# Optical Sensor, Based on an Accelerometer, for Low-Frequency Mechanical Vibrations

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Design of the Optical Sensor and Test Bench

#### 2.1. Vibration Sensor Design

#### 2.1.1. Analytical Accelerometer Modeling

_{eq}, and dissipation effects are included using a damper with damping coefficient $\beta $.

_{eq}, being commonly referenced in the literature as k

_{1/4}.

_{e}and a

_{m}are the force and the acceleration, respectively, to which the system is subjected.

_{n}is the oscillation frequency of the first resonance mode, ξ is the damping ratio, and m = ρV, where ρ is the material density and V is the corresponding mass volume.

_{n}), as can be observed in all hypothetical cases considered (f

_{n}=100, 1000 and 10,000 Hz), only as a reference for the selection of the design bandwidth. From this response, the required sensitivity of the optical detector to detect vibration amplitudes can be determined, as well as the frequency bandwidth, as will be described later (Section 2.1.2). Additionally, note that the $Y/{a}_{m}$ ratio increases abruptly as the vibration frequency $\omega $ approaches the resonance frequency ${\omega}_{n}$. This is the case for the three frequency examples, since the relation $1/{\omega}_{n}^{2}-{\omega}^{2}$ is undefined for $\omega ={\omega}_{n}$.

#### 2.1.2. Optical Detection

_{0}, and a change in this position ΔX as a function of this sinusoidal vibration is as follows:

_{0}is the wavelength of the light, ω

_{s}is the assumed vibrational frequency of the accelerometer, when a displacement change is applied, and φ is the initial phase shift.

_{0}= 1 µm), the maximum vibration amplitude of the sensor will be X

_{max}= 0.5 µm. For a length in green (λ

_{0}= 500 nm), the amplitude would be X

_{max}= 250 nm.

#### 2.1.3. Silicon Microstructure Design

^{2}to allow reflection of a laser with a spot diameter of 1 mm or less. For a ratio area/spot diameter, larger values provide a better tolerance for the laser positioning. This fact provides the first design criterion for the size and shape of the seismic mass of the accelerometer.

#### 2.2. Finite Element Analysis

#### Modelling

- i.
- A steady-state study to verify the maximum displacement amplitude at a given acceleration and the static effects, mainly the sensitivity to gravitational attraction as a function of the orientation of the microstructure with respect to the Earth (Section 2.3.1).
- ii.
- A modal study to determine the natural resonance modes and their frequencies is performed in Section 2.3.2.
- iii.
- A harmonic study to find the frequency response of the microstructure subjected to vibrations at a known frequency (Section 2.3.3).

#### 2.3. Simulation Results

#### 2.3.1. Steady-State Study

#### 2.3.2. Modal Analysis

#### 2.3.3. Harmonic Analysis

- Its vibration amplitude is the largest, reaching a displacement of 1 μm at 10 g, which will allow the performance of the optical detection system to be evaluated.
- It has its second vibration mode of 2481 Hz further away from the frequency of the first one of 1007 Hz, in comparison with other designs, which allows more stability in the experimental test range.
- The seismic mass can be adjusted to various dimensions, and even consider circular shapes.

#### 2.4. Vibration Sensor Package Design

- (a)
- A support for the microstructure (in green)
- (b)
- The encapsulation made in Steel SAE 304 (in metallic color)

#### 2.5. Test Bench Design

#### 2.5.1. Interferometer Scheme

- Light source: Linearly polarized He-Ne laser, with wavelength of 633 nm, power of 2 mW, divergence of 1.3 mrad, and polarization ratio of 500:1.
- Beam splitter: 50:50 in non-polarizing cube in the range of 400 to 700 nm.
- Microestructure mirror: Monocrystalline silicon microstructure under test
- Mechanical support: Two-degree-of-freedom circular mirror assembly and 8.3 mrad/rev. resolution adjustment.
- Projection objective: With 10× magnification and 0.25 aperture
- Reference mirror: Dielectric fused silica mirror with reflective coating for a 400 to 700 nm range.
- Photodiode: With integrated preamplifier OPT101, detection area of 2.29 mm × 2.29 mm, sensitivity of 0.45 A/W (at 650 nm), and bandwidth of 14 kHz.

