m}}$, respectively. The compact inline MZI has the advantages of small size, good spectrum, and insensitivity to temperature, strain, and refractive index [30]. It can be flexibly embedded in SMF and is an excellent reference arm.
In this structure, NCF is used as beam splitter and beam combiner. Light travels along the wall and air core of HCF. The intensity of the transmission spectrum of the MZI can be expressed as follows:
where ${I_1}$ and ${I_2}$ are the light intensity of the core mode and the wall mode, respectively. $\Delta {\varphi _1}$ is the phase difference between the two modes, which can be described as follows: where ${L_1}$ is the length of the HCF, $\lambda$ is the wavelength of the light, ${n_{co}}$ and ${n_{wa}}$ are the refractive index of the core mode and the wall mode, respectively. The FSR can be described as follows: where ${\lambda _m}$ is the wavelength of ${m_{th}}$ order interference dip, and m is an integer.The structural diagram and the microscopic images of the FPI is shown in Fig. 1(b). The structure consists of an SMF connected with a HCF, the inner and outer diameters of which are 40 and 125${\mathrm{\mu} \mathrm{m}}$, respectively. The other end of the HCF is splice with a HSC, and the inner and outer diameters of the HSC are 100 and 150 ${\mathrm{\mu} \mathrm{m}}$, respectively. Figures 5(d) and (e) show the microscopic images of the cross section of HCF and HSC, respectively. The TD-SMF with a diameter of 95 ${\mathrm{\mu} \mathrm{m}}$ was corroded by hydrofluoric acid (40% concentration, corrosion time is 20 minutes). The motor of fiber fusion splicer was used to control the TD-SMF insert into HSC, until the gap between the TD-SMF and the HCF was 0 ${\mathrm{\mu} \mathrm{m}}$, the TD-SMF was fixed by discharging at the end of the HSC. Figure 2(b) shows the fabrication process of the FPI. The TD-SMF -in-HSC structure has a long active length to enhance the strain sensitivity. This finding will be verified by experiments in Chapter 3.
In this structure, the end face of the lead-in SMF is the first reflection face ${M_1}$ of the FP cavity, and the end face of the TD-SMF is the second reflection face ${M_2}$; their reflection coefficients are ${R_1}$ and ${R_2}$, respectively. According to Eq. (1), when the light passes through MZI, the reflected light intensity of FPI can be expressed as follows:
where ${I_3} = {I_T}{R_1}$ and ${I_4} = {I_T}({1 - {R_1}} )({1 - k} ){R_2}$; k is the transmission loss of the FP cavity. ${\varphi _2}\textrm{ = }{{4\pi {n_0}{L_{FP}}} / \lambda }$, ${n_0}$ is the refractive index of air. When ${\varphi _2}\textrm{ = }({2m + 1} )\pi$, ${m_{th}}$ order interference dip appears, as follows: where ${L_{FP}}$ is the length of FP cavity. The FSR of the interference spectrum of FPI can be expressed as follows:When the strain force F is applied on the FPI with a total length L, the strain $\varepsilon$ can be expressed as follows [29]:
According to Eq. (8), the strain sensitivity is proportional to ${{{L_{HSC}}} / {{L_{FP}}}}$. Therefore, the strain sensitivity can be greatly improved by increasing ${L_{HSC}}$ and shortening ${L_{FP}}$. When the temperature changes, the lengths of the HCF, HSC and the TD-SMF are changed when thermally expanded, causing the interference spectrum to shift. The partial derivative of Eq. (5) can be obtained as follows [29]:
The strain measuring device after cascade MZI and FPI is shown in Fig. 3. After light passes through MZI, it generates interference spectrum, and then passes through circulator and enters FPI. The FSR of the spectra of the two interferometer is similar but not equal, and the superimposed spectrum produces Vernier effect. The FSR of the envelope is given by the following [31]:
The amplification effect of the Vernier effect is converting the shift of the tracking interference spectrum into the shift of the tracking Vernier envelope to amplify the sensitivity. The amplification factor M can be expressed as follows:
According to Eqs. (1) and (8), the strain sensitivity can be improved by increasing the amplification factor M and ${{{L_{HSC}}} / {{L_{FP}}}}$. Increasing M can be achieved by narrowing the difference of FSR between MZI and FPI, whereas increasing ${{{L_{HSC}}} / {{L_{FP}}}}$ can be achieved by increasing ${L_{HSC}}$ or shortening ${L_{FP}}$.
