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Abstract
We previously designed a dual-axis piezoelectric MEMS mirror with a low crosstalk gimbal structure, which is utilized as the key device for further research for laser beam scanning. This paper mainly focuses on studying the Lissajous scanning resolution of this MEMS mirror with frequency ratio and phase modulation. For accurately evaluating the scanning resolution, the center angular resolution of Lissajous scanning is redefined by theoretical calculation and verified with experimental measurement. Meanwhile, the scanning nonlinearity of MEMS mirror is studied carefully. Finally, the MEMS mirror works at the state of pseudo-resonance, and the center angular resolution better than 0.16° (H) × 0.03° (V) is achieved at a scanning Field of view (FoV) of 35.0° (H) × 16.5° (V). Moreover, a feasible route of resolution adjustable Lissajous scanning is provided by optimization of frequency ratio and phase modulation, which is helpful for high definition and high frame rate (HDHF) laser scanning imaging with the dual-axis mirror.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the development of auto-driving technologies, the demand for LiDAR is rapidly increasing. MEMS scanner has gradually become one of the main LiDAR solutions due to its advantages of high scanning speed, small size, and low cost [1]. Raster scanning and Lissajous scanning are usually used in MEMS scanners. The primary advantages of raster scanning are the constant velocity and simple image reconstruction which is due to regularly sampled data appearing on a square-shaped grid [2]. However, the frequency of triangular grating scanning is usually limited to 1-10% of the first resonance frequency of the positioner, which greatly limits the scanning speed [3]. Sinusoidal raster scanning involves driving the x-axis (fast-axis) with a sinusoidal trajectory while shifting the sample in steps or continuously in the y-axis (slow-axis) [4–6]. The addition of sinusoidal trajectory improves the scanning frequency of Sinusoidal raster scanning, but at the same time introduces the scanning inhomogeneity in the fast axis direction [7–11]. Lissajous scanning methods require sinusoidal trajectories on both the x-axis and y-axis and have been widely used in many optical imaging systems due to their advantages, such as single-tone spectrum [12] and high-precision angle measurement [13]. With the development of LiDAR, the requirements for resolution are getting more and more attention. Therefore, Lissajous scanning with the high angular resolution has become one of the most frequently used scanning modes in the field of LiDAR, and a lot of the Lissajous scanning theories are studied [14–18]. In 2016, Du et al proposed a theory of Lissajous scanning based on the diamond pixel [19]. In 2017, Hwang et al proposed a frequency selection rule for high definition and high frame rate Lissajous scanning [20,21]. In 2018, Lee et al presented the calculation method for the interval of the Lissajous scanning [22]. Wang et al systematically present three design principles of the Lissajous scanner for the first time, which makes the Lissajous scanning theory gradually complete, and gives a method for calculating the maximum interval Fill Factor ($FF$) at the center of the Lissajous scanning trajectory [23]. In 2021, For the application of Lissajous scanning at regions of interest (ROI), the phase modulation algorithm was proposed to further improve the local spatial resolution without changing the scanning Field of view [24].
Above all, the published Lissajous scanning theory is rarely aimed at a self-developed MEMS mirror. Meanwhile, the $FF$ value is an overall value that represents the worst angular resolution of the Lissajous scan. So, the $FF$ value does not represent the angular resolution in the vertical and horizontal direction. Therefore, a general calculation of the $FF$ value is given, and the angular resolution in the vertical and horizontal directions at the center is redefined in this paper. Based on an earlier self-developed piezoelectric MEMS mirror, we carried out the device’s characterization and optimization of the driving parameters, which verified the effect of phase modulation on the scanning trajectory of Lissajous. Then the validation of the definition of the angular resolution in vertical and horizontal directions at the center was measured and proved. Finally, we optimized the frequency ratio and phase modulation of the MEMS mirror to make the MEMS mirror work at steady Lissajous scanning with high angular resolution.
