## Abstract

We introduce a simple residual pressure self-measurement method for the Fabry-Perot (F-P) cavity of optical MEMS pressure sensor. No extra installation is required and the structure of the sensor is unchanged. In the method, the relationship between residual pressure and external pressure under the same diaphragm deflection condition at different temperatures is analyzed by using the deflection formula of the circular plate with clamped edges and the ideal gas law. Based on this, the residual pressure under the flat condition can be obtained by pressure scanning process and calculation process. We carried out the experiment to compare the residual pressures of two batches MEMS sensors fabricated by two kinds of bonding process. The measurement result indicates that our approach is reliable enough for the measurement.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Extrinsic Fabry-Perot interferometric (EFPI) pressure sensors have been extensive studied and widely used in various fields such as biomedical sensing [1], environmental monitoring in aerospace application [2,3] and downhole application [4,5], due to their advantages of immunity to electromagnetic interference, compact size and high sensitivity. With the development of micro/nanofabrication technology [6–10], the Fabry-Perot (F-P) cavity can be sealed in a variety of ways. So far, Micro-electromechanical Systems (MEMS) technology is the most effective way to manufacture these pressure sensors with a sealed vacuum cavity [7–10]. However, some gases will still be trapped in the cavity because of chemical reaction and leakage. The thermal expansion of the residual gas induces an undesirable pressure on the inner surface of diaphragm with an increase of the environment temperature, leading to a strong temperature dependence of the sensor [11]. In addition, vacuum packaging is also very important for other types of MEMS devices such as resonators, gyroscope sensors and pressure sensors because they need to maintain the frequency response and Q-factor [12]. Thus, an accurate measurement of the residual pressure is critical for evaluating the MEMS sensor quality such as hermeticity and vacuum maintenance.

So far, several works have been performed for measuring the residual pressure inside the cavity of MEMS devices. In 1993, Michael A. Huff et al. calculated the residual gas pressure inside the bonded silicon wafers by using the theoretical geometric deflection of edge clamped circular diaphragm and ideal gas state equation [13]. However, the accuracy of the method is strongly dependent on the measurement accuracy of the parameters like cavity radius, diaphragm thickness, diaphragm deflection and the exact Young’s modulus of the material. In 1998, H. Kapels et al. drilled a channel into a sealed cavity with the help of the focused ion beam and compared the pressure dependent resonance frequency of the sealed and opened cavities respectively to realize the residual pressure in it [14]. This method will destruct the sensor and therefore is usually employed to measure several samples of sensors to approximately evaluate the whole batch of sensor’s performance. In 2014, Jinde Yin et al. measured the residual pressure inside the MEMS chip of fiber-optic pressure sensor by placing two fibers at different position of the cavity. The deflection responses of the two positions with the pressure were analyzed to obtain the residual pressure [15]. However, this method increases the complexity for assembling the two fibers with the MEMS chip and can only realize sampling inspection.

In this paper, we present a novel method for measuring the residual pressure of the F-P pressure sensors fabricated by MEMS technology. We establish a model which describes the relationship among diaphragm deflection, residual pressure and external pressure at different temperatures by combining the deflection formula of the circular plate theory and the ideal gas law. Based on this model, the residual pressure can be measured without the influence of the varied material properties and thermal stress of the diaphragm caused by temperature variation. We carried out the experiment by using polarization low-coherence interference (PLCI) demodulator to measure the cavity length information with the external pressure and successfully demonstrated the reliability of the method by comparing the residual pressures of two batches of MEMS sensors which were fabricated by two kinds of bonding process. The maximum measured standard deviation is 0.172 kPa. Our method allows the residual pressure to be achieved only based on the sensor itself and thus realize a self-measurement. It will not affect the structure and the use of the sensor comparing to the pervious approaches. The possibility for evaluating the long-term tightness of the cavity can be provided.

## 2. Theory of residual pressure measurement method

The configuration of the optical MEMS pressure sensor is shown as Fig. 1. The F-P cavity structure is composed of silicon diaphragm and glass substrate where a shallow cylindrical cavity is etched into the surface. Light injects into the multimode fiber and is partially reflected on the reflective layer on the glass surface and the inside surface of the diaphragm. The light then propagates back through the same fiber with an optical path difference (OPD) of double cavity length which varies along with the deflection of the diaphragm. The gas sealed in the cavity is the residual pressure which has an undesired effect on the diaphragm deflection. Hence we expect to measure it. In this paper, we aim to measure the residual pressure at 0 $\xb0\text{C}$ when the diaphragm under flat condition.