3. Exploration of ER and strain sensitivity
Generally, sensitivity is an important factor to evaluate the performance of sensors, but the accuracy of sensors affects the measurement results. Therefore, improving the accuracy of sensors is very important. In the fiber sensor based on Vernier effect, the measurement is carried out by tracking the shift of Vernier envelope; thus, the extinction ratio of envelope ($E{R_{en}}$) affects the measurement accuracy. $E{R_{en}}$ is related to the extinction ratios of the reference interferometer ($E{R_r}$) and the sensing interferometer ($E{R_s}$), and their difference. MATLAB was used to explore the ER of Vernier envelope, considering $FS{R_r}$=3.4$\textrm{nm}$, $FS{R_s}$=3.69nm, and the wavelength range is 1500–1600nm. When either or $E{R_s}$ is fixed, the smaller indicates larger $E{R_{en}}$, as shown in Figs. 4(a) and 4(b). When the $|{E{R_r} - E{R_s}} |$ is fixed, larger $E{R_r}$ and $E{R_s}$ indicate larger $E{R_{en}}$, as shown in Figs. 4(a) and 4(c). Therefore, to improve $E{R_{en}}$, $E{R_r}$ and $E{R_s}$ should be increased, and $|{E{R_r} - E{R_s}} |$ should be reduced.
Fig. 4. Simulated superposition spectra (a) $E{R_r}$=12.5$\textrm{dB}$, $E{R_\textrm{s}}$=12.5$\textrm{dB}$, $E{R_{en}}$=10.1$\textrm{dB}$. (b) $E{R_r}$=12.5$\textrm{dB}$, $E{R_\textrm{s}}$ = 4.2$\textrm{dB}$, $E{R_{en}}$=4$\textrm{dB}$. (c) $E{R_r}$=4.2$\textrm{dB}$, $E{R_\textrm{s}}$=4.2$\textrm{dB}$, $E{R_{en}}$=3.8$\textrm{dB}$.
The energy distribution of air-FP cavity (Fig. 5(a)) and HCF-FP cavity (Fig. 5(b)) structures was simulated in Beam PROP using the BPM algorithm to verify that the HCF in this structure can effectively improve the ER. The 2D mode is used to speed up the calculation; the wavelength is 1.55${\mathrm{\mu} \mathrm{m}}$, and the built-in computed mode is used as the launch field of SMF model, the detailed parameters of the model are shown in Table 1. As shown in Fig. 6(b), the HCF-FP cavity still effectively restricts light propagation in the “core” of the HCF after the cavity length exceeds 300${\mathrm{\mu} \mathrm{m}}$. However, the light in air-FP cavity “diverges” after the cavity length exceeds 300${\mathrm{\mu} \mathrm{m}}$, as shown in Fig. 6(d). As shown in Fig. 6(a), when the inner diameter of HCF is 20${\mathrm{\mu} \mathrm{m}}$, light is transmitted in two parts at the welding point of SMF and HCF: one part is the leakage mode propagating along the air core, while the other part is the cladding mode propagating along the cladding [32]. The light transmitted along the cladding will also be reflected at the end face of HCF, interfering with the reflected light of ${M_1}$. This part of the interference signal will deteriorate the quality of the interference spectrum of FPI, affecting the ER of the envelope. Therefore, such interference signal should be minimized. However, when the inner diameter of HCF is greater than 40${\mathrm{\mu} \mathrm{m}}$, the light can be confined well to the core of HCF. As shown in Figs. 6(b) to 6(d), only a small amount of light enters the cladding, and thus, the influence of light reflected at the end face of HCF to the interference spectrum can be disregarded. With comprehensive consideration, choosing an inner diameter of 40${\mathrm{\mu} \mathrm{m}}$ can ensure a high ER and reduce unnecessary clutter signals.