2. Methods
2.1 Definition of Lissajous angular resolution
The traditional Lissajous scan mode is obtained by oscillating two monophonic frequencies along two orthogonal axis. Therefore, the trajectories of the dual axis can be defined as
The amplitude of the Lissajous scanning depends on the value of ${A_x}$ and ${A_y}$, and the frame rate and density of the Lissajous scanning depend on the relationship between two single-tone frequencies. If ${f_x}$, ${f_y}$ are integers and there exists a maximum common factor ${f_0}$, then Eq. (2) holds [23]. ${f_0}$ is the frame rate of the Lissajous scanning. ${n_x}$ and ${n_y}$ are called leaf numbers, which represent the elapsed number of the x-axis or y-axis periods [24]. By definition, it is clear that ${n_x}$ and ${n_y}$ are a pair of coprime numbers (Except for the special case where ${f_x} = {f_y}$. In this case, the image of Lissajous scanning is a standing circle). It is worth noting that the frame rate here is the inverse of the minimum time required by Lissajous scan to complete a periodic scan, and has no essential relationship with the scanning resolution of Lissajous.
If t in Eq. (1) is reduced, the trajectory equation can be obtained by sorting out the equation.
Then the parameter that depends on the Lissajous scanning trajectory is $({{n_x}{\varphi_y} - {n_y}{\varphi_x}} )$ so a parameter k is introduced [23], which is defined as
We can use the simulation of the scan trajectory of Lissajous to show the effect of the k value. For example, taking the 1958Hz and 801 Hz as the frequency parameters of the axis of the Lissajous scanning, which is the ${f_0} = 89$ Hz and ${n_x}:{n_y} = 22:9$. For the convenience of calculation, the initial phase ${\varphi _x}$ of the x-axis is set as 0, then Eq. (4) can be simplified asAccording to simulation, the image of Lissajous scanning with different k values can be drawn as ${\varphi _x} = 0,\frac{\pi }{{88}},\frac{\pi }{{44}}$ (corresponding $k = 1,2,3$), ${A_x} = 1$ and ${A_y} = 1$. The results are shown in Fig. 1. When $k = 0$, there is an overlap of lines in the Lissajous image, resulting in a lack of lines in the image. When $k = 1$, the lines in the image are complete but poorly distributed. When $k = 2$, the lines in the image are complete and highly symmetrical (symmetry about the x-axis, y-axis, and origin).
Fig. 1. Lissajous scanning simulation images with different k values (${n_x}:{n_y} = 22:9$). (a) $k = 0$. (b) $k = 1$. (c) $k = 2$.
When the initial phase remains unchanged, take ${\varphi _x} = 0$, and substitute Eq. (5) into Eq. (1) to obtain Eq. (6).
The $FF$ value can be used to validate the scan quality of the Lissajous scanning. In the following, the general method for calculating the $FF$ values was given. Two typical Lissajous images were obtained, as shown in Fig. 2. According to Eq. (2), ${n_x}$ and ${n_y}$ are prime numbers of each other (Except for the special case where ${f_x} = {f_y}$), then there are two cases as follows: (1). ${n_x}$ and ${n_y}$ are all odd numbers; (2). One of ${n_x}$ and ${n_y}$ is an even number. For the case without an even number, $FF$ is equal to the height of the diamond shown in Fig. 2(a); for the even case, we cannot find a single biggest diamond symmetric about the origin near the origin, so in this case, the $FF$ value is equivalent to half the height of the large diamond consisting of a, b, c, d in Fig. 2(b). Taking the scanned image of Lissajous in Fig. 2(a) as an example. The distance from a and c to the origin was defined as ${d_1}$; The distance from b and d to the origin as ${d_2}$. In Fig. 2(a), the values of ${d_1}$ and ${d_2}$ can be calculated by Eq. (7a) and Eq. (7b).
For the case shown in Fig. 2(a), the $FF$ value can be calculated by Eq. (8a), and for the case shown in Fig. 2(b), the $FF$ value can be calculated by Eq. (8b).
