# The Design, Modeling and Experimental Investigation of a Micro-G Microoptoelectromechanical Accelerometer with an Optical Tunneling Measuring Transducer

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

^{2}at a calculated eigenvalue of the MSE. The studies cover the selection of the dimensions, mass, eigenfrequency and corresponding stiffness of the spring suspension, gravity-induced cross-displacements. The authors propose and experimentally test an optical transducer positioning system represented by a capacitive actuator. This approach allows avoiding the restrictions in the fabrication of the transducer conditioned by the extremely high aspect ratio of deep silicon etching (more than 100). The designed MOEMA is tested on three manufactured prototypes. The experiments show that the sensitivity threshold of the accelerometers is 2 µg. For the dynamic range from minus 0.01 g to plus 0.01 g, the average nonlinearity of the accelerometers’ characteristics ranges from 0.7% to 1.62%. For the maximum dynamic range from minus 0.015 g to plus 0.05 g, the nonlinearity ranges from 2.34% to 2.9%, having the maximum deviation at the edges of the regions. The power gain of the three prototypes of accelerometers varies from 12.321 mW/g to 26.472 mW/g. The results provide broad prospects for the application of the proposed solutions in integrated inertial devices.

## 1. Introduction

## 2. Designing Micromechanical Sensing Element of Accelerometer

#### 2.1. Functional Scheme

_{y}, the proof mass displaced along the Y axis; the gap between the moving and fixed waveguides changed (Figure 1), which altered the output optical power P

_{opt}that was proportional to the measured acceleration a

_{y}[34].

#### 2.2. Mechanical Characteristics of the MSE

#### 2.2.1. Dimensions, Mass, and Eigenfrequencies of the MSE

_{b}is the Boltzmann constant equal to 1.38 × 10

^{−23}J/K; T is absolute temperature, K; Q is the mechanical Q-factor of the accelerometer; ω is the eigenfrequency of the MSE on the spring suspension, s

^{−1}; M is the mass of the MSE, kg; and K is the stiffness of the MSE’s spring suspension, N/m. Brownian noise contributes the most to the Noise Equivalent Acceleration and can be reduced by increasing the mass and the Q-factor of the accelerometer’s MSE and by decreasing the eigenfrequency. To decrease the transient time, during acceleration measurement, the Q-factor should be minimal. In our case, the Q-factor was equal to 20.

_{i}is proportional to the measured acceleration:

_{i}(Hz) is the eigenfrequency of the MSE along the i-th axis (Y or Z).

_{y}of the MSE ranging from 10 to 1000 Hz.

_{y}= (10, 20, 50, 100, 200) Hz.

^{−6}kg, the area occupied by the MSE with additional mass was 2.53 times less than that of the MSE without it. For a mass of M = 1 × 10

^{−6}kg, the area of the Type 2 MSE was 10.66 times less than that of the Type 1 MSE.

_{BR}on the mass and stiffness, which correspond to the set frequency. A tenfold increase in the mass (regardless the eigenfrequency) decreased the noise 3.16 times while increasing the occupied area 6.53 times. This affects the number of sensors fabricated from a single SOI wafer.

^{−6}m/s

^{2}. Following all the above, the MSE’s mass at eigenfrequencies from 10 to 200 Hz should exceed 2.3 × 10

^{−6}kg to measure a minimal acceleration of 1 μg.

#### 2.2.2. Spring Suspensions of the MSE

_{y}) and along the transverse Z axis (f

_{z}).

_{y}of the proof mass corresponded to 10, 20, 50, 100 and 200 Hz. The dimensions b and b

_{1}were fixed and amounted to 4 and 20 μm, respectively; the height of the elastic elements was 35 μm. For each mass and frequency f

_{y}, the corresponding frequencies along the transverse axis fz and displacement Δz were calculated. The results are presented in Table 2 and Table 3.

_{y}should exceed 100 Hz. At the same time, the reasonable length of the elastic elements should be proportional to the dimensions of the proof mass, so the eigenfrequency f

_{y}of 200 Hz was selected for the further design and experimental studies.

_{1}and G of the elastic elements (Figure 7) on the displacement along the transverse Z axis for a maximum mass of 1.00 × 10

^{−5}kg and eigenfrequency f

_{y}of 200 Hz. At different values of b, we selected an eigenfrequency f

_{y}of 200 Hz through the alteration of the length of the elastic element G. Then, we determined d

_{z}by varying b

_{1}and matching G with the eigenfrequency f

_{y}. The results are presented in Figure 8.