Fig. 5. Microscopic image of three different structures of the FP cavity- (a) air-FP cavity, (b) HCF-FP cavity, (c) SMF–HCF–SMF; (d) and (e) microscopic images of the cross section of HCF and HSC, respectively.
Fig. 6. Energy distribution diagrams of HCF with inner diameters of (a) 20, (b) 40, (c) 75, and (d) 100${\mathrm{\mu} \mathrm{m}}$ are simulated using the BPM algorithm; (b) and (d) show the energy distribution of Figs. 5(a) and 5(b), respectively.
Table 1. The detailed parameters of the model.
The above simulation is verified by experiments. First, the air-FP cavity was investigated experimentally with cavity lengths of 250, 302, 351, 402, 454, and 501 ${\mathrm{\mu} \mathrm{m}}$. When the cavity length becomes longer, its FSR decreases, and the ER also decreases continuously. The related interference spectra are shown in Fig. 7(a1). Then, the air-FP cavity was investigated experimentally with cavity lengths of 316, 332, 374, 416, 443, and 492 ${\mathrm{\mu} \mathrm{m}}$. The FSR decreased with the increase in cavity length, but the ER remained at approximately 13$\textrm{dB}$, as shown in Fig. 7(b1). Figures 8(a) and (b) show the statistical graphs of Figs. 7(a1) and (b1). The above simulation and experimental analysis concluded that the HCF can effectively improve ER and play an important role to improve the evident degree of Vernier effect and the accuracy of the sensor.
Fig. 7. Interference spectra of FPI with different cavity lengths – (a1) air-FP cavity, (b1) HCF-FP cavity; (a2) and (b2) are microscopic images corresponding to different sizes in (a1) and (b1), respectively.
FPI was fixed between two 3D displacement platforms (with an accuracy of 0.02 ${\mathrm{\mu} \mathrm{m}}$) with UV glue at room temperature (25$^\circ \textrm{C}$) to compare the strain sensitivity of HCF–FP cavity (Fig. 5(b)) and SMF–HCF–SMF (Fig. 5(c)), and the fixed length is 20$\textrm{cm}$. The elongation of 5 ${\mathrm{\mu} \mathrm{m}}$ (25$\mathrm{\mu} \mathrm{\varepsilon}$) is changed each time. The measuring range is 0–20 ${\mathrm{\mu} \mathrm{m}}$, and the corresponding strain range is 0–100$\mathrm{\mu} \mathrm{\varepsilon}$. The input light is emitted by ASE light source (1525–1610$\textrm{nm}$, Fiber Lake, China), and the output spectrum is monitored by optical spectrum analyzer (OSA, YOKOGAWA-AQ6385B, Japan) with resolution of 20$\textrm{pm}$. As shown in Fig. 9, the strain sensitivity of HCF-FP cavity is 57.12 ${\mathrm{pm}/\mathrm{\mu} \mathrm{\varepsilon} }$, whereas the strain sensitivity of SMF–HCF–SMF structure is only 1.76 ${\mathrm{pm}/\mathrm{\mu} \mathrm{\varepsilon} }$. Therefore, the method of fixing SMF by discharging at the end of HSC without direct splice can increase the effective length of stress and significantly increase the strain sensitivity.
According to the theoretical analysis in Chapter 2, strain sensitivity can be improved by increasing the M and the ${{{L_{HSC}}} / {{L_{FP}}}}$. Therefore, we experimentally verify the feasibility of the two methods. The size of MZI is invariant, and the relevant parameters are shown in Fig. 1(b); $FS{R_r}$ = 3.4$\textrm{nm}$, and $E{R_r}$=13.5$\textrm{dB}$. The MZI and four FPIs of different sizes are cascaded according to Fig. 3, and four sensors of S1, S2, S3, and S4 are designed. The related parameters are shown in Table 2. Figures 10(a)-(d) show the microscopic images of S1-S4, and Figs. 10(e)-(h) show the spectra corresponding to Figs. 10(a)-(d), indicating that they have high spectral contrast.
Table 2. Parameter values of five fabricated structures.