**Figure 11.**(

**a**) Microstructure Test Bench Design, (

**b**) Interferogram Projected from Test Bench on (7), the Selected Photodiode, which is aligned with (5), the Projection Objective.

#### 2.5.2. Photodiode Response

_{Ph}is rotated by an angle ${\theta}_{Ph}$, to model non-ideal alignment with the fringes. The rotation is performed using a rotation matrix in Euclidean space as follows:

_{f}due to a linear change in position of the microstructure.

## 3. Manufacturing

#### 3.1. Manufacture of Accelerometers

- i.
- A 100 mm diameter, 400 μm-thick monocrystalline silicon wafer with a polished top side (with a thickness roughness less than 10 nm) and <100> crystalline orientation is used.
- ii.
- Silicon nitride (300 nm) and silicon oxide (1.5 μm) layers are deposited through a sputtering process on both faces. Their roughness is not relevant because these layers will be used as sacrificial material.
- iii.
- A pattern is etched on the oxide and nitride layers of both faces by the next steps:
- a.
- A chromium deposit (300 nm) is made by electron-beam deposition
- b.
- By means of a photolithography process, a pattern is etched on the chromium with wet etching by chrome etchant.
- c.
- The oxide pattern is etched with hydrofluoric acid. The chromium acts as a protective layer for the regions not to be etched, until the silicon is exposed.

- iv.
- Deep reactive ion etching (DRIE) is performed on both sides of the wafer, one after the other. The back side is etched first; the duration of the etching sets the proof mass thickness of the microstructure. Subsequently, the pattern of the beam supports is etched on the top (polished) side until it is transferred to the cavity formed on the bottom side.
- v.
- The sacrificial layers, made of nitride, oxide, chromium, and photoresist residues, were removed. For the sake of simplicity, the photo resin process is not shown. The liberated microstructure was obtained.

#### 3.2. Vibration Measurement Probe Housing

#### 3.3. Implementation of the Test Bench

## 4. Experimental Tests

#### 4.1. Optical System Evaluation

_{piezo}= 0.2 μm/V (Figure 24). This measurement was obtained by measuring the focusing distance change in the silicon surface over the piezoelectric for different voltage stimulus with a Keyence VHX6000 digital microscope.

- The polished monocrystalline silicon substrate has optical properties in the visible wavelength range, which is sufficient for use in an interferometric vibration detection scheme.
- The fringe shift coincides with the ratio of λ/2 = 316.5 nm. Consequently, from Figure 24, where the amplitude of the applied triangular signal is A
_{piezo}= 1 V, the vibration generated has an amplitude of ±A_{piezo}∙S_{piezo}= ±200 nm. - From the photodetector response to the measured deflection amplitude of ±200 nm, the sensitivity of the entire system to the sensor can be calculated by a linear approximation as S
_{sys}= 200 nm/0.115 V = 1.74 μm/V. - The optical system cannot be used to measure signals with frequencies that are higher than 14 kHz due to the bandwidth of the photodetector selected.
- To measure the relative position of the microstructure, the measurement system must have a vibration in phase with its reference frame.
- With this setup, a peak-to-peak noise voltage (${V}_{n}$) lower than 10 mV is achieved without further signal processing. Therefore, vibration resolution can be calculated using the linear sensitivity approximation as:

#### 4.2. Microstructure Performance Test

- There is a small distortion of the reference accelerometer signal due to the adaptation of the accelerometer manufactured, which is not concentric and is made by means of a screw that does not ideally approximate a rigid solid. The effect is negligible and can be reduced by placing the accelerometer concentrically to the microstructure support for future evaluations.
- To obtain the vibration signal of the microstructure with reference to its local coordinate system, it is required that the interferometer be vibrating next to the reference of the support to avoid the modification of the trajectory with the main vibration. Therefore, the result of this test corresponds to a composite signal between the vibration of the support plus the vibration of the microstructure.
- Because of the previous point, there is a limitation on the spectrum and amplitude that can be measured, because to produce a significant vibration in the microstructure, a minimum acceleration, which depends on the used frequency, is required. This acceleration cannot be increased arbitrarily, since if the total amplitude (of the support plus the microstructure) exceeds the maximum range and the change of fringes produces a saturation of peaks in the response, it is not possible to distinguish between them.