#### 2.2.3. Optical Transducer Positioning System

_{y}are the mass, Q-factor, stiffness of the spring suspension and the eigenfrequency of the micromechanical structure (${\omega}_{y}=2\pi {f}_{y}=\sqrt{\frac{K}{M}}$); ${F}_{e}=\frac{1}{2}\frac{\partial C}{\partial x}{U}^{2}$ is the electrostatic force generated by the comb electrode structures, which are used to set the working gap; a

_{y}is the measured acceleration; C is the capacity of the electrodes; and U is the electrode voltage.

_{y}of 200 Hz and determination of the minimal mass from the Brownian noise (2.3 × 10

^{−6}kg), the accelerometer’s MSE was designed (Figure 9).

_{l}and C1

_{r}are the comb electrode structures used to set the initial working gap in the positive direction along the sensitivity Y axis; C2

_{l}and C2

_{r}, C2

_{lS}and C2

_{rS}, C3

_{l}and C3

_{r}, C3

_{lS}and C3

_{rS}are the comb electrode structures used to simulate the acceleration in the positive and negative directions along the sensitivity Y axis; and the comb electrode structures C1

_{l}and C1

_{r}, C1

_{lS}and C1

_{rS}are used to compensate for the MSE’s gravity-induced displacement along transverse axis Z.

## 3. Optical Measuring Transducers

_{through}is the output optical power at the through port, P

_{input}is the input optical power at the input port, λ is the wavelength, L is the coupling length, and ∆n

_{eff}is the difference between the effective indices of the even and odd modes.

_{eff}in Equation (5) or to obtain data on the transmission coefficient in a selected waveguide (waveguide of the through port in our case) for a necessary coupling length immediately in the solver.

_{through}≈ 0.5 and should be set by the OTPS.

_{−80}) nm to (360

^{+140}) nm, where the OTC varies from 0.002 to 0.906. The OTC changes asymmetrically, which should be taken into account during the output signal processing. To linearize the working section of the OTC, the dynamic range should be reduced to a range from (339

_{−39}) nm to (339

^{+39}) nm, where the OTC varies in a range of 0.32 ± 0.27.

^{6}m

^{−1}.

## 4. Fabrication of the Accelerometer

_{2}/Si

_{3}N

_{4}/SiO

_{2}films were formed using the methods of plasma chemical deposition and plasma chemical etching (Figure 15a). The results are presented in Figure 16.

_{2}film through a metallic 100-nm Al film was performed.

_{2}mask with a thickness of 100 nm. At the next stage, the device wafer with the formed functional elements of the accelerometer’s MSE was bonded to the base wafer (Figure 15f). The bonding was performed using a Benzocyclobutene (BCB) polymer adhesive. At the last stage, the photoresist HT10.11 was lifted off in toluene and the carrier wafer was removed (Figure 15g). The presented technology allowed for fabricating the prototype of the accelerometer’s MSE (Figure 17).

## 5. Experimental

#### 5.1. Setup

#### 5.2. Testing of the Accelerometer

_{1}—equal to 0.8 of the calculated value (13.5 V)—was fed from an E36313A voltage source to the electrodes (Figure 19). This generated the electrostatic force that displaced the MSE along the Z axis. To find the actual voltage U

_{1}, we increased its value step by step and at each step altered the voltage U

_{2}to reach the minimal power at the optical power meter, which would mean that the waveguides lie in a single plane and the absolute maximum of optical power is transferred to the drop port and disperses. For further tests, voltage U

_{1}was fixed.

_{2}at the electrode structures, the MSE was displacing along the Y axis, which was altering the gap between the waveguides and, hence, the output optical power. From the E36313A power source, the sum of voltages U

_{1}and U

_{2}was fed to electrode structures C1

_{ls}and C1

_{rs}. With changing voltage U

_{2}, the Pout was recorded, and the dependence of output power on voltage U

_{2}was plotted. On the obtained plot, the most linear section was selected, and voltage U

_{2}was set so that it corresponded to the middle of this section. This voltage corresponded to the working gap at which further acceleration testing was performed.

_{1acc1}= 14.769 V; U

_{1acc2}= 14.391 V; U

_{1acc3}= 13.696 V; U

_{2acc1}= 8.831 V; U

_{2acc2}= 8.462 V; U

_{2acc3}= 7.891 V.