In the case of a certain scanning angle, the smaller the value of $FF$ means the better the angular resolution of Lissajous scanning. However, $FF$ does not directly indicate the horizontal angular resolution and vertical angular resolution which are that important parameter in the field of LiDAR. Therefore, the $FF$ values need to be decomposed in both horizontal and vertical directions. When the scanning of the Lissajous was simulated with different sampling times, Lissajous scanning images with the same trajectory but different pixel networks were obtained as shown in Fig. 3(a-c). When the sampling interval satisfies the equation shown in Eq. (9), then the sampling points per second species are $4{f_0}{n_x}{n_y}$, and the points per frame are $4{n_x}{n_y}$. In this case, the points in a frame of Lissajous such as the scanned image are almost all distributed at the intersection of the scan trajectory [25]. We assume that ${f_0} = 1$, ${n_x} = 23$, ${n_y} = 9$. Then the number of Lissajous scan points in one frame is 828. We then obtain the image shown in Fig. 3(c).
A higher resolution rectilinear pixel grid can be obtained if an initial shift of half a sampling time interval is applied as shown in Fig. 3(d). The horizontal and vertical decomposition of $FF$ were approximated by the distances of mn and mp. Compared to Fig. 3(c), the points in Fig. 3(d) were shifted by half the sampling time units, so that m, n, p, and q can be approximated as the midpoints of ab, bc, ad, and cd. Therefore, the distances of mn and mp can be calculated as follows.
In addition, the angular resolution of Lissajous scanning is expressed as
where $\Delta {a_h}$ and $\Delta {a_v}$ are the center angular resolution of the horizontal direction and vertical of the Lissajous scanning, respectively. Similarly, when ${n_x}$ and ${n_y}$ one of them is an even number, the angular resolution is expressed asIn the rectilinear pixel grid shown in Fig. 3(d), the number of columns is $2{n_y}$ and the number of rows is $2{n_x}$ therefore the average angular resolution can be expressed as
where $\Delta {a_{h\_avg}}$ and $\Delta {a_{v\_avg}}$ are the average resolution in the horizontal direction and vertical of the Lissajous scanning, respectively.Fig. 2. Two typical Lissajous simulation images. (a) ${n_x}:{n_y} = 23:9$, ${n_x}$ and ${n_\textrm{y}}$ are all odd numbers. (b) ${n_x}:{n_y} = 22:9$, one of ${n_x}$ and ${n_\textrm{y}}$ is an even number.
Fig. 3. Effect of different sampling rates on the image of Lissajous scanning (when ${n_x}:{n_y} = 23:9$, ${f_0} = 1$). (a) the sampling rate is 600 points per second (b) the sampling rate is 1000 points per second. (c) the sampling rate is 828 (sampling time is $\Delta {t_s}$) points per second. (d) All points are offset by 1/1656 second ($0.5\Delta {t_s}$) second based on (c).
2.2 MEMS mirror devices
The MEMS mirrors are piezoelectric driven, based on AlScN piezoelectric film. The design and fabrication process of the MEMS mirror is described in detail elsewhere [26]. The piezoelectric film of the AlScN micro-mirror is a sandwich structure, in which Mo is used as the upper and lower electrodes. The MEMS mirror adopts a gimbal structure to reduce mechanical coupling between two axis resonances, as shown in Fig. 4. The far-end of the ring-shaped cantilevers connects the mirror with springs and makes the mirror rotate about the fast-axis by the actuator. While the mirror rotates about the fast axis, the inner frame remains essentially stationary, as shown in Fig. 4(a). The actuator connecting with the outer frame transfers the displacement to the inner frame through a hinge, then the mirror follows the entire inner frame rotating about the slow axis, as shown in Fig. 4(b). Figure 5 shows MEMS mirrors in ceramic packages with two different diameters (3.0 mm and 1.5 mm, named DS-3.0 and DS-1.5, respectively).
Fig. 4. The diagram of the MEMS mirror rotates about two different axis. (a) fast axis. (b) slow axis.
2.3 Optical measurement system
In order to measure the deflection angle of the MEMS mirror and demonstrate the trajectory of the Lissajous scanning, an optical measurement system was set up as shown in Fig. 6. The whole system consists of a laser source (Thorlabs, S1FC635), signal generator (KEYSIGHT, 33600A), signal amplifier (Aigtek, ATA-2032), MEMS mirror, and a screen. The measurement of the optical scan angle of the MEMS mirror is achieved, according to the displacement of the light on the light screen in Fig. 6(a).
