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Miniature fiber-optic tip pressure sensor assembled by hydroxide catalysis bonding technology

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Abstract

A miniature fiber-optic tip Fabry-Perot (FP) pressure sensor with excellent high-temperature survivability, assembled by hydroxide catalysis bonding (HCB) technology, is proposed and experimentally demonstrated. A standard single-mode fiber is fusion spliced to a fused silica hollow tube with an outer diameter (OD) of 125 µm, and a 1-µm-thick circular silicon diaphragm with a diameter slightly larger than the OD is bonded to the other endface of the hollow tube by HCB technology. The ultrathin silicon diaphragm is prepared on a silicon-on-insulator (SOI) wafer produced by microelectromechanical systems (MEMS), providing the capability of large-scale mass production. The HCB technology enables a polymer-free bonding between diaphragm and hollow tube on fiber tip with the obvious advantages of high alignment precision, normal pressure and temperature (NPT) operation, and reliable effectiveness. The static pressure and temperature response of the proposed sensor are discussed. Results show that the sensor has a measurable pressure range of 0∼100 kPa, which is well consistent with the measurement range of biological blood pressure. The pressure sensitivity is up to 2.13 nm/kPa with a resolution of 0.32% (0.32kPa). Besides, the sensor possesses a unique high-temperature resistant capability up to 600 °C, which can easily survive even in high-temperature sterilization processes, and it has a low temperature dependence of 0.09 kPa/°C due to the induced HCB bonding technology and the silicon-based diaphragm. Thus, the proposed fiber tip pressure sensor is desirable for invasive biomedical pressure diagnostics and pressure monitoring in related harsh environments.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Miniature pressure sensors have attracted great interest for pressure monitoring in extremely narrow spaces, such as pressure measurement in automotive/aerospace engines [1,2], and oil/gas pipeline [3]. They are also playing a key role in the invasive biomedical diagnosis for in-vivo blood pressure monitoring in stenosis detection area [4,5]. Compared with pressure sensors based on electronic devices, fiber-optic pressure sensors have the advantages of compact size, high sensitivity, corrosion resistance, electromagnetic interference (EMI) immunity, and geometrical flexibility, are ideal for the invasive measurements of biological blood pressure [6,7]. Among them, the diaphragm-based fiber-optic Fabry-Perot interferometric (DFPI) sensor with the cavity formed by a fiber endface and a reflective diaphragm is one of the most common configurations. In principle, the length of the Fabry-Perot (FP) cavity changes according to the attached sensing diaphragm that deformed by the ambient pressure variations. The dimensions, biocompatibility, and especially the stability of the sensor are the key factors for its practicability in clinical medicine. However, the fabrication of conventional DFPI sensors typically involves several organic materials [810]. The bonding adhesive or sensing diaphragm containing polymers, such as commonly used epoxy/cyanoacrylate, has a larger thermal expansion coefficient and its structure will creep with time. Moreover, polymer materials often decompose at high temperatures, severely degrading the performance of DFPI sensors [11]. To avoid introducing polymers in DFPI sensors, some methods for polymer-free assemblies have been developed, and are listed in Table 1.

Tables Icon

Table 1. Different methods for polymer-free assemblies.

As shown in Table 1, direct bonding is an efficient bonding method that mainly relies on Van der Waals force, but it does not provide sufficient structural strength and long-term stability. The weak bonding will easily lead to sensor dissociation [12]. Thermal bonding typically offers high strength, but requires extremely high-temperature tools, such as propane torch, to slightly melt the two bonding interfaces. It will result in low alignment precision and low efficiency between diaphragm and hollow tube [13,14]. For anodic bonding, high voltage up to 1000 V is often applied at the two interfaces to be bonded, making it difficult to precisely align micro-size devices [15]. Both fusion splicing and laser welding are also based on thermal effects. When the diaphragm’s thickness is only a few micrometers, these two methods are inconvenient [16,17].

