Highly sensitive temperature and pressure fiber optic sensor based on harmonic Vernier effect

Abstract

We have developed a highly sensitive fiber optic sensor that can measure temperature and pressure. The sensor comprises two Fabry–Perot interferometers (FPIs), FPI₁ and FPI₂, connected in parallel. FPI₁ is composed of single mode fiber (SMF)—capillary—SMF spliced together, with a ventilation hole on the capillary. FPI₂ is composed of SMF—capillary—SMF, and the capillary is filled with polyimide (PI). It is found that FPI₁ is almost insensitive to temperature, but sensitive to air pressure. FPI₂ is sensitive to temperature, but not sensitive to pressure. When the free spectra range (FSR) of FPI₁ and FPI₂ approaches twice the relationship, they are connected in parallel to form a first-order harmonic vernier effect (HVE). When the HVE sensor is used for air pressure sensing, FPI₁ is the sensing cavity and FPI₂ is the reference cavity. When the HVE sensor is used for temperature sensing, FPI₂ is the sensing cavity and FPI₁ is the reference cavity. The experimental results showed that the sensitivities of the pressure and temperature of the HVE sensor are 76.71 nm/MPa and 113.29 nm/ °C, respectively. This is currently the highest temperature sensitivity reported in literature. The accuracy of the obtained intersection point in HVE sensor is higher and than that of that of traditional Vernier effect. In addition, the FSR relationship required to form HVE is easier to achieve. By utilizing the temperature and air pressure sensitivities of FPI₁ or FPI₂, as well as that of the HVE sensor, a sensitivity measurement matrix can be formed to achieve simultaneous measurement of temperature and air pressure.

Introduction

Temperature and pressure are two important physical quantities. In various fields such as petrochemicals, food engineering, medical pharmaceuticals, and seabed exploration, it is usually necessary to measure both temperature and pressure simultaneously. Fiber optic sensors have been widely used in temperature and pressure measurements due to their compact structure, remote sensing measurement, and resistance to electromagnetic interference. At present, sensor structures for simultaneous measurement of temperature and pressure mainly include the following types: two FPIs cascaded or paralleled [1, 2, 3, 4, 5–6], FBG and FPI cascaded [7, 8], two MZIs cascaded [9, 10], etc. Although these sensors have simple structures and strong robustness, their sensitivity is still relatively low. Seeking new structures to further improve the sensitivity of sensors is one of the goals pursued by people.

The optical vernier effect (VE) is one of the main methods to enhance the sensitivity of the fiber optic sensor [11]. In the past few years, the VE method has been used to improve the sensitivity of temperature and pressure [12, 13–14]. The traditional VE requires two interferometers to meet strict optical path length (OPL) matching conditions, but due to the inevitable matching errors in actual preparation processes such as fiber cutting, splicing, and etching, high sensitivity amplification rates are still limited. In 2019, Gomes et al. first proposed the harmonic Vernier effect (HVE), which is formed by increasing the OPL of one interferometer to i times the OPL of the second interferometer (i is the harmonic order, usually taken as a positive integer) [15]. HVE does not require strict FSR matching conditions like traditional VE, it allows for larger manufacturing tolerances without sacrificing sensitivity [15].

The HVE, as a newly developed sensitivity amplification tool, has been widely developed and applied in the field of fiber optic air pressure [16, 17, 18–19] and temperature [20, 21, 22, 23–24] sensing due to its better manufacturing tolerance and higher performance advantages. Recently, the high-order HVE sensors have also been applied for simultaneous measurement of temperature and pressure. Dan et al. used a compact fiber optic HVE sensor based on cascade FPIs formed by three reflection surfaces to simultaneously measure air pressure and temperature [25]. Chen et al. combined a FPI based on high-order HVE with a FBG to achieve simultaneous measurement of pressure and temperature [26]. But the sensitivity they obtained is not yet very high.

In this study, two new FPIs were constructed using single-mode fiber (SMF), quartz capillary (QC) and polyimide (PI) etc. Among them, FPI₁ was only sensitive to gas pressure, and FPI₂ was only sensitive to temperature. When their FSR is approximately twice, they form an HVE sensor in parallel. Then, the experimental exploration was conducted on the HVE sensor, and it was found that the sensitivity amplification of gas pressure and temperature measurement was achieved using HVE. Then, the experimental exploration was conducted on the HVE sensor, and it is found that the sensitivities of gas pressure and temperature are greatly enhanced. Pressure and temperature sensitivities of the HVE sensor are as high as 76.71 nm/MPa and 113.29 nm/ °C, respectively. It is currently the highest known temperature sensitivity. If it is combined with a single FPI (FPI₁ or FPI₂) to form a sensitivity matrix, it can be used to simultaneously measure gas pressure and temperature. In addition, the proposed HVE sensor has good repeatability and stability in measuring temperature and pressure.