_{l}and C2

_{r}(Figure 9) to create electrostatic force F

_{e}equal to the inertia force at a corresponding acceleration. This electrostatic force induced the same displacement of the MSE as the measured acceleration. Then, the simulated acceleration could be determined as

_{i}is the capacity of the electrode structures, F; and U

_{3}is the voltage at electrodes C2

_{l}and C2

_{r}, V. The values of the simulated acceleration and corresponding calculated voltages are presented in Figure 20.

_{out}

_{0}is the output power at g = 0, μW; and k is the linear approximation coefficient, or power gain.

_{out}is the output power, μW.

_{2}to simulate the acceleration induced by those factors. The averaged data from the optical power meter over a period from 10 ms to 1 s also prohibited reaching a definite conclusion on the sensitivity threshold below 2 μg.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**MSE displacements along the Y axis; (

**a**) Frequency range from 10 to 500 Hz; (

**b**) Frequency range from 500 to 1000 Hz.

**Figure 3.**Dependence of the spring suspension stiffness on the MSE’s mass at different frequencies; (

**a**) Mass change from 1 × 10

^{−9}to 1 × 10

^{−5}kg; (

**b**) Mass change from 1 × 10

^{−9}to 1 × 10

^{−7}kg.

**Figure 5.**Dependence of an MSE’s dimensions on its mass; (

**a**) Mass change from 1 × 10

^{−8}to 1 × 10

^{−5}kg; (

**b**) Mass change from 1 × 10

^{−8}to 1 × 10

^{−6}kg.

**Figure 6.**Dependencies of the a

_{BR}on the MSE’s mass; (

**a**) Mass change from 1 × 10

^{−8}to 1 × 10

^{−5}kg; (

**b**) Mass change from 1 × 10

^{−8}to 1 × 10

^{−7}kg; (

**c**) Mass change from 1 × 10

^{−7}to 1 × 10

^{−6}kg; (

**d**) Mass change from 1 × 10

^{−6}to 1 × 10

^{−5}kg.

**Figure 8.**Dependence of the displacement Δz and length G of the elastic elements on the dimensions b and b

_{1}.

**Figure 11.**Dependence of the optical transmission coefficient of the directional coupler’s through port on the coupling length and gap at a wavelength of 1550 nm.

**Figure 13.**Dependence of the OMT’s S21 on the gap at a wavelength of 1550 nm obtained via the FDTD method.

**Figure 14.**S-parameters of the OMT for a gap of 360 nm; the waveguide cross-section is 350 nm × 850 nm and the coupling length is 100 µm; (

**a**) S21, S31, S42, S43, S12, S13, S24, S34; (

**b**) S11, S22, S33, S44, S41, S14, S32, S23.

**Figure 15.**MSE fabrication process flow: (

**a**)—formation of the optical waveguides and metallic plates; (

**b**)—separation of functional elements; (

**c**)—bonding to the carrier wafer; (

**d**)—thinning of the device wafer; (

**e**)—etching of the reverse side of the device wafer; (

**f**)—bonding to the base wafer; and (

**g**)—debonding of the carrier wafer.

**Figure 18.**Experimental setup: 1—accelerometer prototype; 2—triaxial positioner; 3—six-axis automatic positioner; 4—six-axis manual positioner; 5—fiber holder; 6—laser; 7—joystick; 8—personal computer; 9—power meter; 10—sources of electric voltage; 11—sources of electric voltage; 12—polarizer.

**Figure 21.**Dependence of the accelerometer’s output optical power on the acceleration; (

**a**) Range from minus 0.03 g to plus 0.05 g; (

**b**) Range from minus 0.01 g to plus 0.01 g.

**Figure 22.**Dependence of the accelerometer’s output optical power on the acceleration; (

**a**) Range from minus 0.001 g to plus 0.001 g; (

**b**) Range from minus 0.0001 g to plus 0.0001 g.

**Figure 23.**Dependence of the accelerometer’s output optical power on the acceleration in a range from minus 0.00001 g to plus 0.00001 g.

