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Abstract
This paper proposes what we believe to be a novel linearization signal conditioning circuit for a tri-axial micro-grating micro-opto-electro-mechanical systems (MOEMS) accelerometer. The output of a micro-grating accelerometer varies as a sine/cosine function of the acceleration. The proposed circuit utilizes a subdivision interpolation technique to process these nonlinear intensity variations and render a linear digital output across the full range. Such a linearization circuit was achieved through a 90-degree phase-shift circuit, high-precision DC bias-voltage and subdivision interpolation circuits to reduce the influence of phase, magnitude, and offset errors of the sine-cosine signals on the interpolation factor, improving the resolution and accuracy of acceleration detection. Experimental results demonstrated that the micro-grating MOEMS accelerometer achieves a resolution of sub-mg, cross-axis errors of 3.57%, 1.22% and 0.89% for x-, y- and z-aixs, respectively. The bias instabilities and velocity random walks for the vertical and lateral accelerometer are superior to 26 µg and 38.7 µg/√Hz. The tri-axial micro-grating MOEMS accelerometer exhibits significant potential for applications requiring high sensitivity and large operation ranges, including the automotive industry and military equipment.
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1. Introduction
High-precision tri-axial MEMS accelerometer is crucial for inertial navigation [1], automotive industries [2], military equipment [3] and mechanical vibration measurement [4–8], due to the merits of small volume, low power consumption and compact integration. To fully meet the demands of those applications, the pursuit for tri-axial accelerometer with even higher sensitivity and lower noise floor has never stopped. The conventional method for implementing a tri-axial MEMS accelerometer involves assembling three uni-axial accelerometers orthogonally in a discrete package, with the disadvantages of difficult assembly, orthogonal misalignment, and large volumes. Although monolithic integrated tri-axial accelerometer with a single proof-mass results in extremely small volumes, but provides limited performance due to cross-axis interference and the complexity of sensing three different axes. To address some of these limitations, individual lateral and vertical axis accelerometer can be fabricated on a single substrate, thereby suppressing alignment errors inherently during the fabrication process.
Great efforts have been made to develop high-performance tri-axial MEMS accelerometer, with various design concepts and sensing techniques being explored. Capacitive [9–13], piezoresistive [14–18], piezoelectric [19–21], convective [22,23], and other sensing mechanisms [24–26] have all been considered. Among these, capacitive sensing has gained significant attention due to its high sensitivity, low power consumption, and compatibility with integrated circuits. Nevertheless, it is still inevitably affected by parasitic capacitance and fringe effects, especially when three-axis integration is involved. Piezoelectric and piezoresistive sensing suffer from low sensitivity, material fatigue, and large nonlinear error, which cannot meet the requirements of high-resolution acceleration detection. Thermal convective sensing is based on heat transfer in a fluid-filled cavity, but it has issues of slow response and narrow dynamic range. Optical sensing can overcome some of these challenges and limitations, with better immunity to electromagnetic interference and higher sensitivity compared to other types [27–32]. However, the complex fabrication process and the necessity for additional linearization circuitry have hindered the development of tri-axial MOEMS accelerometer, resulting in most reported optical accelerometer being single-axis at present, and not providing a linear output for the full range [27–29,33,34].
In this paper, we presented the design, fabrication, and demonstration of a monolithic integrated tri-axial MOEMS accelerometer based on Talbot effect of micro-gratings. We proposed a novel signal conditioning circuit to linearize the sine output of the micro-grating MOEMS accelerometer. Our approach involved the use of the 90-degree phase-shift technique, in conjunction with high-precision DC bias-voltage and subdivision interpolation circuits. These were implemented to convert the sine signal output into standard incremental digital signals. The proposed linearization circuit not only reduces the impact of phase, amplitude and offset errors on the interpolation factor, thereby improving the resolution and accuracy of the acceleration detection, but also renders a linear digital output across the entire range. Experimental results demonstrated that the proposed accelerometer exhibits excellent sensitivity and linearity when subjected to tri-axial directional accelerations.
