Abstract
This paper analyzes anti-phase parametric excitation for a resonant MEMS mirror by independently driving two out-of-plane electrostatic comb-drive actuators positioned at the left and right sides of the MEMS mirror, enabling a fast and reliable start-up from zero amplitude. Both the angular derivative of the comb drives’ capacitance and the square wave driving signals are approximated by complex Fourier series, leading to a nonlinear model that describes the slow evolution of the amplitude and the phase of the MEMS mirror. The proposed model is validated through measurements, demonstrating strong agreement with the analytical results. A detailed discussion on injected and dissipated energy provides an intuitive understanding of the response curve for in-phase and anti-phase excitation signals with various duty cycles. Additionally, the initial start-up behavior of conventional in-phase parametrically excited MEMS mirrors is analyzed and compared to that of MEMS mirrors operated with anti-phase excitation, revealing an improvement of the start-up time by a factor between 8 to 50, depending on the operating point and condition.
Subject terms: Electrical and electronic engineering; Mechanical engineering; Electronics, photonics and device physics
Introduction
Micro-electromechanical systems (MEMS) are an intriguing technology for laser beam scanning (LBS) applications. This technology enables the fabrication of resonant MEMS mirrors, which are regarded as a promising solution for LBS systems due to their high resonant frequencies, compact size, large scan angles, low power consumption, and extended lifetimes resulting from minimal mechanical wear1. They are particularly suitable for application requiring continuous and periodic scanning, ranging from LiDAR2,3 over laser projection displays for Augmented Reality applications4–6 to biomedical applications such as Optical Coherence Tomography7,8.
Among various actuation methods, electrostatic actuation is the most widely discussed for MEMS mirrors, primarily due to its compatibility with conventional CMOS fabrication processes1. Electrostatic actuation typically employs comb drives that generate a torque to actuate the MEMS mirror. This torque depends on the overlap area of the comb drives, which in turn varies with the angle of the mirror. As a result, typical resonant mirrors actuated by comb drives are operated by parametric excitation, including a nonlinear, angle dependent torque9–12. Comb drives can be categorized by the relative positioning of the stator and the rotor combs into out-of-plane comb drives, and in-plane comb drives. Out-of-plane comb drives feature an asymmetric shape, an offset or a tilt between the stator and rotor combs13,14, allowing both a wide-band quasistatic scanning motion and a resonant operation. In contrast, in-plane comb drives lack this offset and, therefore, cannot generate any torque at zero angle. An oscillation is still achieved from rest because of the instability of the equilibrium point with zero amplitude15,16 along with small, unavoidable manufacturing tolerances or external excitation from the environment. The start-up time of in-phase comb drives depends on the driving frequency and is significantly longer17, which can be reduced by incorporating an out-of-plane starting electrode15.
Accurate modeling of MEMS mirrors provides numerous advantages, serving as a crucial tool to guide the design of MEMS mirrors and design closed-loop controller to ensure robust performance under external vibration18. It also provides insights into the influence of various design parameters of the MEMS mirror, improving the optimization of MEMS mirrors during the development phase. However, simulating resonant MEMS mirrors with a high Q-factor is computationally intensive, due to the rapid oscillation of the mirror combined with the relatively slow evolution of its amplitude and phase, arising from the fact that the torque generated by the comb drive is relatively small. This challenge is often addressed by separating the system dynamics into fast and slow components using a multiple-scale analysis. Perturbation theory-based approaches have been successfully applied to resonant MEMS mirrors in numerous cases to efficiently analyze the influence of parameters16,19,20.
The contribution of this paper is the modeling and multiple-scale analysis of a resonant MEMS mirror with two individually controllable out-of-plane comb drives, and two separated driving signals for the left and right sides of the comb drive. In contrast to the Ref16., the comb-drive torques and driving voltages are represented by complex Fourier series, significantly reducing the required algebraic formulas and simplifying the derivation. The method of averaging21 is applied to the MEMS mirror’s equations of motion to obtain a model of the slow dynamics of the mirror, which is verified by measurements of a resonant MEMS mirror with high Q-factor, excited with in-phase and anti-phase driving signals of different duty cycles. The start-up behavior of two forms of parametric excitation, in-phase and anti-phase excitation, is analyzed and verified. Lastly, a model-based transition between anti-phase and in-phase excitation is investigated and validated experimentally.