Since 1998, a bonding technology: hydroxide catalysis bonding (HCB), or named silicate bonding, has been developed for astronomical applications, especially in the assembly of optical components (patented [18] by D.-H. Gwo). The bonding operates between any materials that can form silicate-like networks or covalently attach to a silicate-like network [19] (or aluminate-like network [20]) through hydroxide-catalyzed hydration and dehydration. The HCB technology has been used in NASA satellite mission to joint fused silica components in star tracking telescope (Gravity Probe B [18,19,21]), ground-based gravitational wave detectors (GEO 600 [22], Advanced LIGO [23], and Virgo [24]), space-based gravitational wave detectors (LISA Pathfinder [25,26]). In these astronomical applications, the HCB technology shows great potential in performing at normal pressure and temperature (NPT) conditions, achieving high alignment precision, improving optical properties, and increasing bonding strength [1826]. With respect to previously developed polymer-free assembly methods, the HCB technology is more suitable for assembling DFPI fiber sensors due to its good bonding characteristics. Thanks to its high precision alignment, the diaphragm can be smaller in size (about one hundred microns) and thinner in thickness (only a few microns) which makes the sensor miniature and more sensitive.

In this paper, we present a miniature DFPI pressure tip sensor fabricated directly onto the tip of commercial silica fiber by simple fusion splicing and HCB technology. A hollow tube with the same outer diameter (OD) as fiber is fusion spliced to a fiber endface, and an ultrathin silicon diaphragm with a diameter slightly larger than the OD of fiber is precisely bonded to the other endface of the hollow tube by HCB technology. The entire structure of the proposed sensor is assembled with no organic components, shows biocompatibility, structural stability, and high-temperature resistant capability. The sensing diaphragm is the key element in the sensor, whose materials and dimensions work together to determine the pressure sensitivity. Silicon is commonly used in the preparation of sensing diaphragms due to its advantageous properties, such as good stability, and the thermal expansion coefficient is similar to the fiber [15,27,28]. Therefore, the fabrication method of silicon diaphragm greatly affects the sensor’s performance. With the aid of the microelectromechanical systems (MEMS) technology, the fabrication of ultra-thin silicon diaphragm in desired geometries is simple and of high quality. And the processing technology has the ability of large-scale mass production, greatly improving the efficiency and reducing the cost. In addition, the HCB technology for diaphragm bonding offers high alignment precision, NPT operation, and reliable effectiveness. The extensive production of the proposed sensor can be realized by both MEMS and HCB technologies. The pressure and temperature response of the proposed sensor were experimentally studied. Results show that the sensor has a measurable pressure range of 0∼100 kPa, which is well consistent with the measurement range of biological blood pressure. The sensor has a temperature insensitivity and high-temperature survivability because no polymer is included, resulting in characteristics of good structural stability and resistance to high-temperature sterilization. Thus, the fiber-optic tip sensor has a great potential to work as an invasive medical pressure sensing probe for in-vivo cardiovascular, cerebrovascular, pulmonary, and vesical pressure measurements or other applications in related harsh environments.

2. Sensor design and fabrication

2.1 Sensor design and principle

The structure of the DFPI pressure sensor is shown in Fig. 1(a). It consists of a commercially available single-mode fiber (9 µm/125 µm, YOFC), a silica hollow-core fiber (80 µm/125 µm, FiberHome) as a hollow tube, and an ultrathin silicon diaphragm (thickness of 1 µm) with a thermal oxide layer on its surface. The hollow tube is connected to a fiber endface through an electric arc fusion splicer (KL-300T, JILONG), and a silicon diaphragm is bonded on the other endface of hollow tube by HCB technology. The top view and the side view of a well-bonded sensor probe under an optical microscope are shown in Figs. 1(b) and 1(c).

 figure: Fig. 1.

Fig. 1. Configuration of the proposed fiber-optic tip pressure sensor, (a) structure of the sensor probe, (b) the top view of a well-bonded sensor probe, and (c) the side view of the sensor probe.

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For diaphragm bonding, the HCB technology has been utilized to achieve a polymer-free bonding interface. This method is to introduce aqueous hydroxide solution or silicate solution as a catalyst (e.g. sodium hydroxide, potassium hydroxide, sodium silicate, potassium silicate, ammonia water, and sodium ethoxide) between two surfaces of oxide or oxidizable materials, forming a three-dimensional silicate-like network after settled. When the two surfaces to be bonded are in contact, the hydroxide catalysis commences. The first step is hydration and etching, in which hydroxide ions in the bonding solution act as a catalyst and etch the two surfaces in contact. The process releases silicate ions as described:

$$\textrm{Si}{\textrm{O}_\textrm{2}}\textrm{ + O}{\textrm{H}^{-}}\textrm{ + 2}{\textrm{H}_\textrm{2}}\textrm{O} \to \textrm{Si}({\textrm{OH}} )_{5}^{-} $$
During silicate ions liberation, active hydroxide ions are consumed and the pH of the solution decreases. Once pH is <11, the silicate ions disassociate [29], and polymerization commences:
$$\textrm{Si}({\textrm{OH}} )_\textrm{5}^{-} \to \textrm{Si}{({\textrm{OH}} )_\textrm{4}}\textrm{ + O}{\textrm{H}^{-}}$$
And three-dimensional siloxane chains are formed:
$$\textrm{2Si}{({\textrm{OH}} )_\textrm{4}} \to {({\textrm{HO}} )_\textrm{3}}\textrm{SiOSi}{({\textrm{OH}} )_\textrm{3}}\textrm{ + }{\textrm{H}_\textrm{2}}\textrm{O}$$
With the siloxane chains lengthening, these chains form networks and bond the two surfaces together. In the last step: dehydration, the water in the bonding interface migrates or evaporates until the bond is completely rigid. During this time the bond severely reduces in thickness and increases in strength [30]. The settling time can be varied by appropriately adjusting the ambient temperature and humidity where the bond is formed or changing the concentration of bonding solution [31]. The full bonding procedure takes several weeks at NPT conditions. After completely cured, the bonding interface will yield high mechanical strength and high thermal stability [30,32].

In the structure of the proposed sensor, single-mode fiber is a lead-in fiber for introducing light into the hollow tube. The inner surface of the diaphragm and the fiber endface form a low-finesse FP interferometer. When incident light is illuminated into the FP cavity, the reflections occur on two interfaces of the fiber-cavity interface and the inner surface of silicon diaphragm. The reflection spectrum can be estimated as a two-beam interferometer model. The FP cavity length can be extracted from its spectrum and act as a function of detected parameters. The reflection spectrum of the FP interferometer is regarded as an intensity modulation signal caused by the optical phase difference between two reflections. The intensity of the reflection spectrum can be expressed as

$$I = {I_\textrm{1}} + {I_\textrm{2}} + \textrm{2}\sqrt {{I_\textrm{1}}{I_\textrm{2}}} \cos \delta$$
where I1 and I2 are intensities of the two reflected beams, δ is the phase difference between two reflected beams, and $\delta = \frac{{4\pi nL}}{\lambda } + \pi$, n is the refractive index (RI) of the medium filled in the FP cavity, L is the length of the FP cavity, λ is working wavelength.

Silicon diaphragm is the sensing element in the proposed sensor. According to surrounding pressure changes, the circular silicon diaphragm deforms and result in FP cavity length variations. Because the diaphragm is flat and uniform in thickness, it deforms from the center to the surrounding area. According to the elastic deformation theory of circular diaphragm, its center deflection Δy, under applied pressure ΔP, is determined by [33]:

$$\Delta y = \frac{{3(1 - {v^2}){a^4}}}{{16E{h^3}}}\Delta P$$
where $v$ and E are Poisson’s ratio ($v$=0. 23 at room temperature) and Young’s modulus (E = 190 GPa) of silicon, respectively, a is the effective radius, and h is the thickness of the diaphragm.

In theory, the DFPI sensor has a linear pressure limit that is determined by the diaphragm’s elastic deformation limit (no more than 30% of its thickness). Throughout the deformation range, the sensitivity is uniform (linear) and the pressure limit can be determined by:

$${P_{limit}} = \frac{{8E{h^4}}}{{5(1 - {v^2}){a^4}}}$$
According to Eq. (5) and Eq. (6), the sensitivity and pressure measurement range of the sensor only depend on the effective radius and thickness of the sensing diaphragm. If a diaphragm with a larger effective radius or thinner thickness is used, higher sensitivity will be obtained. However, a larger effective radius or a thinner thickness might result in a lower linearity pressure limit. Therefore, it is important to determine the diameter and thickness of the diaphragm, and make the sensitivity and pressure limit appropriate. Figures 2(a) and 2(b) shows the sensitivity and pressure limit of the sensing silicon diaphragm with an effective radius of 40 µm calculated by Eq. (5) and Eq. (6) under different diaphragm thickness. When the thickness of the diaphragm is 1 µm, the theoretical sensitivity of the sensor is calculated as 2.39 nm/kPa and the pressure measurement range is 125.38 kPa.

 figure: Fig. 2.