**Figure 26.**(

**a**) Assembly of Microstructure on Test Bench. (

**b**) Output Waveforms from Vibration Sensor Design C and the Reference Accelerometer (B&K 4513-B). The waveforms are obtained with a 100 Hz and 0.2 g vibration.

_{sh}is the shaker’s sinusoidal vibration with amplitude a

_{sh}, angular frequency ω

_{sh}and phase φ

_{sh}, and γ

_{s}is the sinusoidal vibration of the sensor with amplitude a

_{s}, angular frequency ω

_{sh}and phase φ

_{sh}

_{+}

_{π}. The vibration of the microstructure has the same frequency as the excitation but is out-of-phase by π radians. Therefore, applying trigonometric identities:

_{ph}), considering the area coverage analysis is given by:

_{off}is the DC offset voltage of the signal. Then, the measured response of the photodiode V

_{ph}(𝛾(𝑡)) is given by:

_{ph}(γ(t)) can be expressed approximately by the first three terms of the series:

_{sh}, a signal V

_{ph}

_{-}

_{fillt}is obtained, which can be expressed as:

_{sh}, which for this test corresponds to 100 Hz, as shown by the peak acceleration in Figure 27. Thus, the acceleration amplitude seen in the signal spectrum of the sensor under test is proportional to the acceleration of the reference sensor, given by:

_{V}has integrated photodiode response information relative to the vibration displacement, any nonlinearity or error, and its frequency sensitivity, which can be considered constant within the working range. ω

_{ph}is the frequency response of the area coverage effect of the photodiode and it depends on the working distance (or, alternatively, to the geometry of the interferogram projection), and the area of the photodiode. This causes the difference between acceleration amplitudes corresponding to ${A}_{s}\left(0.8801{A}_{v}{\omega}_{ph}-1\right)$, which can be seen in Figure 27.

- The proposed methodology can be used to detect the vibrational spectrum of the microstructure with a 17.5 nm amplitude resolution in a frequency range from 5 Hz up to 13 kHz due to the shaker limitation.
- The response of the microstructure to inertial stimuli can be adjusted by geometric modifications to improve the resolution in a smaller frequency range or to increase the frequency range but reducing the resolution.
- An operational probe can be designed and manufactured from materials that can be used at higher than 85 °C without significant variations in response
- It is feasible to design and manufacture a vibration measurement system for testing under relevant conditions.

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

% Data initialization lw = 8.36; % Light width in pixels dw = 2.79; % Dark widht in pixels fr = 6; % Fringe repetitions wp = 4; % Frige projection width % Varible initiallization cx = 0; cy = 0; a1 = 0; a2 = 0; a3 = 0; a4 = 0; a5 = 0; a6 = 0; % Sensor size = 2.29; % Size (square) of sensor xs = wp/2; % Sensor center x point ys = fr*(lw + dw)*0.7; % Sensor center y point theta = 0.5; % Sensor rotation angle refp = [xs ys;xs ys;xs ys;xs ys;xs ys]; sensor_nr = [ xs-size/2 ys-size/2 xs + size/2 ys-size/2 xs + size/2 ys + size/2 xs-size/2 ys + size/2 xs-size/2 ys-size/2 ]; % Sensor area rotation R = [cos(theta) − sin(theta); sin(theta) cos(theta)]; sensor = refp + (sensor_nr-refp)*R; % Step calculation samples = 1000; range = 4*(lw + dw); step = range/samples; av = zeros(samples,1); at = av; at2 = av; av2 = av; index = 1; % Base shadow sh0 = [ 0 0 ; wp 0 ; wp dw ; 0 dw ; 0 0]; for j = 0:step:range % Shadows generation sh1 = [sh0(:,1) sh0(:,2) + 0*(lw + dw) + j]; sh2 = [sh0(:,1) sh0(:,2) + 1*(lw + dw) + j]; sh3 = [sh0(:,1) sh0(:,2) + 2*(lw + dw) + j]; sh4 = [sh0(:,1) sh0(:,2) + 3*(lw + dw) + j]; sh5 = [sh0(:,1) sh0(:,2) + 4*(lw + dw) + j]; sh6 = [sh0(:,1) sh0(:,2) + 5*(lw + dw) + j]; % Intersection [rx1,ry1] = oc_polybool(sensor,sh1,’and’); i = numel(rx1); if(i > 0) a1 = polyarea(rx1(2:i-1),ry1(2:i-1)); endif [rx2,ry2] = oc_polybool(sensor,sh2,’and’); i = numel(rx2); if(i > 0) a2 = polyarea(rx2(2:i-1),ry2(2:i-1)); endif [rx3,ry3] = oc_polybool(sensor,sh3,’and’); i = numel(rx3); if(i > 0) a3 = polyarea(rx3(2:i-1),ry3(2:i-1)); endif [rx4,ry4] = oc_polybool(sensor,sh4,’and’); i = numel(rx4); if(i > 0) a4 = polyarea(rx4(2:i-1),ry4(2:i-1)); endif [rx5,ry5] = oc_polybool(sensor,sh5,’and’); i = numel(rx5); if(i > 0) a5 = polyarea(rx5(2:i-1),ry5(2:i-1)); endif [rx6,ry6] = oc_polybool(sensor,sh6,’and’); i = numel(rx6); if(i > 0) a6 = polyarea(rx6(2:i-1),ry6(2:i-1)); endif av(index) = a1 + a2 + a3 + a4 + a5 + a6; at(index) = j; a1 = 0; a2 = 0; a3 = 0; a4 = 0; a5 = 0; a6 = 0; index = index + 1; endfor |