Sensor configuration and sensing principle

Design and fabrication of the sensor

Based on our previous research results [13, 14], two low-fineness FPIs (namely FPI₁ and FPI₂) were designed using SMF, QC and PI. FPI₁ is composed of SMF—QC—SMF splicing, and a gas-hole is drilled in the QC wall by using femtosecond (fs) laser pulse. FPI₂ is also composed of SMF—QC—SMF, but the QC is filled with the heat sensitive material polyimide (PI), and both sides of the QC are sealed with AB glue. SMF is come from Wuhan Changfei Company in China (with 9/125 μm). QC is come from Polymicro Technology Company in the United States. The QC of FPI₁ is TSP075150 with the inner/outer diameter of 75/150 μm, and the QC of FPI₂ is TSP150375 with the inner/outer diameter of 150/375 μm. Figure 1 shows the configuration method of FPI₁ and FPI₂. The capillary is a hollow-core structure. In the manufacturing process of FPI₁, in order to prevent collapse during the fusion splicing process between the capillary and the optical fiber, we manually set the following fusion splicing parameters: clean discharge time of 300 ms, pre fusion strength of 30 bit, pre fusion time of 150 ms, discharge strength of 82 bit, and discharge time of 2200 ms. The FPI₁ structure produced by fusion splicing capillary and optical fiber fully meets the requirements.

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Fig. 1

The configuration method of a FPI₁ and b FPI₂

The key to making sensor manufacturing repeatable is the repeatability of the two FPI production processes. The key to achieving this goal is the fiber optic precision cutting and alignment system. We have built a fiber optic precision cutting system and a fiber optic precision alignment system, which can control the length of the cut fiber and capillary, and ensure that the two segments of SMF are completely aligned inside the capillary. Figure 2a and b show the physical images of the two constructed systems. The fiber optic cutting system is composed of a fiber optic cleaver, a 3D precision displacement platform, a fiber optic fixture, and an imaging CCD, which can accurately cut the length of capillaries of FPI₁ and FPI₂. The fiber optic precise alignment system is composed of two 3D precision displacement platforms and an imaging CCD, and an optical spectrum analyzer (OSA, real-time spectral observation) is used to precisely control the length of the SMF inserted into the capillary and obtain the designed F-P cavity length of FPI₂.

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Fig. 2

a Precision cutting system for optical fiber length, b accurate alignment system for optical fiber and quartz capillary

Principle and spectrum of the sensor

In Fig. 1, M₁ and M₂ form an air-cavity, defined as FPI₁. M₃ and M₄ form a PI cavity, defined as FPI₂. L₁ and L₂ are the F-P cavity lengths of FPI₁ and FPI₂, respectively. Based on the theory of FPI, the dip wavelengths of the spectra of FPI₁ and FPI₂ respectively are deduced as:

1

$${\lambda }_{FPI1}=\frac{4n_1{L}_1}{2m+1},(m=1,2,3,\dots )$$

2

$${\lambda }_{FPI2}=\frac{4n_2{L}_2}{2k+1},(k=1,2,3,\dots )$$where n₁ is the air refractive index (RI) in FPI₁ and n₂ is the RI of PI in FPI₂.

In addition, the free spectral range (FSR) of FPI₁ and FPI₂ spectrum can be approximated as:

3

$$FS{R}_{FPI1}=\frac{{\lambda }^2}{2n_1{L}_1}$$

4

$$FS{R}_{FPI2}=\frac{{\lambda }^2}{2n_2{L}_2}$$

The F-P cavity of FPI₁ is an air cavity, which is connected to the external gas through micro-hole. When the external air pressure changes, the air RI in FPI₁ changes, causing the dip wavelength of the FPI₁ spectrum to shift. The F-P cavity length of FPI₁ remain almost unchanged under the influence of external air pressure. Therefore, according to Eq. (1), FPI₁ has the following gas pressure sensitivity:

5

$$S_{FPI{1}_P}=\frac{d{\lambda }_{FPI1}}{\mathit{dP}}={\lambda }_{FPI1}\left(\frac{d{n}_1}{n_{1}dP}+\frac{d{L}_1}{L_{1}dP}\right)\approx {\lambda }_{FPI1}\frac{d{n}_1}{n_{1}dP}$$where P is air pressure, T is ambient temperature, and n₁ is air RI ($n_1=n_{\mathit{air}}=1+2.8783\times {10}^{-9}P/(1+0.00367\times T)$ [27]). Therefore, when T is constant, FPI₁ has the following simplified gas pressure sensitivity formula:

6

$$S_{FPI{1}_P}=\frac{d{\lambda }_{FPI1}}{\mathit{dP}}\approx {\lambda }_{FPI1}\frac{d{n}_1}{n_{1}dP}={\lambda }_{FPI1}\frac{2.8783\times {10}^{-9}}{n_1(1+0.00367\times T)}$$

In addition, FPI₁ is an air cavity composed of quartz optical fiber and capillary. Due to the extremely small thermal expansion coefficient (TEC) and thermal optical coefficient (TOC) of quartz optical fiber and capillary, the air medium in the air cavity also has extremely small TOC and TEC. Therefore, FPI₁ has very little sensitivity to temperature, and it can be considered that FPI₁ is almost insensitive to temperature.

FPI₂ is composed of PI cavity. Due to large TEC and TOC of PI, FPI₂ is very sensitive to temperature. Utilizing Eq. (2), FPI₂ has the following temperature sensitivity:

7

$$S_{FPI{2}_T}=\frac{d{\lambda }_{FPI2}}{\mathit{dT}}={\lambda }_{FPI2}\left(\frac{d{n}_2}{n_{2}dT}+\frac{d{L}_2}{L_{2}dT}\right)={\lambda }_{FPI2}({\alpha }_2+{\beta }_2)$$where ${\alpha }_2=d{n}_2/n_{2}dT$ presents the sum of TOC of silica and PI, and ${\beta }_2=d{L}_2/L_{2}dT$ presents the sum of TEC of silica and PI. Usually, the TEC and TOC of silica are very small, but the TOC and TEC of PI are very large. Therefore, FPI₂ has high temperature sensitivity.

In addition, FPI₂ is an F-P cavity entirely composed of PI material, which is sealed by a quartz capillary tube. Due to the transparency of PI, the reflection light of two PI surfaces form a low-fineness FPI₂. When the external gas pressure acts on FPI₂, it cannot effect the PI medium, so the length and RI of the PI cavity will not change. Therefore, FPI₂ is not sensitive to air pressure, and it can be inferred that its air pressure sensitivity is zero.

When the FSR of FPI₂ is about twice of that of FPI₁, they form a first-order (i = 1) HVE after being connected in parallel. Based on the theory of HVE [15], the FSR of the inner-envelope of the HVE sensor can be represented as follows:

8

$$FS{R}_{\text{inner - envelope}}^i=\frac{(i+1)FS{R}_{FPI1}·FS{R}_{FPI2}}{\left|FS{R}_{FPI2}-(i+1)FS{R}_{FPI1}\right|}$$

In pressure experiments, because FPI₁ is sensitive to air pressure and FPI₂ is not, FPI₁ serves as the sensing cavity and FPI₂ serves as the reference cavity. FPI₁ only needs to be placed in the pressure tank. When the intersection of the two inner-envelopes of the HVE spectrum is selected for physical quantity measurement, according to the theory of HVE, the air pressure sensitivity amplification factor of the HVE sensor are expressed as follows:

9

$$A_P^i=\frac{FS{R}_{\text{inner - envelope}}^i}{(i+1)FS{R}_{FP{I}_1}}=\frac{S_P^i}{(i+1)S_{FPI{1}_P}}$$

In temperature experiments, because FPI₂ is sensitive to temperature and FPI₁ is not, FPI₂ serves as the sensing cavity and FPI₁ serves as the reference cavity. FPI₂ is only placed in a temperature furnace for temperature measurement. When the intersection of the two inner envelopes of the HVE spectrum is selected for physical quantity measurement, according to the theory of HVE, the temperature sensitivity amplification factor of the HVE sensor are expressed as follows:

10

$$A_T^i=\frac{FS{R}_{\text{inner - envelope}}^i}{FS{R}_{FP{I}_2}}=\frac{S_T^i}{S_{FPI{2}_T}}$$