Fig. 2. Theoretically calculated pressure sensitivity and pressure limit of the silicon diaphragm with an effective diameter of 80 µm under different diaphragm thickness, (a) the relationship between the theoretical sensitivity and diaphragm thickness, and (b) the relationship between the theoretical pressure limit and the diaphragm thickness.

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2.2 Sensor fabrication

The sensor fabrication process consists of three parts: sensing diaphragm preparation, the splicing between fiber and hollow-core fiber, the bonding process of the hollow-core fiber endface and diaphragm. The preparation of silicon diaphragm is shown in Fig. 3, including common MEMS lithography and etching processes. The silicon diaphragm was prepared using a commercially available 4-inches double-polished silicon-on-insulator (SOI) wafer, with a device layer’s thickness of 1 µm, a buried oxide of 1.8 µm, and a silicon base layer of 430 µm. Before the diaphragm preparation process, the SOI wafer was thermally oxidized. We first spin coated the positive photoresist (S1805, Shipley) on the SOI wafer and after that soft baking was done for 1 min at 90 °C. Next, a mask and the SOI wafer were placed into lithography equipment for exposure. After the exposure, the wafer was immersed in a developer solution. Then the wafer was etched by hydrofluoric acid (HF) to remove redundant parts on the surface of the thermal oxide layer. Thereafter we rinsed the wafer with acetone to remove the photoresist and introduced potassium hydroxide (KOH) to etch bare silicon surface and form silicon diaphragms. Finally, all remaining oxide layer was etched away by HF. The acquired silicon diaphragm is uniform in shape and has high surface flatness, as shown in Figs. 1(b) and 1(c). The entire fabrication procedure by MEMS technology is simple, clean, and very effective for producing hundreds of well-shaped diaphragms each time.

 figure: Fig. 3.

Fig. 3. Fabrication procedure of silicon diaphragms by MEMS technology.

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To further compact the entire sensor structure, we selected fused silica hollow-core fiber as a hollow tube to be fusion spliced at the fiber endface, as shown in Fig. 4(a). The other end of the hollow tube was cut to a specified length (175 µm) using a fiber cleaver (KL-21B, JILONG) under a microscope, as shown in Fig. 4(b). To smooth the endface of the hollow tube, it was then slightly polished using 0.3-µm grit size polishing paper for surface flatness. Prior to the HCB process, the silicon diaphragm must be thermally oxidized to grow an oxide layer of at least tens of nanometers. Thereafter, the silicon diaphragm was rinsed successively with methanol, acetone, and deionized water to keep the bonding surface clean. And the fiber tip spliced with a hollow tube was ultrasonically cleaned for five minutes in methanol, acetone and deionized water, respectively. Then the bonding solution was prepared for the HCB process. A sodium silicate solution (typically 10∼14% NaOH, 25∼30% SiO2, in aqueous solution) was diluted with deionized water in a volume ratio of 1:6 and filtered through a 0.22 µm micropore filter. In class 100 ultra-clean working environment, a little bonding solution was deposited in the center of a plastic substrate by a micro-pipettor. We dipped the bonding solution on the substrate with the splicing structure of the fiber tip, then quickly aligned and compressed the silicon diaphragm slightly for 5 mins at room temperature until the bond was settled, as shown in Figs. 4(c)–4(e). Since the bonding may take several weeks to cure at room temperature [18,19], elevating the ambient temperature of the bonding process appropriately can decrease the curing duration [30]. The sensor was placed in an oven and heat treated at 200°C for 24 hours to ensure that the moisture in the bonding interface was evaporated completely. After the HCB process, the diaphragm was completely bonded to the hollow tube with clear boundaries and no impurities, as shown in Fig. 4(f).

 figure: Fig. 4.

Fig. 4. Fabrication of the fiber-optic tip pressure sensor, (a) fusion splice fiber endface to a hollow tube, (b) cut the other endface of hollow tube, (c) dip bonding solution, (d) align and slightly compress the silicon diaphragm, (e) lift the silicon diaphragm, and (f) the picture of a typical sensor after HCB process under a microscope.

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3. Experimental results and discussion

Figure 5 is the schematic diagram of the experimental setup for pressure tests. We employed an optical sensing interrogator (OSI, sm125, Micron Optics) with a wavelength range of 1510-1590 nm, a spectral resolution of 5 pm, and a sampling rate of 2 Hz to acquire reflection interference spectrum of the fiber-optic tip pressure sensor. The proposed sensor was sealed in a pressure chamber whose pressure was supplied by a pressure generator (113A, ConST) and monitored by a digital pressure gauge (211, ConST). The output spectrum signal was processed by a cross-correlation signal processing method [34,35] to demodulate the FP cavity length as a function of ambient pressure.

 figure: Fig. 5.