The diaphragm deflection is affected by the following factors: the residual pressure ${P}_{R}$ in the cavity, the external pressure ${P}_{E}$ applied on the diaphragm, the flexural rigidity of the diaphragm and the diameter of the cavity. Besides, for the silicon-glass bonding situation, the lateral thermal stress on the bonding interface due to thermal expansion mismatch among the materials, also have an effect on the behavior of the diaphragm. The out-of-plane deflection of the diaphragm resulting from the applied pressure for an edges clamped round diaphragm of uniform thickness can be expressed as [16],

*t*and

*a*are the thickness and the effective radius of the silicon diaphragm respectively.

*E*and $\nu $ are Young’s modulus and Poisson’s ratio of the diaphragm, which are both affected by temperature,

*r*is the radial distance from the center of diaphragm, and 1 + $\xi $ is the compensation factor for the deflection due to a lateral load on the diaphragm. When the diaphragm is subjected to a lateral load, the compensation factor $\xi $ is given by [16],$\xi =\sigma t{a}^{2}/\left(14.68D\right)$, where $\sigma $ is the thermal stress which depends on the difference between the bonding temperature and the operating temperature. The deflections of diaphragm are not a fixed value under different operating temperatures since $D$ and $\xi $ are all related to temperature. Thus, the deflections of the diaphragm ${\omega}_{1}$ and ${\omega}_{2}$ at temperature ${T}_{1}$ and ${T}_{2}$ can be written as Eqs. (2) and (3),

*T*can be described by the ideal gas law since the F-P cavity is sealed,where

*V*is the volume of the cavity,

*n*is the number of gas molecules,

*R*is the specific gas constant and

*T*is the absolute temperature of the gas. Under certain external pressures ${P}_{E1}$ and${P}_{E2}$, the diaphragm deflections ${\omega}_{1}$and ${\omega}_{2}$ will be equal, which means that the current cavity volume${V}_{1}={V}_{2}$. According to Eq. (4), we getAssume${\omega}_{1}={\omega}_{2}={\omega}_{eq}$, according to Eqs. (2) and (3), we can get the expressions of residual pressure at ${T}_{1}$ and ${T}_{2}$ by the following equations,

*D*and $\xi $as the temperature changes. When under the flat diaphragm condition${\omega}_{eq}=0$, the second part can be removed. As a consequence, a simplified residual pressure calculation formula group is given as

*D*and $\xi $.

Figure 2 is an illustrative graph interpretation for calculate process of our method. Through scanning external pressure at ${T}_{1}$, ${T}_{2}$ and then demodulating the cavity length information at each pressure, we will obtain the relationship between cavity length information *d* and external pressure ${P}_{E1}$, ${P}_{E2}$ respectively as shown in Fig. 2(a). Find every satisfied group of external pressure ${P}_{E1}$and ${P}_{E2}$ that make the current cavity lengths equal, like examples showed in Fig. 2(a). Substitute all the groups into Eq. (9),${P}_{R1}=({P}_{E2}-{P}_{E1}){T}_{1}/({T}_{2}-{T}_{1})$, and the calculation results are presented in Fig. 2(b). The curve of ${P}_{R1}={P}_{E1}$ in Eq. (9), is plotted to identify the satisfied${P}_{R1}$. Obviously, the intersection of the two curves indicates the flat diaphragm condition ${\omega}_{eq}=0$and the current external pressure ${P}_{E1}$ is the residual pressure ${P}_{R1}$ at ${T}_{1}$. In this way, when ${T}_{1}$ is chosen to be 273 K, the residual pressure ${P}_{R1}$ what we expect to measure can be figured out.