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**Figure 1.**Conceptual model for optical vibration detection, with its elements numbered. Movement is considered in the Y axis.

**Figure 3.**(

**a**) Accelerometer with four arms composed by an arrangement of folded beams. (

**b**) Equivalent diagram.

**Figure 4.**Displacement and Acceleration Ratio vs. Vibration Frequency for Three Different Natural Frequencies f

_{n}= ω

_{n}/2π.

**Figure 5.**Geometries of accelerometers designed to meet the established specifications, with (

**a**) Z, (

**b**) double Z, and (

**c**) a series of three folded beam arrangements.

**Figure 7.**Steady-state FEA of designs A, B and C subjected to a static Acceleration of 10 g. (

**a**) Design A with 1.41 µm mass displacement. (

**b**) Design B with 0.47 µm mass displacement. (

**c**) Design C with 2.64 µm mass displacement.

**Figure 9.**(

**a**) Vibration amplitude of the three designs subjected to an acceleration of 10 g relative to the vibration frequency. Data are shown in a logarithmic scale. (

**b**) Frequency bandwidth for A, B and C designs using a maximum vibration amplitude variation of +3 dB from the flat response. (

**c**) Proof mass acceleration in relation to vibration frequency for the three designs. (

**d**) Percentage of vibration amplitude variation between 25 °C and 200 °C temperature for design C.

**Figure 12.**(

**a**) Geometric reconstruction of the alignment of photodiode and interferogram. It shows the fringes (red), the photodiode area with an angle φ from the fringe (blue) and the area exposed to incident light of the interferogram fringes (green). (

**b**) Estimated exposed area change by fringe displacement y

_{f,}which can be approximated by a sinusoidal function.

**Figure 14.**(

**a**) Mask Design for the Entire 400 mm Wafer. (

**b**) Lower and (

**c**) Upper Photolithographic Masks.

**Figure 15.**Wafer during manufacturing. (

**a**) Wafer during manufacturing (step #4). (

**b**) Wafer after release of sacrificial layers in (step #5). (

**c**) Single chip after dicing using laser ablation. (

**d**) Zoom in of the groove produced by laser ablation.

**Figure 18.**(

**a**) Test Bench with Michelson Interferometer. (

**b**) Interferogram to Photodetector Projection Adjustment Showing the Match between the Size and Angle of the Fringe and the Area of the Photodetector.

**Figure 19.**Test of Optical Detection System with Piezoelectric Actuator. (

**a**) Silicon Substrate on Piezoelectric Actuator. (

**b**) Alignment of Silicon Substrate on Piezoelectric Actuator with Interferometer Laser.

**Figure 20.**(

**a**) Microphotography of fabricated microstructure on wafer. (

**b**) Mount assembly with attached microstructure, reference accelerometer and alignment camera. (

**c**) View of laser spot alignment on microstructure.

**Figure 21.**Measurement of microstructure with a digital optical microscope. (

**a**) Microstructure dimensions measurement. (

**b**) Measurement of bottom face reduction of proof mass (

**c**) Top and left: 3D recreation model, cross-section view, using an optical profiler. Top and right: Cross-section line (blue) for height measurement. Bottom: Height profile corresponding to the blue line in the top and right image.