3. Results and discussion
3.1 Characterization of MEMS mirrors
The frequency response curves of the MEMS mirror were measured by Laser Doppler Velocimetry (LDV, Polytec MSA-500) system, as shown in Fig. 7. The resonance frequencies of the fast-axis and slow-axis of the DS-3.0 mirror are 2090.6 Hz and 806.3 Hz, respectively. The quality factors (Q factor) of the fast-axis and slow-axis of the DS-3.0 mirror are 251.1 and 58.6, respectively. The resonance frequencies of the fast-axis and slow-axis of DS-1.5 are 8186.4 Hz and 3151.6 Hz, respectively. The Q factors of the fast axis and slow axis of the DS-1.5 mirror are 488.3 and 132.0, respectively. The resonance frequency of the DS-1.5 is higher than that of the DS-3.0, in addition, the frequency of the fast axis is approximately 2.59 times that of the slow axis at the same size.
Combined with the results of the LDV test, the resonance frequencies of the two types of mirrors are measured. Afterward, the maximum scanning angle of the MEMS mirror at different voltages was tested by the optical measurement system. The results are shown in Fig. 8(a). The Optical scan angles of the fast axis of DS-1.5 and DS-3.0 are capable of achieving 31.5° and 32.0°, respectively, at a sinusoidal driving voltage of 30 V. However, the scan angle of the slow axis is poor, that is, the slow axis of DS-3.0 and DS-1.5 reaches a maximum of 6.7° and 1.0° separately, at a driving voltage of 30 V. To address the problem of the small scanning angle of the slow axis, the mirror was tested with vacuum conditions (4 Pa) based on the work of Hoffman et al [27]. Figure 8(b) shows that the scan angle of the slow axis of the MEMS mirror increases dramatically at an air pressure of 4 Pa. The maximum optical scan angle of 20° of the slow axis of the DS-1.5 is achieved at the sinusoidal driving voltage of 30 V; the slow axis of the DS-3.0 achieves an optical scan angle of more than 20° at the sinusoidal driving voltage of only 1.1 V. Air damping is reduced by evacuation process, and thus the sharp increment in Q factor, which is the major reason for the increasement in the optical deflection angle of the MEMS mirror [28].
Fig. 8. Response curve of MEMS mirror scanning angle with driving voltage. (a) Under atmospheric pressure. (b) Under vacuum condition (4 Pa).
When the voltage increased, the scanning angle increased and its resonance frequency shifted to a higher frequency, as shown in Fig. 9. At the same time, the MEMS mirror exhibits a nonlinear scan profile as shown in Fig. 10, that is, when the different initial frequencies were applied to excite the MEMS mirror, two different frequency response curves are obtained. The results are illustrated as spring hardening dominance [29]. As shown in Fig. 10, when the initial excitation frequency is on the left side of ${F_1}$ and swept upward, the response of the MEMS mirror scan angle conforms to the “Up” curve. When the initial excitation is on the right side of ${F_3}$ and swept downward, the response of the MEMS mirror scan angle conforms to the “Down” curve. In the “Up” curve, once the frequency of ${F_2}$ is crossed, there is a sudden drop (${F_3}$) in the scan angle, and it cannot be returned unless the drive frequency is lowered to the left of ${F_1}$ to restore the normal “Up” curve. Ataman et al have systematically explained the phenomenon of MEMS scanning nonlinearity using the Floquet theory solutions of the Mathieu equation [30–32].
Fig. 9. Shift of MEMS mirror resonance frequency with driving voltage. (a) fast axis of DS-3.0. (b) fast axis of DS-1.5.
Fig. 10. Frequency response curves at different initial excitation frequencies. (a) fast axis of DS-3.0. (b) fast axis of DS-1.5.
The nonlinearity effect of MEMS mirrors can be attributed to two factors, the driving voltage and the Q factor [27,32]. Figure 11(a) and Fig. 11(b) show the level of influence on the scanning nonlinearity of the MEMS mirror at different voltages. As shown in Fig. 11(a), the MEMS mirror shows almost no nonlinearity at low voltage (4 V), as the voltage increases, the phenomenon of nonlinearity of the MEMS mirror starts to appear, and the higher the voltage, the more obvious the nonlinearity becomes. As shown in Fig. 11(b), the MEMS also exhibits severe nonlinearity even at a low voltage of 1 V under 4 Pa.