In the proposed method, the effect of thermal expansion of the material on the cavity length of the sensor has been ignored. In fact, the composition material of the cavity, Pyrex glass, has a slight expansion from ${T}_{1}$ to ${T}_{2}$. The ignored cavity length change can be approximately express as

where $h$ is the depth of the cylindrical cavity,${\alpha}_{g}$ is the thermal expansion coefficient of Pyrex glass. Substitute Eq. (10) into Eq. (9), the compensation part of the measurement result can be written as,## 3. Sensor fabrication and experimental setup

In order to demonstrate the method, we carried out the experiment by comparing two batches of fiber-optic MEMS pressure sensors fabricated by our laboratory. The sensors were fabricated as the following steps. We first etched out the shallow cylindrical cavity array in a 4-inch Pyrex 7740# glass wafer. Then, reflective coating with 30% reflectivity is plated on the bottom of cylindrical cavity. After that, 4-inch silicon on insulator (SOI) wafer is bonded onto the glass wafer. After etching out the handing layer and the buried oxide layer, the wafer is finally diced into hundreds of small pieces to become independent FPI chips. The chip is connected to the glass ferrule. A multimode fiber is inserted into the glass ferrule and fixed to contact with the glass substrate surface. The difference of the two batches is that they are produced by using two kinds of bonding method. The first batch used the anodic bonding [17]. The interface of silicon and glass forms chemical bonds during chemical reaction under the force of electric power to bond them. A small amount of gas is produced during that process.The second batch used the Au-Au thermal-compress bonding [18] through patterning Au on the glass wafer and SOI wafer. The Au layers diffused and contacted with each other in the atomic level under the environment of heat and force. For batch 1, the thickness of the silicon membrane is 50 μm, and the depth and the radius of the cylindrical cavity are 26 μm and 1200 μm respectively. For batch 2, the thickness of the silicon membrane is 40 μm, and the depth and the radius of the cylindrical cavity are 26 μm and 1000 μm respectively. The thicknesses of Pyrex glass of the two batches are both 500 μm. The samples of the sensor are shown in Fig. 3.

The schematic diaphragm of the residual pressure measurement system is shown in Fig. 4. We put the optical MEMS F-P pressure sensors in an air-pressure chamber, in which the pressure can be precisely tuned by a controller with a control precision of 0.02 kPa. The chamber and the pressure sensor are placed in a thermostat with precision of 0.5 K to change the temperature. We can demodulate the information of optical path difference (OPD) of the sensor by a PLCI demodulator. The light from a white-light source is launched into the sensor via a 3 dB coupler, and then the reflected signal from the sensor is guided into PLCI. In the PLCI, light passes througha polarizer, a birefringent optical wedge and a polarized analyzer in turn.When the OPD caused by the F-P cavity matched with the OPD caused by the thickness of the birefringent wedge, the interference fringes will appear and a linear CCD array is used to receive it. CCD converts the optical signal into an electrical signal and digitized by data acquisition card for further processing in computer. The absolute phase recovery algorithm [19] is used to obtain the absolute phase correspond to the cavity length from the interference fringes. Through scanning the pressure in the chamber at ${T}_{1}$ and ${T}_{2}$, we can obtain the demodulation results of absolute phase with external pressure by utilizing the system mentioned above.

## 4. Experiment result and discussion

Two sensors of each batch were placed in the pressure chamber together. In consideration of the possible range of the residual pressure, we scanned the pressure from 10 kPa to 50 kPa with a step of 1 kPa at the constant temperature provided by a thermostat. The scanning process was repeated twice at temperature ${T}_{1}$ and ${T}_{2}$ respectively. Since the ratio ${T}_{1}/({T}_{2}-{T}_{1})$in Eq. (9) has an enlargement effect on the error of the scanning and demodulation process, it would be better to set ${T}_{1}$ as low as possible and set temperature difference ${T}_{2}-{T}_{1}$ as large as possible to reduce the ratio. Thus, we choose ${T}_{1}=273\text{K}$and ${T}_{2}=323\text{K}$. For each change of the pressure, the interference fringe was recorded after 2 minutes to ensure reliable data. Each temperature was kept at least 60 min before changing the pressure to eliminate the influence of the temperature fluctuation.

The demodulated results of the absolute phase of the four sensors are shown in Fig. 5. Sensor 1-A and sensor 1-B belong to batch 1, sensor 2-A and 2-B belong to batch 2. There is a good linearity of the curves between external pressure and absolute phase at each temperature with the linear correlation coefficients not greater than −0.999945. The residual pressure expands in the sealed cavity when temperature rises, thus the cavity length becomes longer and the curve rises under the same external pressure. Note that the curves of one sensor at ${T}_{1}$ and ${T}_{2}$ are not completely parallel due to the effect of the Young's modulus of silicon and the thermal stress at different temperatures.

