# Analysis of Error Sources in the Lissajous Scanning Trajectory Based on Two-Dimensional MEMS Mirrors

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## Abstract

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## 1. Introduction

## 2. Methods

#### 2.1. Lissajous Theory

#### 2.2. MEMS Scanning Lissajous Trajectory Test Platform Overview

## 3. Experimental Validation

#### 3.1. Frequency Response Error of a Two-Dimensional MEMS Mirror

#### 3.2. AD Acquisition Synchronization Error

#### 3.3. Drive Source Error

#### 3.4. Cross-Coupling Error between the MEMS Mirror Axes

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**The schematic diagram of the MEMS-based Lissajous trajectory test platform. PSD: position sensitive detector. AD: analog-to-digital module. PC: personal computer. DC power: direct current power. Red arrows indicate the optical path. Blue arrows signify the signal paths generated by the signal generator and observed by the oscilloscope. Finally, the black arrows represent the signal paths generated by the ZYNQ control board and the X/Y trajectory paths.

**Figure 4.**The MEMS mirror frequency response characteristics. Red line: X axis. Blue line: Y axis. (

**a**) Measured amplitude of the PSD output signals versus the input amplitude ratio curves. The input amplitude ratio varied from 0 to 0.4. (

**b**) Amplitude frequency response curves. The frequency varied from 50 Hz to 2600 Hz in 50-Hz increments. The input amplitude ratio was set as 0.2. (

**c**) Phase frequency response curves. The frequency varied from 50 Hz to 2600 Hz in 50 Hz increments. The input amplitude ratio was set as 0.2.

**Figure 5.**Overall and zoomed-in diagrams of two theoretical Lissajous trajectories. (

**a**) Overall view. (

**b**) Zoomed-in view of the green dashed box labeled in (

**a**).

**Figure 6.**Overall and zoomed-in diagrams of the measured and theoretical trajectories. (

**a**) Overall view of the uncompensated measured and theoretical trajectory 1. (

**b**) Zoomed-in view of the green dashed box labeled in (

**a**). (

**c**) Overall view of the compensated measured and theoretical trajectory 1. (

**d**) Zoomed-in view of the green dashed box labeled in (

**c**).

**Figure 7.**The characteristics of the drive source errors at three different frequencies. (

**a**) Amplitude difference $\u2206{A}_{X2-Y2}$ versus the input amplitude ratio curves. The input amplitude ${V}_{X2}={V}_{Y2}$. (

**b**) Phase difference deviation $\u2206{Phasedifference}_{X2-Y2}$ versus input phase difference curves. The input amplitude ${V}_{X2}={V}_{Y2}=0.2{V}_{max}$.

**Figure 8.**Low-pass filter circuit frequency response characteristics. Frequency ${f}_{X2}$ vary from 50 Hz to 2600 Hz in 50 Hz increments. (

**a**) Amplitude frequency response curve. (

**b**) Phase frequency response curve.

**Figure 9.**The overall and zoomed-in diagrams of the Lissajous trajectories. (

**a**) The reference Lissajous trajectory. (

**b**) The zoomed-in view of the white box in (

**a**). (

**c**) The uncompensated measured Lissajous trajectory. (

**d**) The zoomed-in view of the white box in (

**c**). (

**e**) The compensated measured Lissajous trajectory. (

**f**) The zoomed-in view of the white box in (

**e**).

**Figure 10.**Cross-coupling error characteristics of the MEMS mirror’s X axis. Yellow line: X axis. Blue line: Y axis. They indicate the results of the PSD waveform signals from the X and Y axes captured in the oscilloscope’s YT mode at different drive conditions. (

**a**,

**b**) ${V}_{X}=0.1{V}_{max}$, ${f}_{X}=2300\mathrm{H}\mathrm{z}$. (

**c**,

**d**) ${V}_{X}=0.1{V}_{max}$, ${f}_{X}=1200\mathrm{H}\mathrm{z}$. (

**e**,

**f**) ${V}_{X}=0.1{V}_{max}$, ${f}_{X}=600\mathrm{H}\mathrm{z}$. (

**g**,

**h**) ${V}_{X}=0.2{V}_{max}$, ${f}_{X}=2300\mathrm{H}\mathrm{z}$. (

**i**,

**j**) ${V}_{X}=0.3{V}_{max}$, ${f}_{X}=2300\mathrm{H}\mathrm{z}$.

**Figure 11.**Source drive signal waveforms captured in the YT mode. Yellow line: X axis. Blue line: Y axis. (

**a**) Uncompensated X and Y axis waveforms. ${V}_{X}=13.6\mathrm{V}$, ${V}_{Y}=14\mathrm{V}$, ${f}_{X}={f}_{Y}=1200\mathrm{H}\mathrm{z}$. X and Y axes’ phase difference of 89.8°. (

**b**) Compensated X and Y axis waveforms. ${V}_{X}=14V$, ${V}_{Y}=14V$, ${f}_{X}={f}_{Y}=1200\mathrm{H}\mathrm{z}$. X and Y axes’ phase difference of 90°.

**Figure 12.**Signal waveforms after eliminating the effect of MEMS mirror frequency response characteristics captured in the YT mode. Yellow line: X axis. Blue line: Y axis. (

**a**) Uncompensated X and Y axis waveforms. ${V}_{X}=0.88\mathrm{V}$

**,**${V}_{Y}=1.04\mathrm{V}$, ${f}_{X}={f}_{Y}=1200\mathrm{H}\mathrm{z}.$ X and Y axes’ phase difference 92.6°. (

**b**) Compensated X and Y axis waveforms. ${V}_{X}=0.88\mathrm{V}$, ${V}_{Y}=1.04\mathrm{V}$, ${f}_{X}={f}_{Y}=1200\mathrm{H}\mathrm{z}$. X and Y axes’ phase difference 90°.

**Figure 13.**Compensated trajectory waveforms. Yellow line: X axis. Blue line: Y axis. (

**a**) X and Y axis waveforms in the YT mode. ${V}_{X}={V}_{Y}=1.04\mathrm{V}$, ${f}_{X}={f}_{Y}=1200\mathrm{H}\mathrm{z}$. X and Y axes’ phase difference 90°. (

**b**) Combined trajectory in the XY mode.

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## Share and Cite

**MDPI and ACS Style**

Zhang, X.; Wang, C.; Han, Y.; Wang, J.; Hu, Y.; Wang, J.; Fu, Q.; Wang, A.; Feng, L.; Hu, X.
Analysis of Error Sources in the Lissajous Scanning Trajectory Based on Two-Dimensional MEMS Mirrors. *Photonics* **2023**, *10*, 1123.
https://doi.org/10.3390/photonics10101123

**AMA Style**

Zhang X, Wang C, Han Y, Wang J, Hu Y, Wang J, Fu Q, Wang A, Feng L, Hu X.
Analysis of Error Sources in the Lissajous Scanning Trajectory Based on Two-Dimensional MEMS Mirrors. *Photonics*. 2023; 10(10):1123.
https://doi.org/10.3390/photonics10101123

**Chicago/Turabian Style**

Zhang, Xiulei, Conghao Wang, Yongxuan Han, Junjie Wang, Yanhui Hu, Jie Wang, Qiang Fu, Aimin Wang, Lishuang Feng, and Xiaoguang Hu.
2023. "Analysis of Error Sources in the Lissajous Scanning Trajectory Based on Two-Dimensional MEMS Mirrors" *Photonics* 10, no. 10: 1123.
https://doi.org/10.3390/photonics10101123