The experimental strain sensitivities of S1, S2, S3, and S4 are 152.49, 203.01, 347.11, and −210.04 ${\mathrm{pm}/\mathrm{\mu} \mathrm{\varepsilon} }$. Figure 11 shows the relationship between the wavelength shift of their envelope dip and strain. According to Table 2, without considering the amplification effect of Vernier effect, the strain sensitivity of single FPI (${{{S_\varepsilon }} / M}$) of S1–S4 is 37.01, 39.04, 47.68, and −52.51 ${\mathrm{pm}/\mathrm{\mu} \mathrm{\varepsilon} }$, respectively. With the increase in ${{{L_{HSC}}} / {{L_{FP}}}}$, the strain sensitivity of the sensor gradually increases, consistent with the theoretical analysis in Chapter 2.
However, the value of ${{{L_{HSC}}} / {{L_{FP}}}}$ should not be extremely large, because under the condition that $FS{R_r}$ is fixed, if ${L_{FP}}$ is extremely small, then M and ${S_\varepsilon }$ decreases, as shown in the comparison of S3 and S4. At the same time, increasing ${L_{HSC}}$ increases the size of the sensor remarkably, causing difficulty to adapt to strain measurement in the special environment, such as narrow space. M should not be extremely large given the limit of the bandwidth of the ASE; otherwise, the range of strain measurement becomes extremely narrow.
4. Experiment and results
4.1 Strain sensitivity
According to the above analysis, S5 is designed to obtain greater strain sensitivity. Its related parameter is shown in Table 2, and its microscope image is shown in Fig. 1. Figure 12(a) shows the interference spectrum of MZI when the length of NCF is 750 ${\mathrm{\mu} \mathrm{m}}$ and the length of HCF is 1570 ${\mathrm{\mu} \mathrm{m}}$. Its $FS{R_r}$ and $E{R_r}$ are 3.4 $\textrm{nm}$ and 13.5$\textrm{dB}$, respectively. Figure 12(b) shows the interference spectrum of FPI when ${L_{FP}}$=330.0 ${\mathrm{\mu} \mathrm{m}}$; $FS{R_s}$=3.69$\textrm{nm}$, and $E{R_s}$=12.3$\textrm{dB}$. As shown in Fig. 12(c), evident Vernier effect and high ER can be observed in the superimposed spectrum. Fast Fourier transform (FFT) filter is applied to the Vernier effect to avoid interference from irregular structures. After extracting the main signal from the spectrum, Fig. 12(d) is presented, and the shape of the filtered spectrum is consistent with Fig. 12(c). The upper envelope curve obtained by curve fitting is shown by black lines in Fig. 12(d) to observe the shift of the Vernier effect envelope more clearly.
Fig. 12. (a) Transmission spectrum of MZI; (b) reflection spectrum of FPI; (c) envelope of Vernier effect obtained by superimposing MZI and FPI; (d) FFT filter result corresponding to (c).
The strain of S5 is measured, MZI remains unchanged, and FPI is fixed on the 3D displacement platform with UV glue. The detailed steps are the same as those introduced in the previous chapter. Figures 13(a)-(e) are superimposed spectra under different strains. It can be found that with the increase of strain, the envelope of the superimposed spectrum has a blue shift. The strain sensitivity of the S5 is –649.18${\mathrm{pm}/\mathrm{\mu} \mathrm{\varepsilon} }$, and the linearity is 99.94$\%$. The relationships between the envelope dip wavelength shift and the strain for S5 is shown in Fig. 13(f).
Fig. 13. (a)–(e) show the superimposed spectra of S5 under different strains; (f) relationships between the envelope dip wavelength shift and the strain for S5.
S5 is also combined with Terfenol–D (Huizhou South Rare Earth Functional Material Research Institute Co., Ltd., China) for magnetic field measurement to verify the strain performance of the sensor. When the Terfenol–D is placed in the magnetic field, the change in magnetic field strength ($H$) causes deformation of the Terfenol–D, which drives the cavity length of the FP cavity to change and causes the wavelength of the spectrum to shift. Therefore, the measurement of magnetic field combined with Terfenol–D is essentially a strain measurement.