Figure 6 shows the calculation results of the residual pressure ${P}_{R1}$ by Eq. (9), in which one curve describes the formula ${P}_{R1}=({P}_{E2}-{P}_{E1}){T}_{1}/({T}_{2}-{T}_{1})$, and we plot ${P}_{R1}$ varies with ${P}_{E1}$. The other curve describes the constraint condition of it, i.e.${P}_{R1}={P}_{E1}$. The coordinate value of the intersection point of the two curves indicates the residual pressure value under the flat diaphragm condition at 273 K before compensation. As the inset of Fig. 6 shows, the intersection position of the sensor 1-A and 1-B are 30.063 kPa and 29.412 kPa respectively, the sensor 2-A and 2-B are 12.278 kPa and 12.136 kPa respectively. It is reasonable that the sensors produced from batch 1 have a larger residual pressure than sensors from batch 2 because of the production of gas during anodic bonding. The consistency of measurement results for the sensors of the same batch verifies the reliability of the introduced method. The little measurement difference of the same batch sensors happens highly because of the etching process that brings error to the diameter and length of the cylindrical cavities and result in a difference of the cavity volumes.

In order to analyze the random error of the measurement, experiments have been carried out for 50 times at the same experimental conditions. The measurement results are presented in Fig. 7 and Table 1. The standard deviations of the four sensors are 0.046 kPa, 0.095 kPa, 0.117 kPa and 0.172 kPa respectively.

The compensation part $\u25b3{P}_{R1}$ which owing to the expansion of Pyrex glass are calculated according to Eq. (11), as the depth of the cylindrical cavity$h=26\text{\mu m}$, the thermal expansion coefficient of Pyrex #7740 glass ${\alpha}_{g}=3.23\times {10}^{-6}/\text{K}$ [20]. As shown in Table 1, the parts are all less than 1 kPa.

In general, the proposed method is not only applicative for measuring the residual pressure of optical F-P pressure sensors based on MEMS technology, but also theoretically suitable for measuring that of other kinds of diaphragm based F-P pressure sensors. It should be noted that the cavity length of sensor must be larger than the coherence length of light source to avoid overlap of adjacent orders low-coherence interference fringes and the pressure sensitivity should be large enough to distinguish the residual pressure.

## 5. Conclusion

A novel non-destructive self-measurement method of residual pressure for MEMS F-P pressure sensor was presented. We built a model that related to external pressure and temperature for calculating the residual pressure. Two batches of pressure sensors with different MEMS bonding processes were measured for the residual pressure by simple pressure scanning and calculation process. The results showed there is a reasonable consistency between the sensors of the same processing technology and the maximum measurement standard deviation is 0.172 kPa which indicate that our approach is reliable for the measurement. No special installation is required and the structure of the sensor is unchanged. The residual pressure can be measured any time to evaluate the long-term tightness of the structure.

## 6. Funding

This work was supported by National Natural Science Foundation of China (Grant Nos. 61505139, 61735011, 61675152, 61378043, 61735011 and 61475114), National Instrumentation Program of China (Grant No. 2013YQ030915), the open project of Key Laboratory of Opto-electronics Information Technology (Grant No. 2017KFKT003), the open project of Key Laboratory of Micro Opto-electro Mechanical System Technology (Grant No. MOMST2016-3), Ministry of Education, and the Seed Foundation of Tianjin University.

## References and links

**1. **P. Roriz, O. Frazão, A. B. Lobo-Ribeiro, J. L. Santos, and J. A. Simões, “Review of fiber-optic pressure sensors for biomedical and biomechanical applications,” J. Biomed. Opt. **18**(5), 50903 (2013). [CrossRef] [PubMed]

**2. **M. J. Gander, W. N. Macpherson, J. S. Barton, R. L. Reuben, J. D. C. Jones, R. Stevens, K. S. Chana, S. J. Anderson, and T. V. Jones, “Embedded micromachined fiber-optic Fabry-Perot pressure sensors in aerodynamics applications,” IEEE Sens. J. **3**(1), 102–107 (2003). [CrossRef]

**3. **W. J. Pulliam, P. M. Russler, and R. S. Fielder, “High-temperature, high-bandwidth, fiber optic, MEMS pressure-sensor technology for turbine-engine component testing,” Proc. SPIE **4578**, 229–238 (2002). [CrossRef]

**4. **H. Choi, A. Cantrelle, C. Bergeron, and P. Tubel, “Minimization of temperature cross-sensitivity of EFPI pressure sensor for oil and gas exploration and production applications in well bores,” Proc. SPIE **5589**, 337–344 (2004). [CrossRef]