2. Design, simulation and fabrication of MOEMS accelerometer
Figure 1 depicts the schematic diagram of a tri-axial MOEMS accelerometer utilizing the Talbot effect of micro-gratings, illustrating the implementation of lateral and vertical accelerometer on the same substrate through a glass-silicon bonding structure. It comprises an upper grating layer and a sensitive structure layer. The upper fixed grating is fabricated by depositing Al film onto the glass substrate using magnetron sputtering. The sensitive mechanical structures of each accelerometer include a proof mass (the bottom grating is fabricated on the proof mass), symmetric cantilever beams, and a support substrate. To enable the device to measure acceleration along all three axes simultaneously, the laser beam is divided into four parts, with three being utilized for tri-axial acceleration measurement and one for monitoring the power stability of the laser. The light intensities of the four channels are not necessarily the same, which only affects the sensitivity of the accelerometer along three axes. When subjected to an external acceleration, the proof mass along with the bottom grating undergoes either in-plane or out-of-plane movement, resulting in proportional displacements relative to the applied acceleration. The Talbot effect of the micro-grating refers to the phenomenon of self-imaging of grating pattern at periodic intervals (∼$\textrm{n}{\textrm{d}^2}/\mathrm{\lambda }$). The diffracted light intensity varies sinusoidally with the relative lateral or vertical motion between the double-layer gratings, exhibiting advantages of high sensitivity, compact size, and simple optical path. Consequently, the acceleration can be determined by the variation of diffracted light intensity, which are converted into voltage by photodetectors and processing circuits.
Fig. 1. Schematic diagram of a tri-axial MOEMS accelerometer utilizing the Talbot effect of micro-gratings.
In order to evaluate the mechanical performance of the tri-axial accelerometer, a stationary analysis using finite element analysis (FEA) was conducted. The resonance frequencies of both the lateral and vertical axis accelerometer were simulated using Comsol analysis. The geometrical parameters of the tri-axial accelerometer are listed in Table 1. Both the in-plane and out-of-plane accelerometers exhibit high bandwidth and low mechanical noise of sub-µg/√Hz. During the design of the tri-axial accelerometer, it can be challenging to achieve identical performance parameters for both the lateral and vertical accelerometers. Based on this simulation, the estimated lateral and vertical mechanical sensitivities S1 are approximately 0.2 µm/g and 1 µm/g, respectively.
Table 1. Mechanical and geometrical parameters of the micro-grating MOEMS accelerometer.
After configuring the meshing, boundary conditions and the light source in the optical FDTD software, we optimized the grating parameters to maximize the optical diffraction sensitivity of the micro-grating, resulting in a grating pitch d of 4 µm, a duty ratio Δ of 0.5, and a thickness t of 500 nm. The Talbot carpet patterns of the double-layer diffraction gratings is illustrated in Fig. 2(a). The diffraction laser passing through the micro-gratings is focused on the detector. This detection offers advantages of high precision and fast response compared with 2D-plane image detectors and can also suppress the influence of partial defects in the micro-gratings. When the distance between the the double-layer gratings equals to half of the Talbot distance or its multiples, self-imaging of the grating is formed, resulting in maximum optical diffraction sensitivity. Figure 2(b) illustrates the diffraction efficiency as a function of transverse displacement at half of the Talbot distance (i.e. ${\textrm{d}^2}/\mathrm{\lambda }$∼10.3µm). The relationship between diffraction efficiency and longitudinal displacement between the micro-gratings is depicted in Fig. 2(c). Through linear fitting, the lateral and vertical optical diffraction sensitivities S2 were determined to be 36.1%/µm and 6.5%/µm, respectively. Therefore, the total sensitivity of the accelerometer S can be expressed as:
which is the product of acceleration-displacement sensitivity S1, optical diffraction sensitivity S2, and photoelectric conversion sensitivity S3. Here, S3= Pin·η·G·R, the laser intensity Pin is 10 mW, G = 5 represents the current gain of photoelectric detector, the photoelectric conversion efficiency of photodiode η is 0.89A/W, and transresistance resistor R is 1.8 kΩ. Thus, the system can achieve an acceleration-voltage amplification output sensitivity of 5.78 V/g (lateral) and 5.2 V/g (vertical), respectively.Fig. 2. (a) The Talbot carpet patterns of the double-layer diffraction gratings. (b) The simulation diffraction efficiency versus transverse displacement at the half of Talbot distance of 10.3 µm. (c) The simulation diffraction efficiency versus longitudinal displacement. Linear fitting of the curves from the optimal design yields the lateral and vertical optical diffraction sensitivities of 36.1%/µm and 6.5%/µm, respectively.