**Figure 23.**Microphotograph of microstructure bottom view with an angle of 60°. The variation in proof mass perimeter due to fabrication process is shown.

**Figure 24.**Photodetector Signal with System Aligned to Silicon Substrate Actuated by a Piezoelectric Buzzer.

**Figure 25.**Approximation of the photodetector response to the ideal shape by an area calculation algorithm.

**Figure 27.**Comparison between Signal Spectra between Vibration Sensor and Reference Accelerometer. The acceleration amplitudes at 100 Hz are labeled and a band pass filter range is shown for comparison.

Parameter | Description | Size (µm) | ||
---|---|---|---|---|

Design A | Design B | Design C | ||

H_{m} | Mass thickness | 378 | 362 | 490.36 |

L_{m} | Mass length | 1500 | 1500 | 1500 |

W_{1} | Beam width | 106.9 | 69.55 | 86.02 |

W_{2} | Joint width between supports and mass | 83.6 | 94.15 | 96.21 |

L_{2} | Length of the joint between supports and mass | 86.6 | 124.2 | 86.6 |

H_{b} | Beam thickness | 19.4 | 11.4 | 18.9 |

Device | Solver Target | Element Type/Mesh | Convergence | |||
---|---|---|---|---|---|---|

Total Number of Nodes | Total Number of Elements | Change % | Total Mass (Kg) | |||

Design A | Mechanical APDL | SOLID187/Refinement controlled program (Tet10) | 83,842 | 54,922 | 4.02 | 1.9868 × 10^{−6} |

Design B | 157,411 | 100,226 | 3.03 | 1.8877 × 10^{−6} | ||

Design C | 19,845 | 8577 | 0 | 1.7624 × 10^{−6} |

Vibration Mode | Frequency (Hz) | ||
---|---|---|---|

Design A | Design B | Design C | |

1 | 1390.6 | 2445.4 | 1007.7 |

2 | 2232.6 | 3672.8 | 2481.3 |

3 | 2235.4 | 3676.3 | 2481.6 |

4 | 49,157 | 53,924 | 7307.7 |

5 | 49,212 | 53,943 | 7308.4 |

6 | 75,224 | 95,190 | 16,444 |

Parameter | Description | Design C | Error, % | |
---|---|---|---|---|

Designed Sizes (µm) | Fabricated Sizes (µm) | |||

H_{m} | Mass thickness | 490.36 | 401.98 | 18.02 |

L_{m} | Mass length | 1500 × 1500 | 1500.13 × 1497 | 0.01 × 0.20 |

W_{1} | Beam width | 86.02 | 89.43 | 3.96 |

W_{2} | Joint width between supports and mass | 96.21 | 96.87 | 0.69 |

L_{2} | Length of the joint between supports and mass | 86.6 | 81.46 | 5.94 |

H_{b} | Beam thickness | 18.9 | 51.14 | 180.99 |

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## Share and Cite

**MDPI and ACS Style**

Sánchez-Fraga, R.; Tecpoyotl-Torres, M.; Mejía, I.; Mañón, J.O.; Riestra, L.E.; Alcantar-Peña, J.
Optical Sensor, Based on an Accelerometer, for Low-Frequency Mechanical Vibrations. *Micromachines* **2022**, *13*, 1462.
https://doi.org/10.3390/mi13091462

**AMA Style**

Sánchez-Fraga R, Tecpoyotl-Torres M, Mejía I, Mañón JO, Riestra LE, Alcantar-Peña J.
Optical Sensor, Based on an Accelerometer, for Low-Frequency Mechanical Vibrations. *Micromachines*. 2022; 13(9):1462.
https://doi.org/10.3390/mi13091462

**Chicago/Turabian Style**

Sánchez-Fraga, Rodolfo, Margarita Tecpoyotl-Torres, Israel Mejía, Jorge Omar Mañón, Luis Eduardo Riestra, and Jesús Alcantar-Peña.
2022. "Optical Sensor, Based on an Accelerometer, for Low-Frequency Mechanical Vibrations" *Micromachines* 13, no. 9: 1462.
https://doi.org/10.3390/mi13091462