Device Type | Sensitivity | Eigenfrequency | Intrinsic Noise | Measuring Range | Bandwidth |
---|---|---|---|---|---|

Photonic-crystal nanocavity [14] | 10 µg Hz^{−1/2} | 20 kHz | - | - | 50 dB |

Sub-wavelength gratings [15] | 2033 nm/g | 379 Hz | - | 0.12 g | - |

Interferometric position sensor [16] | - | From 80 Hz to 1 kHz | 40 ng/rt Hz | - | 85 dB at 40 Hz |

Fiber Bragg grating [17] | ~100 pm/g cross-axis anti-interference degree < 5% | (10–120) Hz | - | - | - |

Fiber grating [18] | - | - | - | - | Up to 200 Hz |

Electron tunneling transducers [21] | - | - | 20 ng/Hz | - | 5 Hz–1.5 kHz |

Ring resonators [22] | 18.9/g | - | 4.874 μg | From −23.5 g to 29.4 g | - |

Micro-grating-based [24] | 169 μm/g 60 V/g | - | 15 ng/√Hz for 1 Hz | - | - |

Fabry–Pérot interferometer (FPI) [25] | (1.022–1.029) mV/(m/s ^{2}) | 1274 Hz | - | 7 g | - |

Fiber-free optical [26] | 0.156 mA/g, resolution of 56.2 µG | - | - | - | |

Fabry–Perot (FP) cavities [27] | X-axis—309 μg Y-axis—313 μg | X-axis—1382.5 Hz Y-axis—1398.6 Hz | - | 1 g | - |

Michelson interferometer structure [28] | 3.638 nm/g | 1742.2 Hz | - | ±500 g | - |

Talbot effect [29] | 0.14 μm/g 0.74 V/g | 1878.9 Hz | 2.0 mg | - | - |

Intensity modulation of light [30] | 600 nm/g | 560 Hz | - | 3 g | - |

Photonic crystal [31] | 0.0750 nm/g | 17.7 kHz | - | ±200 g | - |

Optical tunneling effect [32] | (6.25 × 10^{6} m^{−1}) | (10–200) Hz | - | - | - |

Optical tunneling effect [33] | 9 pm/g | - | - | ±130 g | 0–1500 Hz |

M, kg | 1.00 × 10^{−8} | 1.00 × 10^{−7} | 1.00 × 10^{−6} | 1.00 × 10^{−5} | |
---|---|---|---|---|---|

f_{y}, Hz | f_{z}, Hz | G, μm | |||

10 | 87.5 | 23,510 | 10,910 | 5060 | 2370 |

20 | 175 | 14,770 | 6860 | 3200 | 1505 |

50 | 437.5 | 8040 | 3760 | 1760 | 836 |

100 | 875 | 5094 | 2374 | 1111 | 539 |

200 | 1750.04 | 3210 | 1509 | 720.3 | 353.7 |

M, kg | 1.00 × 10^{−8} | 1.00 × 10^{−7} | 1.00 × 10^{−6} | 1.00 × 10^{−5} | |
---|---|---|---|---|---|

f_{y}, Hz | f_{z}, Hz | Δz, μm | |||

10 | 87.5 | 32.8831 | 32.7483 | 32.5152 | 33.2363 |

20 | 175 | 8.16708 | 8.12145 | 8.20043 | 8.52342 |

50 | 437.5 | 1.3078 | 1.32746 | 1.35832 | 1.5075 |

100 | 875 | 0.3331137 | 0.333466 | 0.355366 | 0.440739 |

200 | 1750.04 | 0.0826125 | 0.085916 | 0.0983281 | 0.149845 |

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**MDPI and ACS Style**

Barbin, E.; Nesterenko, T.; Koleda, A.; Shesterikov, E.; Kulinich, I.; Kokolov, A.; Perin, A.
The Design, Modeling and Experimental Investigation of a Micro-G Microoptoelectromechanical Accelerometer with an Optical Tunneling Measuring Transducer. *Sensors* **2024**, *24*, 765.
https://doi.org/10.3390/s24030765

**AMA Style**

Barbin E, Nesterenko T, Koleda A, Shesterikov E, Kulinich I, Kokolov A, Perin A.
The Design, Modeling and Experimental Investigation of a Micro-G Microoptoelectromechanical Accelerometer with an Optical Tunneling Measuring Transducer. *Sensors*. 2024; 24(3):765.
https://doi.org/10.3390/s24030765

**Chicago/Turabian Style**

Barbin, Evgenii, Tamara Nesterenko, Aleksej Koleda, Evgeniy Shesterikov, Ivan Kulinich, Andrey Kokolov, and Anton Perin.
2024. "The Design, Modeling and Experimental Investigation of a Micro-G Microoptoelectromechanical Accelerometer with an Optical Tunneling Measuring Transducer" *Sensors* 24, no. 3: 765.
https://doi.org/10.3390/s24030765