The measuring device diagram is shown in Fig. 14. The photo of the Terfenol–D is in the black dotted box, and its size is 6.5${\times} $0.7${\times} $0.5$\textrm{cm}$. Its magneto-strictive coefficient is $\le $2000 × 10−6 (240$\textrm{kA/m}$), the operating temperature is −40 $^\circ \textrm{C}$ to 150 $^\circ \textrm{C}$, and the response time is less than 0.1 $\mathrm{\mu} \textrm{s}$. FPI is fixed by UV glue on the Terfenol–D, and the fixed distance is 5$\textrm{cm}$. The DC power supply model is DC-305D (Dongguan DingCe Technology Co., Ltd., China), the current adjustment range is 0–3$\textrm{A}$, and the voltage adjustment range is 0–120$\textrm{V}$. The Gauss meter model is WT10C (Weite Magnetic Technology Co., Ltd., China). The magnetic field sensor is placed in the center of the Helmholtz coil parallel to the direction of the magnetic field.
The change in H is adjusted by changing the output current. The measuring range is 0–10 $\textrm{mT}$ with a step of 2 $\textrm{mT}$. Figures 15(a)–(f) show the superimposed spectra under different H. The figures show that with the increase in H, the wavelength of the spectrum has a blue shift, which is consistent with the moving direction of the previous strain measurement.
Figure 16 shows the relationships between the envelope dip wavelength shift and the H. Figure 16 shows that when H is less than 4 $\textrm{mT}$, the magnetic field sensitivity of the sensor is low. This condition is because only a few magnetic domains inside the Terfenol–D start to turn over when the H is low, and the deformation of the Terfenol–D is small. With the increase in H, the magnetic field sensitivity can reach –7.53${{\textrm{nm}} / {\textrm{mT}}}$ in the range of 4–10 $\textrm{mT}$.
4.2 Temperature cross-sensitivity
The temperature response of a single FPI is tested, and the experimental results are shown in Figs. 17(a1) and (a2). The temperature sensitivity of a single FPI is 0.48$\textrm{pm/}^\circ \textrm{C}$, and the linearity is 98.86$\%$.The temperature response of the proposed sensor was also investigated. The MZI can be isolated from the FPI and the detection environment because the proposed sensor is a separate structure. The FPI is placed horizontally in a temperature control box under free load, and the MZI was placed in a constant temperature environment. Thus, MZI is unaffected by temperature. The SMF, HSC and HCF are silica materials; thus, their thermal expansion coefficients are close to each other. When ${L_{To}}$ and ${L_{TD - SMF}}$ satisfy the matching conditions, Eq. (9) shows that ${{\partial {L_{To}}} / {{\partial _T}}} \approx {{\partial {L_{TD - SMF}}} / {{\partial _T}}}$. Therefore, the temperature sensitivity of FPI is close to 0$\textrm{pm/}^\circ \textrm{C}$. After amplification by the Vernier effect, the temperature sensitivity is also low. When the temperature changes from 45 °C to 95 °C, the wavelength of the envelope dip shifts only to 0.28 $\textrm{nm}$, as shown in Fig. 17(b1). Figure 17(b2) shows the relationship between the wavelength of the envelope dip and the temperature. The sensor has a very low temperature sensitivity, which is only 5.76 $\textrm{pm/}^\circ \textrm{C}$, the cross sensitivity of temperature to strain is only 0.0089 $\mathrm{\mu} \mathrm{\varepsilon} /^\circ \textrm{C}$. The repeatability error is only 0.73$\%$ after repeated measurement for three times.
Fig. 17. (a1) Dip shifts versus the temperature of FPI; (b1) envelope shifts versus the temperature of S5; (a2) and (b2) are the linear fitting of dip wavelengths of (a1) and (b1) around 1590 $\textrm{nm}$ versus temperature.