Figure 3a exhibits the micrograph and corresponding reflection spectrum of FPI₁. Obviously, the F-P cavity length of FPI₁ is about 188 μm, and the low-fineness FPI₁ has an approximately sine wave spectrum, and the FSR of FPI₁ is about 6.4 nm near 1550 nm. Figure 3b presents the micrograph and reflection spectrum of FPI₂. The PI cavity length of FPI₂ is about 61 μm, and the low-fineness FPI₂ has an approximately sine wave spectrum, and the FSR of FPI₂ is about 12.2 nm near 1550 nm. Their FSRs are close to the result calculated according to Eqs. (3) and (4). In their spectra, the Dip 1 and Dip 2 are chosen the tracking point to measure the temperature and air pressure.

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Fig. 3

a Micrograph and reflection spectrum of FPI₁, b micrograph and reflection spectrum of FPI₂

According to the principle of HVE [15], since the FSR of FPI₁ and FPI₂ is close to twice, when they are connected in parallel, a first-order HVE can be formed. Figure 4a shows the spectrum of the first-order HVE sensor. This is a typical first-order HVE spectrum, with an inner-envelope FSR of 196.4 nm and an outer envelope FSR of 90.2 nm, which is close to the result calculated according to Eq. (8). In Fig. 4a, Dip 3 is an intersection point of the two inner-envelope lines of the HVE spectrum. In the following experiment, we chose to track the changes of Dip 3 points with temperature and air pressure to complete the measurement of temperature and air pressure by HVE sensor. The spatial spectrum obtained by performing Fast Fourier Transform (FFT) on the spectrum in Fig. 4a is shown in Fig. 4b. It is evident that Fig. 4b has two main spatial frequencies of 0.0819 and 0.1531, which correspond one-to-one with the spatial frequencies of FPI₂ (1/FSR_FPI2 = 0.82) and FPI₁ (1/FSR_FPI1 = 0.156), indicating that the spectrum of the HVE sensor is composed of the spectra of FPI₁ and FPI₂.

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Fig. 4

a Spectrum of the HVE sensor, b FFT spectrum of the HVE sensor

Experiment results and discussion

Experimental setup

The HVE sensor is formed by connecting FPI₁ and FPI₂ in parallel through a 3dB coupler. This HVE sensor can achieve high performance of temperature and pressure measurement. Figure 5 shows the experimental devices of the HVE sensor measuring temperature and air pressure. The light emitted from broadband source (BBS, FL-ASE, China Beijing) is transmitted to FPI₁ and FPI₂ through a 2 × 2 3dB coupler, respectively. The light reflected back from FPI₁ and FPI₂ is then received by OSA (AQ6370D, Japan) through this 2 × 2 3dB coupler. Because FPI₁ is sensitive to air pressure and FPI₂ is not sensitive to air pressure, FPI₁ is placed into the pressure tank for measuring air pressure and FPI₂ is placed outside the tank at atmospheric pressure during the air pressure experiment. Because FPI₂ is sensitive to temperature and FPI₁ is not sensitive to temperature, FPI₂ is placed into the temperature furnace for measuring temperature and FPI₁ is placed in the room temperature outside the temperature furnace during temperature experiment. In addition, the air pressure measurement of the sensor is finished at room temperature, and the temperature measurement of the sensor is finished at standard atmospheric pressure.

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Fig. 5

Schematic of the experimental device of the HVE sensor measuring the temperature and air pressure

FPI₁ measuring temperature and air pressure

Figure 6 demonstrates the shift and fitting plots of Dip 1 for a single FPI₁ as a function of temperature. It can be observed that with temperature changes, Dip 1 does not drift significantly, and the fitting between temperature and Dip 1 wavelength is basically a horizontal straight line, indicating that FPI₁ is almost insensitive to temperature. This result is consistent with theoretical analysis. Figure 7 shows the drift and fitting plot of Dip 1 for a single FPI₁ as a function of air pressure. It can be observed that as the air pressure increases, Dip 1 shifts significantly to right. The fitting between the air pressure and Dip 1 wavelength is a good linear relationship, with a slope of 4 nm/MPa and a fitting coefficient of 1. This indicates that FPI₁ has the air pressure sensitivity of 4 nm/MPa and no hysteresis error.