Fig. 5. The schematic diagram of the experimental setup for a static pressure test.

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The FP cavity length was monitored as the pressure increased/decreased and the proposed sensor’s pressure response was experimentally studied. In Fig. 6(a), the blue shift in the reflection spectrum shows that the length of the FP cavity decreases with increasing pressure due to the diaphragm’s deformation. It indicates that the ultrathin silicon diaphragm is tightly bonded to the hollow tube by the HCB technology, and the FP cavity is sealed. As shown in Fig. 6(b), after three increase/decrease pressure cycles from 0 to 100 kPa (10 kPa per step) at room temperature, the sensor has a pressure sensitivity of 2.13 nm/kPa with linearity up to 0.99, is similar to the theoretically calculated result in Fig. 2(a). And the pressure response of the sensor has a maximum hysteresis of 0.42%.

 figure: Fig. 6.

Fig. 6. The static pressure response of the proposed sensor from 0 to 100kPa, (a) the shift in the reflection spectrum with pressure, and (b) the pressure response of the sensor for three cycles.

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The OSI we employed has an available internal National Institute of Standards and Technology (NIST) traceable absolute wavelength reference to ensure long-term accuracy and reliability. We utilized the OSI to collect the spectrum of the proposed sensor and used our demodulation algorithm based on the cross-correlation signal processing method to attain the FP cavity length [17,3437]. The resolution was obtained by measuring the standard deviation of FP cavity length results in a period of time. During 10 minutes of continuous testing at a pressure of 50 kPa, the real-time cavity length was recorded once per second. The measurement results of the cavity length are shown in Fig. 7(a), and the standard deviation is approximately 0.34 nm. If the resolution of the cavity length is twice the standard deviation, the pressure resolution of the proposed sensor is about 0.32 kPa, which is 0.32% of the full scale. In addition, the variations in FP cavity length over time after continuous testing at room temperature of 24°C for 10 hours is shown in Fig. 7(b). Results show that the cavity length has good stability, and the initial cavity length does not change much with time. The maximum fluctuation of the FP cavity length is about 0.70 nm, and the maximum pressure fluctuation is 0.33 kPa.

 figure: Fig. 7.

Fig. 7. Variations of the FP cavity length with time at room temperature, (a) experiment results of the standard deviation of the proposed sensor (about 10 mins of time span), and (b) changes of initial cavity length within 10 hours.

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The proposed sensor has the potential to operate at high temperatures for the entirely polymer-free structure. To investigate the high-temperature response characteristics of the sensor, the sensor was inserted into a temperature control case together with a thermocouple for temperature calibration from room temperature (24 °C) to 600 °C. As shown in Fig. 8(a), its temperature sensitivity is 0.21 nm/°C which leads to a temperature dependence of 0.09 kPa/°C. Our proposed sensor exhibits low temperature dependence, thanks to the induced the silicon-based diaphragm and HCB technology. The thermal expansion coefficient of silicon is very low and similar to that of fibers, and the silicate network formed by HCB technology between diaphragm surface and hollow tube endface is an inorganic structure, which is only a covalent bond structure composed of silicon-oxygen atoms. The thermal expansion coefficient of the bonding layer is much lower than that of the commonly used polymer layer. Therefore, the temperature dependence caused by the thermal expansion of materials constituting the sensor is very low. When the surrounding temperature changes, the pressure changes of trapped air in the cavity is almost the major factor. Such error can be neglected in applications with limited temperature variations, such as invasive biomedical pressure monitoring.

 figure: Fig. 8.

Fig. 8. High-temperature response characteristics of the proposed sensor, (a) temperature response from room temperature (24 °C) to 600 °C, and (b) comparison of reflection spectrum before/after a cycle of temperature warming and cooling to room temperature.