4.3 Repeatability and stability
The repeatability of S5 was measured three times, the strain range was 0–100$\mathrm{\mu} \mathrm{\varepsilon}$, and the step was 25$\mathrm{\mu} \mathrm{\varepsilon}$. In each measure, the reversibility test was also carried out, and the result showed that the sensor hardly hysteresis. As shown in Fig. 18(a), the maximum deviation between the experimental strain sensitivity and the average sensitivity of four strains is ${\pm} $3.88${\mathrm{pm}/\mathrm{\mu} \mathrm{\varepsilon} }$, and the average strain sensitivity is –645.30 ${\mathrm{pm}/\mathrm{\mu} \mathrm{\varepsilon} }$; thus, the repeatability error of the sensor is only ${\pm} $0.60$\%$. It can also be concluded that the hysteresis of the sensor and the relative error of measurement are 1.08 $\%$ and 1.6$\%$, respectively.
Fig. 18. (a) Strain repeatability of S5 was tested at 0–100$\mathrm{\mu} \mathrm{\varepsilon}$ for three cycles. (b) Stability of S5 was tested at 0 and 100$\mathrm{\mu} \mathrm{\varepsilon}$.
We also tested the stability of the sensor. The sensor was placed under the strain of 0 and 100$\mathrm{\mu} \mathrm{\varepsilon}$, and the spectral information was recorded every 60 minutes. The relationship between wavelength of envelope dip and time is shown in Fig. 18(b). At 0 and 100 $\mathrm{\mu} \mathrm{\varepsilon}$, the maximum fluctuation of the sensor is 54.30 and 58.20$\textrm{pm}$, respectively, corresponding to 0.084 and 0.089 $\mathrm{\mu} \mathrm{\varepsilon}$ variations. The slight noise may be caused by the vibration of the environment.
4.4 Detection limit
Finally, we also explore the detection limit of the sensor. It is very important for evaluating the quality of a sensor. The detection limit is the ability to accurately detect the tiniest strain changes, depending on the spectral resolution and noise of the sensing system. We tested the microstrain of S5 with each change of 0.1$\mathrm{\mu} \mathrm{\varepsilon}$. The spectral information was recorded every 30 seconds for each microstrain value. As shown in Fig. 19, with the increase in strain, the envelope dip shift to the short-wave direction, and the envelope dip fluctuates slightly around a certain value (46$\textrm{pm}$) when the applied strain is constant. When a strain is applied, the envelope dip changes instantaneously, and the envelope dip completely separates when different strains are applied. This finding shows that the detection limit of the sensor at least 0.1$\mathrm{\mu} \mathrm{\varepsilon}$, and it is suitable for microstrain detection.
Fig. 19. Wavelength of the upper envelope dip fluctuates at 0, 0.1, 0.2, and 0.3$\mathrm{\mu} \mathrm{\varepsilon}$.
Table 3 shows the performance comparison between S5 and the sensor based on Vernier effect mentioned in this paper. As shown in Table 3, S5 has the advantages of high strain sensitivity, high ER, and low temperature cross sensitivity. In Ref. [25], the M of the sensor was designed to be 22, and thus, strain sensitivity after amplification by the Vernier effect is higher than the value proposed in this paper. In accordance with actual demand, the strain structure proposed in this paper can also reach or exceed the sensitivity indicated in Ref. [25] by increase the M.
Table 3. Performance comparison between the S5 and the sensor based on Vernier effect mentioned in Refs.
5. Conclusion
We propose a strain sensor with high sensitivity and high ER based on Vernier effect; it consists of MZI and FPI with high spectral contrast. In this study, two approaches to increase sensitivity are discussed: increasing the amplification factor $M$ of Vernier effect and increasing the sensitivity of FPI. The maximum strain sensitivity of the sensor is –649.18 ${\mathrm{pm}/\mathrm{\mu} \mathrm{\varepsilon} }$, the temperature cross sensitivity is only 0.0089 $\mathrm{\mu} \mathrm{\varepsilon} /^\circ \textrm{C}$, and the ER of the envelope can reach approximately 10$\textrm{dB}$. In addition, the sensor has the advantages of low detection limit, good stability, good repeatability, and excellent application prospect.
Funding
National Natural Science Foundation of China (11274278); Basic Public Welfare Research Project of Zhejiang Province (LGG22A040001).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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