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Fig. 6

a Dip 1 changes with temperature rising, b fitting of the Dip 1 wavelength with temperature changes

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Fig. 7

a Dip 1 varies with air pressure increasing, b fitting of the Dip 1 wavelength with air pressure changes

FPI₂ measuring temperature and air pressure

Figure 8 presents the shift and fitting plot of Dip 2 for a single FPI₂ as a function of temperature. It can be observed that as the temperature increases, Dip 2 shifts significantly to right. The fitting between the temperature and Dip 2 wavelength is a good linear relationship, the slope is 5.27 nm/ °C with the fitting coefficient of 0.9874 when the temperature rises; the slope is 4.77 nm/ °C with the fitting coefficient of 0.9889 when the temperature drops. This indicates that FPI₂ has a sensitivity of 5.27 nm/ °C for heating and 4.77 nm/ °C for cooling, with a certain hysteresis error. Figure 9 shows the drift and fitting plots of Dip 2 for a single FPI₂ as a function of air pressure. It can be observed that with air pressure changes, Dip 2 does not drift significantly, and the fitting between air pressure and Dip 2 wavelength is basically a horizontal straight line, indicating that FPI₂ is approximately insensitive to pressure. This result is consistent with theoretical analysis.

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Fig. 8

a Dip 2 changes with temperature rising, b fitting of the Dip 2 wavelength with temperature changes

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Fig. 9

a Dip 2 varies with air pressure increasing, b fitting of the Dip 2 wavelength with air pressure changes

HVE sensor (FPI₁ + FPI₂) measuring temperature

Figure 10 illustrates the shift and fitting plot of Dip 3 of the HVE sensor as a function of temperature. It can be observed that as the temperature increases, Dip 3 shifts significantly to right. The fitting between the temperature and Dip 3 wavelength is a good linear relationship, the slope is 113.29 nm/ °C with the fitting coefficient of 0.9981 when the temperature rises; the slope is 113.80 nm/ °C with the fitting coefficient of 0.9875 when the temperature drops. This indicates that the heating sensitivity of the HVE sensor is 113.29 nm/ °C and the cooling sensitivity of the HVE sensor is 113.80 nm/ °C, almost no hysteresis error. This is the highest temperature sensitivity achieved so far. Compared to the temperature sensitivity of FPI₂, the HVE sensor amplifies the temperature sensitivity by 21.5 times (113.29/5.27 = 21.5). This result is approximately consistent with theoretical calculations (196.4/12.2 = 16.1). The use of intersection (Dip 3) for physical quantity measurement in the HVE sensor overcomes the problem of inaccurate envelope dip wavelength measurement in the VE sensor.

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Fig. 10

a Dip 3 changes with increasing temperature, b fitting of Dip 3 wavelength with temperature changes

Figure 11a shows the four temperature rising experimental data of Dip 3 of the HVE sensor under the same condition. Clearly, the data from four temperature experiments are basically the same. The average sensitivity of four experiments is 113.82 ± 1.2928 nm/ °C, indicating a maximum fluctuation in sensitivity of ± 1.2928 nm/ °C. The relative error of sensitivity is 0.57% ((1.2928/2)/113.82) = 0.0057), which is relatively small, indicating that the HVE sensor has good repeatability in measuring temperature. The variation of Dip 3 wavelength within 90 min is shown in Fig. 11(b) when the temperature is kept stable at 25.5 °C and 26.5 °C, respectively. At 26.5 °C, the maximum fluctuation of Dip 3 wavelength is ± 1.0 nm; at 25.5 °C, the maximum fluctuation of Dip 3 wavelength is ± 0.4 nm. Therefore, the maximum error in measuring temperature using Dip 3 is only 0.004 °C (0.5/113.82 = 0.004 °C). This result indicates that the sensor has excellent stability.

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Fig. 11

a Four temperature rising experimental data of the HVE sensor, b fluctuation data of Dip 3 of the HVE sensor at two temperature

HVE sensor (FPI₁ + FPI₂) measuring air pressure

Figure 12 indicates the shift and fitting plot of Dip 3 of the HVE sensor as a function of air pressure. It can be observed that as the air pressure increases, Dip 3 shifts significantly to right. The fitting between the air pressure and Dip 3 wavelength is a good linear relationship, the slope is 76.71 nm/MPa with the fitting coefficient of 0.9865 when the air pressure increases; the slope is 74.79 nm/MPa with the fitting coefficient of 0.9762 when the air pressure decreases. This indicates that the boost sensitivity of the HVE sensor is 76.71 nm/MPa and the buck sensitivity is 74.79 nm/MPa, and the difference between them is not significant, indicating that the HVE sensor has small hysteresis error. The HVE sensor amplifies the sensitivity of FPI₁ by 19.1 times (76.71/4 = 19.1), which is approximately consistent with theoretical calculations (196.4/(2*6.4) = 15.3).