Fig. 11. Effect of driving voltage on MEMS mirror scanning at different air pressure (tested by DS-1.5). (a) 1 atm. (b) 4 Pa.
3.2 Phase modulation
Images of the Lissajous scanning were obtained by driving simultaneously the fast and slow axis to operate around the resonance frequency. Special attention should be paid to the process that the frequency should be gradually increased from lower frequencies to near the resonance frequencies when driving the MEMS mirrors. Several smaller ${n_x}$ and ${n_y}$ were chosen to show the effect of phase on the Lissajous scanning. In Fig. 12, the modulation of the phase of the Lissajous scanning is achieved by controlling the phase difference between the two axis. Figure 12(a-c) shows the Lissajous images of DS-3.0 with k values (0, 1, 2), in which the frequency ratio of fast to slow axis is 2070Hz: 805 Hz, ${n_x}:{n_y} = 18:7$ and the scan frame rate is 115 Hz; The images in Fig. 12 (d-e) were scanned by DS-1.5 with a frequency ratio of 8329 Hz: 3210 Hz, ${n_x}:{n_y} = 77:30$ and the scan frame rate is 107 Hz. The results are consistent with the simulation that the Lissajous scanning with $k = 2$ has a higher resolution [23].
Fig. 12. Images of Lissajous scanning at different k values. (a) $k = 0$, scanned by DS-3.0. (b) $k = 1$, scanned by DS-3.0. (c) $k = 2$, scanned by DS-3.0. (d) $k = 0$, scanned by DS-1.5. (e) $k = 1$, scanned by DS-1.5. (f) $k = 2$, scanned by DS-1.5.
3.3 Calculation and measurement of angular resolution
The center angular resolutions of the Lissajous scanning with different ${n_x}:{n_y}$ were calculated and measured in combination with Fig. 3(d) and Eq. (11-12). In Fig. 13(a), when the ${n_x}:{n_y} = 13:5$ and the projection distance is 200 cm, the resulting scanned area is 22.30 cm ${\times} $ 12.40 cm. The calculated value of mn is 3.13 cm and the measured value of mn is 3.10 cm; the calculated value of mp is 0.74 cm and the measured value is 0.80 cm. In Fig. 13(b), when ${n_x}:{n_y} = 18:7$ and the projection of 200 cm, the resulting scanned area is 17.50 cm ${\times} $ 12.50 cm. The calculated value of mn is 1.90 cm and the measured value of mn is 2.00 cm; the calculated value of mp is 0.52 cm and the measured value is 0.50 cm. Considering the measurement error of mn and mp, the measured and calculated values of the angular resolution at the center are almost the same. Hence, the results verify the correctness of the theoretical calculation.
Fig. 13. Measurement of the center angular resolution of Lissajous scanning (DS-3.0, $k = 2$)with different frequency ratios. (a) 2106 Hz: 810 Hz (${n_x}:{n_y} = 13:5$). (b) 2070Hz: 805 Hz. (${n_x}:{n_y} = 18:7$).