Methods
Resonant MEMS mirror with double-layered stators
Figure 1depicts a conceptual diagram of the used MEMS mirror. The MEMS mirror consists of a rotor with an oscillating mirror surface, torsion bars, leaf springs, and comb drives on both sides of the mirror surface. It is a variant of the MEMS mirror design in Ref16,18,22,23., where its detailed design concepts are described in Ref22.. The current mirror surface’s diameter is around 1 mm to achieve a high resonant frequency, and is coated with aluminum for RGB lasers suitable for display applications. The mirror is packaged with a dome-glass window to prevent parasitic reflections24.
Fig. 1.
The conceptual drawing shows the asymmetry of the comb drive design, also visible in the comb drive capacitance C and angular derivative 
of the comb drive capacitance for the right side (
), left side (
), sensing combs (
) and the sum of left and right side.
The stator design is based on a double-layer structure, where the rotor and the bottom layer of the stator are connected as a common node, while the upper layer of the stator is used to apply a voltage and generate the torque, shown in Fig. 1(a). The inherent asymmetry, resulting from the offset between the stator’s and the rotor’s comb drive fingers, enables a pulling torque from the zero-angle position toward the maximum of the comb drive’s capacitance at around 1 degree, if only one side is actuated. Some comb drive fingers of the stator are allocated for the sensing of the mirror motion, measuring the displacement currents. The sensed signals provide crucial feedback used for closed loop control, such as phase, frequency, amplitude, and direction of the motion of the MEMS mirror23. The comb drive capacitance varies with the current angle 
of the MEMS mirror. The capacitance is determined by the overlapping area of the stator and rotor combs, highlighted with a red shadow in Fig. 1(a). Fig. 1(b) shows the capacitance curves 
and 
for the left and right side comb-drive and the corresponding angular derivatives, 
and 
.
The mirror is suspended by torsion bars and leaf springs, creating a nonlinear stiffness. The torsion bars suppress the piston mode of the MEMS mirror while the leaf springs maintain most of the restoring torque of the rotational movements with excessive geometric hardening. This progressive stiffening of the leaf springs causes a behavior similar to the Duffing oscillator16,21,22. This is beneficial for display applications, providing a large range of frequencies with high amplitude, which is necessary for the synchronization of two such mirrors25.
The equation of motion of the resonant MEMS mirror is11,16
 |
1 |
where I denotes the moment of inertia, c is the damping parameter, and 
, 
, and 
represent the nonlinear stiffness parameters as a polynomial function and 
denotes the derivative with respect to time t. 
denotes the torque generated by the comb drives given by
 |
2 |
where 
and 
is the driving voltage of the left side and the right side, respectively, and 
is the constant voltage used on the sensing combs, which are active at all time.
Parametric in-phase and anti-phase excitation
Typically, resonant MEMS mirrors driven by in-plane comb drives operate using a single square wave signal9,15,16,26. A square wave signal is significantly easier to generate by a digital circuit than other wave forms, allowing a small and cost-efficient realization of the driving circuit. The driving voltage is applied to the comb drives on both sides, leading to the term in-phase excitation. To achieve the highest amplitude, the driving voltage is applied at the peak amplitude, pulling the oscillating mirror surface toward the center. It is then turned off when the mirror angle reaches zero. The mirror surface then keeps moving due to the moment of inertia. When the peak amplitude is reached again, the driving voltage is reapplied, and the cycle repeats. As a results, the frequency of the actuation signal is twice that of the mirror’s oscillation frequency, as shown in Fig. 2.
Fig. 2.
Definition of the mirror oscillation and the phase 
of the mirror, i.e. the phase shift between the mirror oscillation and the driving signal. The time-normalized in-phase and anti-phase driving signals 
and 
with different duty cycles in percent are defined such that the driving signal is always switched off at the normalized time 
. In in-phase excitation, the left and right comb drives are driven by the same signal. In anti-phase mode, each driving signal is only high for at most one half-period of the MEMS mirror.
In contrast, during anti-phase excitation, only one of the two driving signal is active in each half-period. Fig. 2 illustrates the driving signals for the left and the right side under anti-phase driving. Due to the phase shift between the left and right side, the anti-phase excitation pulls the mirror alternately to the left and right side.
Figure 2 shows the time-normalized actuation signals 
and 
with various duty cycles for in-phase and anti-phase excitation with a normalized mirror oscillation period of 
. The mirror oscillation is defined as a cosine wave with amplitude a and phase 
. The duty cycle in percent is defined such that the switching off event occurs at 
. When the phase shift between the mirror angle and the driving signal 
, the energy injection is maximized. When 
, the comb drive is switched off before the mirror’s zero-crossing, while 
keeps the comb drive on while the mirror is moving away from the center, both reducing the injected energy.