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Fig. 12

a Dip 3 changes with increasing air pressure, b fitting of Dip 3 wavelength versus air pressure

Figure 13a shows the four air pressure experimental results of Dip 3 of the HVE sensor under the same condition. Obviously, the data from the four air pressure experiments are basically consistent. The average sensitivity of the four experiments is 72.44 ± 3.1690 nm/MPa, indicating a maximum fluctuation in sensitivity of ± 3.1690 nm/MPa. The relative error of sensitivity is 2.2% ((3.1690/2)/72.44) = 0.022), which is relatively small, indicating that the sensor has good repeatability. The variation of Dip 3 wavelength within 90 min is shown in Fig. 13b when the air pressure is kept stable at 0.1 MPa and 0.2 MPa, respectively. At 0.1 MPa, the maximum fluctuation of Dip 3 wavelength is only ± 0.8 nm; at 0.2 MPa, the maximum fluctuation of Dip 3 wavelength is only ± 1.2 nm. Therefore, the maximum error in measuring air pressure using Dip 3 is only 8.3 kPa (0.6/72.44 = 0.0083 MPa). This result indicates that the sensor has excellent stability.

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Fig. 13

a Four air pressure experimental data of HVE sensor, b fluctuation data of Dip 3 of the HVE sensor at two air pressure

Compare with existing similar sensors

Table 1 lists the comparison of the structure and sensitivity of the proposed sensor with other existing sensors that measure temperature and air pressure simultaneously. From Table 1, it can be seen that our sensor has the highest temperature sensitivity, almost an order of magnitude higher than other sensors. Our sensor’s air pressure sensitivity is slightly lower than that of references [9, 10, 13, 14, 25], and [26], but on the same order of magnitude, while it is higher than the sensitivity of other references. In addition, the sensor proposed by us adopts the harmonic vernier effect, which has the advantages of easy manufacturing process, low cost, and good structural repeatability, providing a new alternative solution for simultaneous measurement of temperature and air pressure in industrial production.

Table 1. Comparison of the existing similar sensors

Sensor structure	Temperature sensitivity	Gas pressure sensitivity	Refs
Two cascade F-P cavity	0.01436 nm/ °C	4.04 nm/MPa	[1]
Two cascade F-P cavity	2.62 nm/ °C	20.63 nm/MPa	[2]
Fiber micro-cavity and fiber-tip	0.0108 nm/ °C	4.1587 nm/MPa	[3]
F-P cavity and anti-resonant reflecting waveguide	0.0277 nm/ °C	− 3.76 nm/MPa	[5]
FBG cascades with FPI	0.2234 nm/ °C	24.99 nm/MPa	[7]
Two cascade MZI	8.455 nm/ °C	78.553 nm/MPa	[9]
Integrated MZI and an F-P cavity	0.317 nm/ °C	-199.0 nm/MPa	[10]
Two parallel FPI and VE	10.29 nm/ °C	-36.93 nm/MPa	[12]
Two cascade FPI and VE	16.51 nm/ °C	100.18 nm/MPa	[13]
Two cascade FPI and VE	14.41 nm/ °C	113.82 nm/MPa	[14]
Two cascade FPI and HVE	0.176 nm/ °C	114 nm/MPa	[25]
Two cascade FPI and HVE	– 0.48 nm/ °C	146.64 nm/MPa	[26]
Two parallel FPI and HVE	113. 29 nm/ °C	76.71 nm/MPa	This work

Discussion of the temperature and air pressure simultaneous measurement

FPI₁ and FPI₂ have different responses to air pressure and temperature. When FPI₁ is acted as the sensing unit in the HVE sensor, the air pressure sensitivity of the HVE sensor is 76.71 nm/MPa. When FPI₂ is used as the sensing unit in the HVE sensor, the HVE sensor achieved the highest temperature sensitivity of 113.29 nm/ °C to date. Therefore, the HVE sensor can measure air pressure at room temperature and temperature at atmospheric pressure. Now, if the Dip 1 wavelength (λ_dip1) of FPI₁ and the Dip 3 wavelength (λ_dip3) of the HVE sensor are selected to track the wavelength shift for sensing, and the sensitivity measurement matrix obtained is:

11

$$\left[\begin{array}{c}\mathrm{\Delta }{\lambda }_{dip1}\\ \mathrm{\Delta }{\lambda }_{dip3}\end{array}\right]=\left[\begin{array}{cc}0& 4\\ 113.29& 76.71\end{array}\right]\left[\begin{array}{c}\mathrm{\Delta }T\\ \mathrm{\Delta }P\end{array}\right]$$

After inverse transformation, the sensitivity measurement matrix for temperature and air pressure follows:

12

$$\left[\begin{array}{c}\mathrm{\Delta }T\\ \mathrm{\Delta }P\end{array}\right]={\left[\begin{array}{cc}0& 4\\ 113.29& 76.71\end{array}\right]}^{-1}\left[\begin{array}{c}\mathrm{\Delta }{\lambda }_{dip1}\\ \mathrm{\Delta }{\lambda }_{dip3}\end{array}\right]$$

Similarly, if the Dip 2 wavelength (λ_dip2) of FPI₂ and the Dip 3 wavelength (λ_dip3) of the HVE sensor are selected for tracking wavelength shift, and the sensitivity measurement matrix obtained is:

13

$$\left[\begin{array}{c}\mathrm{\Delta }{\lambda }_{dip2}\\ \mathrm{\Delta }{\lambda }_{dip3}\end{array}\right]=\left[\begin{array}{cc}5.27& 0\\ 113.29& 76.71\end{array}\right]\left[\begin{array}{c}\mathrm{\Delta }T\\ \mathrm{\Delta }P\end{array}\right]$$

After inverse transformation, the sensitivity measurement matrix for temperature and air pressure follows:

14

$$\left[\begin{array}{c}\mathrm{\Delta }T\\ \mathrm{\Delta }P\end{array}\right]={\left[\begin{array}{cc}5.27& 0\\ 113.29& 76.71\end{array}\right]}^{-1}\left[\begin{array}{c}\mathrm{\Delta }{\lambda }_{dip2}\\ \mathrm{\Delta }{\lambda }_{dip3}\end{array}\right]$$

Therefore, leveraging on the matrices (12) or (14), the sensor achieved simultaneous measurement of temperature and air pressure.

Conclusion

In summary, a highly sensitive HVE sensor capable of measuring temperature and air pressure has been developed. The HVE sensor is formed by FPI₁ and FPI₂ in parallel, where FPI₁ is sensitive to air pressure but not temperature, while FPI₂ is sensitive to temperature but not air pressure. The FSR of FPI₁ and FPI₂ approaches twice, their parallel connection forms a first-order HVE sensor. When the HVE sensor measures air pressure, FPI₁ is the sensing cavity and FPI₂ is the reference cavity. When the HVE sensor measures temperature, FPI₂ is the sensing cavity and FPI₁ is the reference cavity. The experimental results show that the air pressure and temperature sensitivities of the HVE sensor are 76.71 nm/MPa and 113.29 nm/ °C, respectively. When FPI₁ or FPI₂, as well as the HVE sensor, are used as the test objects, simultaneous measurement of temperature and air pressure can be achieved. By measuring the response of the inner envelope intersection point generated by HVE, the accuracy of wavelength detection is improved, the limitation of FSR on magnification factor is eliminated, and the sensitivity of VE is further improved.

Acknowledgements

This work was supported in part by Science and Technology Research Project of Hubei Provincial Department of Education (D20232505), and Hubei Provincial Natural Science Foundation Innovation and Development Joint Fund (2024AFD001), and Hubei Provincial Natural Science Foundation of China (2022CFB480), and National Natural Science Foundation of China (12304281).

Author contribution

Huiling Huang and Chao Jiang wrote the main manscript. Huiling Huang, Tingshui Cao, Long Zhang and Yukun Shu had finished the experimental study. Simei Sun and Guozhou Jiang provided financial support for the experiment and polished the manscript. Hong Li discussed the experimental results and provided some suggestions for the experiment. Chao Jiang and Guozhou Jiang provied guidance for the experiments. Huiling Huang prepared Figs. 1–13. All authors reviewed the manuscript.

Data availability statement

No datasets were generated or analysed during the current study.

Declarations

Conflict of interest

The authors declare no competing interests.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

References

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