3.4 Frequency selection of MEMS mirrors in Lissajous scanning
According to Eq. (2), different frame rates, ${n_x}$ and ${n_y}$ can be selected by varying the frequency ratio of the two axis of the Lissajous scanning. At the same time, the selection of the Lissajous scanning frequency must guarantee a certain Field of view (FoV) to meet the requirements of the LiDAR. Taking the DS-1.5 mirror as an example, we chose the scanning frequency of the Lissajous dual-axis within a 3 dB bandwidth the maximum optical angle [20,21,33]. In Fig. 14, the selectable frequency bandwidth for the fast axis is 158 Hz and for the slow axis is 3.5 Hz (The driving voltage of the slow axis is 30 V, the driving voltage of the fast axis is 1.5 V, and the air pressure condition is 4 Pa). The narrow selectable frequency bandwidth for the slow axis makes it difficult to select the sweep frequency for the Lissajous. Despite this, we found several excellent frequency combinations at different frame rates. The Lissajous scanning images with frame rates of 51 Hz, 44 Hz, 30 Hz, and 19 Hz were shown in Fig. 15 respectively. By combining Eq. (11-13), the angular resolution of the scanned image of Lissajous in Fig. 15 was calculated, and the calculation results are shown in Table 1. In Fig. 14(b), when the frequency of the slow axis exceeds 3211 Hz, there is a sudden drop in angle, as we depicted in Fig. 10 and Fig. 11. Whereas both axis work simultaneously, the slow axis can operate at frequencies above 3211 Hz (such as 3212 Hz, 3213 Hz) without amplitude decrease. The shift of resonance frequency of the slow axis is probably caused by the crosstalk with the fast axis. Finally, as shown in Fig. 15(a), Fig. 15(d), and Table 1, two typical sets of results, which were applied to high frames with low resolution and normal frames with high resolution were acquired respectively. First, a scanned FoV of 36.5° (V) × 22.6° (H) at a scan high frame rate of 51 Hz, while $\Delta {a_h} \le 0.46^\circ $, $\Delta {a_{h\_avg}} \le 0.29^\circ$, $\Delta {a_v} \le 0.11^\circ $, $\Delta {a_{v\_avg}} \le 0.07^\circ$. Second, a scanned FoV of 35° (V) × 16.5° (H) at a normal scan frame rate of 19 Hz, while $\Delta {a_h} \le 0.16^\circ $, $\Delta {a_{h\_avg}} \le 0.10^\circ$, $\Delta {a_v} \le 0.03^\circ $, $\Delta {a_{v\_avg}} \le 0.02^\circ$. The second scanning parameter is already applicable to LiDAR.
Fig. 14. The selection region of the Lissajous scanning frequency (4 Pa). (a) fast-axis driven at 1.5 V. (b) slow-axis driven at 30 V.
Fig. 15. The Lissajous scanning of MEMS mirror at different frequency ratios. (a) ${f_x}:{f_y} = 8449:3212$, ${n_x}:{n_y} = 167:63$. (b) ${f_x}:{f_y} = 8517:3213$, ${n_x}:{n_y} = 194:73$. (c) ${f_x}:{f_y} = 8536:3212$, ${n_x}:{n_y} = 283:107$. (d) ${f_x}:{f_y} = 8493:3211$, ${n_x}:{n_y} = 447:169$.
Table 1. Resolution of the Lissajous scanning of MEMS mirror with different frequency ratios
4. Conclusion
We provide a general calculation method for the $FF$ values of the Lissajous scanning and further redefine the center angular resolution ($\Delta {a_h}$ and $\Delta {a_v}$) based on the vertical network pixels. The frequency response and scanning angle of the self-developed piezoelectric MEMS mirror are characterized, and the results show that the mirror exhibits nonlinear scanning phenomena (spring hardening dominance), which are mainly influenced by the driving voltage and Q factor of the mirror. Based on the above characterization results, we have delicately chosen the process and several parameters to drive the mirror well. Finally, through frequency selection and phase modulation, an angular resolution at the center better than 0.16° ($\Delta {a_h}$) × 0.03° ($\Delta {a_v}$) and an average angular resolution better than 0.10° ($\Delta {a_{h\_avg}}$) × 0.02° ($\Delta {a_{v\_avg}}$) in the FoV of about 35° (H) × 16.5° (V) was achieved. In addition, a feasible route of resolution adjustable Lissajous scanning with optimization of frequency ratio and phase modulation for different application scenarios is achieved, which is helpful for the design and characterization of dual-axis resonant MEMS mirror for LiDAR application.
Funding
Science and Technology Commission of Shanghai Municipality (HCXBCY-2021-044); R&D Program of Scientific Instruments and Equipment, Chinese Academy of Sciences (YJKYYQ20190026); National Key Research and Development Program of China (2021YFB3202500).
Disclosures
The authors declare no conflicts of interest.
Data availability
No data were generated of analyzed in the presented research.
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