Standard MEMS bulk silicon technology is used in the fabrication flow, including the silicon-based process, quartz-based process and quartz-silicon bonding process, as shown in Fig. 3. The silicon-based processes are presented in Fig. 3(a)-(i). The proof mass and cantilever beams are fabricated by bulk silicon technology on a 4-inch silicon wafer with a crystal orientation of <100 > and a thickness of 400 µm. Firstly, shallow cavities and slots are wet-etched to a depth of 8 µm to ensure the distance between the double-layer gratings and prevent the proof mass from colliding with the glass. Electrode grooves with a depth of 0.5 µm are etched outside the square cavities through deep reactive ion etching (DRIE), as shown in Fig. 3(a) and (b). Secondly, a highly reflective film composed of aluminum is patterned on the proof mass areas using magnetron sputtering. The bottom grating and electrode wires are formed by reactive ion etching (RIE) after covering the photoresist and photolithography process, as shown in Fig. 3(c)-(f). Then, etching of the front structures of the accelerometer is carried out at an etching rate of 6 µm/min, with an etching depth of 30 µm and 100 µm for the vertical and lateral accelerometers, respectively. For the opposite side of wafer, the structure of the vertical accelerometer is etched due to the difference thickness between the cantilever beams and the proof mass. The final DRIE process is performed on the back of the silicon wafer to release the sensitive structures, as shown in Fig. 3(g)-(i). The quartz based process is presented in Fig. 3(j). A 200-nm-thick Al grating is deposited on the glass by magnetron sputtering. After patterning, the upper grating structures are produced by dry etching. Finally, the top grating layer is bonded with silicon wafer using anodic bonding technology [Fig. 3(k)]. The optical microscope image and the top view of the MOEMS accelerometer are presented in Fig. 4, which demonstrates the details of the suspended silicon structure, symmetry of cantilevers, and consistency of grating processing.
Fig. 3. Process flow for the fabrication of the integrated tri-axial MOEMS accelerometer (including lateral and vertical accelerometer).
Fig. 4. Photograph of (a) the fabricated upper diffraction grating layer and (b) the structure layer of tri-axial MOEMS accelerometer. Scanning electron microscope (SEM) images of the fabricated structure of (c) vertical MOEMS accelerometer, (d) and (e) lateral MOEMS accelerometer, and (f) high-precision grating.
The Talbot effect of micro-gratings has rigorous requirements for fabrication and assembly errors, which may reduce the optical sensitivity and even lead to the failure of Talbot effect. It poses a significant challenge for the accelerometer to feature an excellent acceleration measurement precision and ultra-high sensitivity. Here we utilized optical alignment and anode bonding techniques to reduce the non-parallelism errors, achieving a grid-lines misalignment accuracy of less than 0.5°.
3. Linearization signal conditioning circuit
To achieve high sensitivity detection and linearization over the full range, a linearization signal conditioning circuit has been proposed, as shown in Fig. 5(a). The architecture for the digital system consists of 90-degree phase-shift circuit, high-precision DC bias-voltage, and subdivision interpolation circuit, converting these non-linear diffraction intensity variations into a linear digital output. Figure 5(b) represents the schematic diagram of phase-shift circuit. The adder G1(s) compares the input sinusoidal signal with a 90-degree phase-shifted signal, and compensates for the gain of the input signal. The amplitude of the phase-shifted signal is controlled by the ratio of R2 to R1. An integrator G2(s) is utilized to achieve a 90-degree phase shift, while the offset voltage generated by the integrator can be eliminated by the feedback integrator G3(s). To quantitatively analyze its phase-frequency characteristics, the transfer function of the phase-shift module is derived as [35]:
Let s = jω, the phase angle of the transfer function Arg[H(jω)] is a constant 90°, suppressing the influence of phase error on the subdivision interpolation circuit. Additionally, the phase-shift circuit exhibits high-pass filtering characteristics that effectively eliminate the influence of 1/f noise at low frequency.
Fig. 5. (a) The principle diagram of the linearization signal conditioning circuit for micro-grating MOEMS accelerometer. (b) The circuit schematic of the 90° phase-shift module. (c) The circuit schematic of subdivision interpolation circuit.
Another factor influencing the interpolation circuit is the stability of the input offset voltage. A high-stability DC bias-voltage circuit has been developed for the interpolation circuit. This circuit employs a precision shunt reference with excellent temperature stability over a wide range of voltage, temperature and operating current conditions. The stability of the DC voltage is evaluated by continuously recording the data for a long time, achieving a level of 10−5@1s from the Allan variance analysis.
The schematic diagram of subdivision interpolation circuit is shown in Fig. 5(c). After rectification, two signals with a 90-degree phase shift are processed using inverse tangent to achieve an 8× subdivision process. Digital subdivision process is completed through the interpolation algorithms, resulting in subdivided square wave signals. Specific interpolation factors and operation modes can be configured through external resistors. Finally, the resolution of the acceleration detection is obtained by counting the numbers of square wave signals over the measurement range.
4. Experimental results and discussion
To characterize the performance of the fabricated micro-grating MOEMS accelerometer, an experimental setup was established for static acceleration measurement, as shown in Fig. 6. The frequency-stabilized laser (1.5 µm wavelength, Model MDL-III-1550) is passed through a polarizer to adjust its polarization. As light intensity fluctuation is one of the primary noises for a micro-grating MOEMS accelerometer, we use an acoustic-optic modulator (AOM) to frequency-shift the laser light. The first-order diffraction efficiency of the AOM is adjusted by controlling its RF driving power based on the monitor signal from PD2 for light power stabilization. We also used phase modulation technique to reduce the 1/f noise of circuit detection system and eliminate the DC component of irrelevant stray light. The variation of light intensity was converted into voltage by photodetectors (Model PDA05CF2, Thorlabs Inc), and the voltage signal was collected through a data acquisition board after demodulation by the lock-in amplifier. All aforementioned optical components were fixed on a high-precision three-axis rotary table.