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Otherwise, the proposed sensor can be connected in series with Fiber Bragg Grating (FBG) for temperature compensation, which is an effective method to solve the issue of temperature-pressure cross-sensitivity. The peak wavelength of FBG and the interference spectrum of FP can be collected by OSI at the same time. Because the FBG is isolated from pressure variations, the temperature information can be demodulated independently from the shift of FBG peak wavelength, and it has a linear response to the temperature changes. Besides, the length of FP cavity has a linear response to temperature and pressure respectively at the same time. In our related works [3638], the temperature information is first obtained by monitoring the shift of the FBG peak wavelength, and then the accurate pressure information can be obtained by subtracting the change of FP cavity length caused by temperature from the actual change of FP cavity length.

The reflection spectrum at room temperature recorded as a black curve is shown in Fig. 8(b). After a temperature cycle process of warming and cooling from room temperature to 600 °C, the other reflection spectrum was recorded as a red curve and no significant changes in the spectrum were observed. The results indicate that the proposed sensor has high-temperature resistant capability and good recovery characteristics.

4. Conclusion

A miniature fiber-optic tip pressure sensor fabricated by fusion splicing and HCB technology has been proposed and experimentally demonstrated. The proposed sensor offers the advantages of compact structure, high sensitivity, good biocompatibility, and high-temperature survivability. The preparation of sensing diaphragm by MEMS technology on SOI wafer enables the large-scale mass production of well-shaped ultrathin silicon diaphragms. The diaphragm’s bonded method based on HCB technology involves no high-temperature, high-pressure tools or strong corrosive chemicals, providing high alignment precision and reliable effectiveness. A measurable pressure range of 0∼100kPa, pressure sensitivity of 2.13 nm/kPa, and a resolution of 0.32% (0.32 kPa) were experimentally demonstrated. And the sensor possesses the high-temperature resistant capability up to 600 °C with a low temperature dependence of 0.09 kPa/°C. This proposed tip pressure sensor is significantly suitable for disposable invasive medical diagnostics and pressure monitoring in harsh environments.

Funding

National Natural Science Foundation of China (61520106013, 61727816); Fundamental Research Funds for the Central Universities (DUT18ZD215).

Disclosures

The authors have no relevant financial interests in this article and no potential conflicts of interest to disclose.

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Figures (8)

Fig. 1.
Fig. 1. Configuration of the proposed fiber-optic tip pressure sensor, (a) structure of the sensor probe, (b) the top view of a well-bonded sensor probe, and (c) the side view of the sensor probe.
Fig. 2.
Fig. 2. Theoretically calculated pressure sensitivity and pressure limit of the silicon diaphragm with an effective diameter of 80 µm under different diaphragm thickness, (a) the relationship between the theoretical sensitivity and diaphragm thickness, and (b) the relationship between the theoretical pressure limit and the diaphragm thickness.
Fig. 3.
Fig. 3. Fabrication procedure of silicon diaphragms by MEMS technology.
Fig. 4.
Fig. 4. Fabrication of the fiber-optic tip pressure sensor, (a) fusion splice fiber endface to a hollow tube, (b) cut the other endface of hollow tube, (c) dip bonding solution, (d) align and slightly compress the silicon diaphragm, (e) lift the silicon diaphragm, and (f) the picture of a typical sensor after HCB process under a microscope.
Fig. 5.
Fig. 5. The schematic diagram of the experimental setup for a static pressure test.
Fig. 6.
Fig. 6. The static pressure response of the proposed sensor from 0 to 100kPa, (a) the shift in the reflection spectrum with pressure, and (b) the pressure response of the sensor for three cycles.
Fig. 7.
Fig. 7. Variations of the FP cavity length with time at room temperature, (a) experiment results of the standard deviation of the proposed sensor (about 10 mins of time span), and (b) changes of initial cavity length within 10 hours.
Fig. 8.
Fig. 8. High-temperature response characteristics of the proposed sensor, (a) temperature response from room temperature (24 °C) to 600 °C, and (b) comparison of reflection spectrum before/after a cycle of temperature warming and cooling to room temperature.

Tables (1)

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Table 1. Different methods for polymer-free assemblies.

Equations (6)

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Si O 2  + O H  + 2 H 2 O Si ( OH ) 5
Si ( OH ) 5 Si ( OH ) 4  + O H
2Si ( OH ) 4 ( HO ) 3 SiOSi ( OH ) 3  +  H 2 O
I = I 1 + I 2 + 2 I 1 I 2 cos δ
Δ y = 3 ( 1 v 2 ) a 4 16 E h 3 Δ P
P l i m i t = 8 E h 4 5 ( 1 v 2 ) a 4
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