**5. **S. H. Aref, H. Latifi, M. I. Zibaii, and M. Afshari, “Fiber optic Fabry-Perot pressure sensor with low sensitivity to temperature changes for downhole application,” Opt. Commun. **269**(2), 322–330 (2007). [CrossRef]

**6. **A. Balčytis, D. Hakobyan, M. Gabalis, A. Žukauskas, D. Urbonas, M. Malinauskas, R. Petruškevičius, E. Brasselet, and S. Juodkazis, “Hybrid curved nano-structured micro-optical elements,” Opt. Express **24**(15), 16988–16998 (2016). [CrossRef] [PubMed]

**7. **M. Li, M. Wang, and H. Li, “Optical MEMS pressure sensor based on Fabry-Perot interferometry,” Opt. Express **14**(4), 1497–1504 (2006). [CrossRef] [PubMed]

**8. **C. Pang, H. Bae, A. Gupta, K. Bryden, and M. Yu, “MEMS Fabry-Perot sensor interrogated by optical system-on-a-chip for simultaneous pressure and temperature sensing,” Opt. Express **21**(19), 21829–21839 (2013). [CrossRef] [PubMed]

**9. **Y. Ge, M. Wang, and H. Yan, “Optical MEMS pressure sensor based on a mesa-diaphragm structure,” Opt. Express **16**(26), 21746–21752 (2008). [CrossRef] [PubMed]

**10. **D. C. Abeysinghe, S. Dasgupta, J. T. Boyd, and H. E. Jackson, “A novel MEMS pressure sensor fabricated on an optical fiber,” IEEE Photonics Technol. Lett. **13**(9), 993–995 (2001). [CrossRef]

**11. **W. Wang and F. Li, “Large-range liquid level sensor based on an optical fibre extrinsic fabry-perot interferometer,” Opt. Lasers Eng. **52**(1), 201–205 (2014). [CrossRef]

**12. **Y. Li and Z. Jiang, “An overview of reliability and failure mode analysis of Microelectromechanical Systems (MEMS),” in *Handbook of Performability Engineering*, K. B. Misra, ed. (Springer London, 2008).

**13. **M. A. Huff, A. D. Nikolich, and M. A. Schmidt, “Design of sealed cavity microstructures formed by silicon wafer bonding,” J. Microelectromech. Syst. **2**(2), 74–81 (1993). [CrossRef]

**14. **H. Kapels, T. Scheiter, C. Hierold, and R. Aigner, “Cavity pressure determination and leakage testing for sealed surface micromachined membranes: a novel on-wafer test method,” in *Proceedings of The Eleventh International Workshop on MICRO Electro Mechanical Systems* (Mems 98. IEEE 1998), pp. 550–555. [CrossRef]

**15. **J. Yin, T. Liu, J. Jiang, K. Liu, S. Wang, S. Zou, Z. Qin, and Z. Ding, “Self-referenced residual pressure measurement method for fiber-optic pressure sensor chip,” IEEE Photonics Technol. Lett. **26**(10), 957–960 (2014). [CrossRef]

**16. **S. Timoshenko and S. Woinowsky-Krieger, *Theory of Plates and Shells* (McGraw-Hill, 1959).

**17. **H. Henmi, S. Shoji, Y. Shoji, K. Yoshimi, and M. Esashi, “Vacuum packaging for microsensors by glass-silicon anodic bonding,” Sensor. Actuat. A-Phys. **43**(1), 243–248 (1994).

**18. **H. R. Tofteberg, K. Schjølberghenriksen, E. J. Fasting, A. S. Moen, M. M. V. Taklo, E. U. Poppe, and C. J. Simensen, “Wafer-level Au–Au bonding in the 350–450 °C temperature range,” J. Micromech. Microeng. **24**(24), 084002 (2014). [CrossRef]

**19. **J. Jiang, S. Wang, T. Liu, K. Liu, J. Yin, X. Meng, Y. Zhang, S. Wang, Z. Qin, F. Wu, and D. Li, “A polarized low-coherence interferometry demodulation algorithm by recovering the absolute phase of a selected monochromatic frequency,” Opt. Express **20**(16), 18117–18126 (2012). [CrossRef] [PubMed]

**20. **J. A. Dziuban, *Bonding in Microsystem Technology* (Springer Science & Business Media, 2007).