Figure 7(a) depicts the sine-cosine output after passing through the phase-shift module. The curves illustrate that the acceleration change exhibits a sine and cosine relationship with the output signal of the phase-shift circuit, and the phase of the two signals maintains a precise 90-degree separation. After amplification, bias-voltage offset and subdivision interpolation circuits, the initial sine signal transforms into a series of square pulses, as illustrated in Fig. 7(b). In the subdivision interpolation circuit, differential inputs are employed to eliminate common mode distortion. Additionally, a lookup table (LUT) is utilized in this specific DSP interpolator to correct residual distortions, thus ensuring minimal errors and jitter. When the input acceleration increases from -1 g to +1 g along the x-, y-, and z-axes, the digital outputs from the linearization signal conditioning circuit are recorded with four measurement repetitions as 4924, 5302 and 4054. The outputs obtained from the signal processing circuit are found to be linear across the measurement range. Upon calculation, the resolution of the MOEMS accelerometer for the x, y and z axes can be determined as 406.2 µg, 377.2 µg and 493.2 µg, respectively. Consequently, the acceleration can be measured by combining this resolution with the counting of the square wave signals.
Fig. 7. (a) Sine-cosine signals after the phase-shift module. (b) The standard incremental AB quadrature signals from the subdivision interpolation circuit output.
Figure 8 illustrates the long-term bias stability of the tri-axial MOEMS accelerometer. We utilized a Tektronix DMM6500 6½ digit multi-meter to continuously monitor the output voltage at a sampling rate of 50 Hz. The bias instabilities, obtained from the Allan deviation results, were measured as 26 µg, 20 µg and 6 µg for the lateral (x- and y-axis) and vertical (z-axis) accelerometer with an averaging time of 1 second, respectively. The primary factor influencing accelerometer stability is random walk noise, as indicated by the slope of the Allan deviation. However, the bias stability at longer times beyond 10 seconds exhibited rapid degradation primarily due to thermal-induced effect. From the Allan deviation graphs, the velocity random walks (VRW) for the lateral and vertical accelerometers were also determined to be 38.7µg/√Hz, 23.7µg/√Hz, and 3.4 µg/√Hz for x-, y-, and z-axes, respectively.
The cross-axis interference of the micro-grating MOEMS accelerometer is determined by applying acceleration around one axis while simultaneously recording acceleration data for all three axes. The cross-axis interference is determined from the collected digital data, as shown in Table 2. It can be observed that the cross-axis errors are less than 3.57% within the measurement range. One of the significant contributions to the cross-axis interference of micro-grating accelerometer is the deviation of the geometric center of the cantilever beam from the proof mass center. Another factor is that the incident light spot may not necessarily align with the geometric center of the mass block, consequently leading to large cross-axis interference as the distance from the center increases.
Table 2. The cross-axis interference of the tri-axial MOEMS accelerometer.
To demonstrate the performance of the fabricated three-axis MOEMS accelerometer, we compared it with several previous works in terms of resolution, bias stability, cross-axis interference and other parameters. From Table 3, it can be observed that the capacitive accelerometer is the most widely reported tri-axial accelerometer. Most of the previously reported optical MEMS accelerometers were based on a single-axis design [36,37] or lack of linearization circuitry. Our proposed linearization signal processing method can achieve full-range linear measurement of the three-axis MOEMS accelerometer, offering superior resolution and accuracy performances.
Table 3. Performance characteristics of the accelerometer compared to previous works.
5. Conclusion
We proposed a novel linearization signal processing method for tri-axial micro-grating MOEMS accelerometer, consisting of 90-degree phase-shift, high-precision DC bias-voltage and subdivision interpolation circuits. The output sinusoidal signal of the grating MOEMS accelerometer can be transformed into standard incremental AB quadrature digital signals. It can be effectively suppressed phase, amplitude, and offset errors between the sine-cosine signals, thereby enhancing the accuracy and linear operation range of the accelerometer. The manufactured accelerometer demonstrates a bias instability superior to 26 µg, sub-mg acceleration resolution and cross-axis interference of less than 3.57% for all three axes. Although the linearization signal conditioning circuit was employed to show the practicality of tri-axial MOEMS accelerometer based on Talbot effect of micro-gratings, the proposed method can be easily adopted for acceleration sensing using grating interference type sensing elements as well.
Funding
National Natural Science Foundation of China (62005253).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper may be obtained from the authors upon reasonable request.
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