Modelling of the resonant MEMS mirror with Anti-phase Excitation
To simplify the calculation, the mirror’s equation of motion is normalized using the normalized time 
, and the normalized angle 
, where 
is the maximum reachable amplitude. Substituting into Eq. (1), the equations transforms into
 |
3 |
where 
defines the normalized damping parameters, and 
and 
define the normalized nonlinear stiffness. All constants in front of the normalized torque 
are merged into a single scaling factor F. 
denotes the derivative of x with respect to the normalized time 
. The normalized values for this device are 
, 
, 
, and 
.
The angular derivative of the comb-drives is approximated as a complex Fourier series with 
coefficients by
 |
4 |
where 
. 
defines the period of the Fourier series, and is chosen to be slightly larger than 
to span the full reachable amplitude of the MEMS mirror. The Fourier approximation is only valid for 
. The complex-valued Fourier coefficients 
can be calculated by
 |
5 |
Since the shape of sensing combs is identical to the shape of the actuation combs, the angular derivative of the sensing combs can be represented by a ratio 
of the actuation combs. Then the angular derivative of the sensing capacitance can be expressed as
 |
6 |
Substituting this into (3) allows to merge the contribution of the sensing combs to the driving signals. The equivalent driving signals of the left and the right side comb drives can be approximated by a Fourier series with 
coefficients as
 |
7 |
where the addition of the sensing voltage only changes the value of the coefficient 
, since the sensing voltage is constant. The Fourier coefficient can be calculated by
 |
8 |
The total torque can then be approximated with the Fourier series as
 |
9 |
Using this approximation, a multiple scale analysis is performed, leading to a slow-flow model of the MEMS mirror, i.e. a system of two nonlinear differential equations for amplitude a and phase 
that don’t depend on the fast time 
. The derivation is based on perturbation theory, specifically on the method of averaging21. The detailed derivation is given in the supplementary material and the final result for 
is
 |
10a |
 |
10b |
where the nonlinear functions 
, 
include the coefficients of the Fourier series and are given by
 |
11a |
 |
11b |
where 
denotes the Bessel function of the first kind. The stability of this system can be analyzed by the Jacobian matrix, also given as supplementary material.
Start-up behavior at small amplitudes
To evaluate the start-up behavior of the MEMS mirror system, it is necessary to evaluate the slow flow model in () for amplitude 
. Substituting 
into (10a) allows the simplification to
 |
12 |
where 
and 
denotes the magnitude and the phase of the complex Fourier coefficient 
, respectively. A detailed derivation is given in the supplementary materials. Eq. (12) gives two important insights into the start-up behavior. First, the component of the Fourier series 
, which denotes to the component of the driving signal at the fundamental frequency, needs to be different from 0. For in-phase excitation, the frequency of the driving signal is twice the oscillating frequency, therefore in that case 
, leading to 
. This means, that for in-phase excitation the operating point with 
is an unstable equilibrium point16. Second, a difference between the left and right side of the comb drives, i.e. 
is necessary. Eq. (12) shows that for anti-phase driving, the amplitude starts to increase regardless of the frequency. This is not the case for in-phase excitation, where the mirror can only be started in a narrow frequency band16. (cf. frequency range marked with double-ended arrow in Fig. 6). An analytical derivation of the start-up frequency range is provided in the supplementary materials.
Figure 3 shows the start-up behavior of in-phase and anti-phase excitation. For each excitation scheme, two trajectories, starting from an amplitude of 3 millidegree and different phase values are included. At zero amplitude, the phase rapidly approaches the stable phase equilibrium at zero amplitude 
, and then the amplitude starts to increase, as indicated by the arrows of the quiver plot. The length of the arrows of the quiver plot is normalized, but the color of the arrows is used to indicate the absolute value of 
, i.e. the change in amplitude. For in-phase excitation, the absolute value of 
around 
is small, since 
is a saddle point, which results in a high start-up time. In contrast, for anti-phase excitation 
is large around 
, leading to a fast increase in amplitude, which is also indicated by (12).
Fig. 3.
Phase portrait with start-up trajectories for (left) anti-phase and (right) in-phase excitation, for two initial states with very small amplitude and different phase, starting with a frequency of 4985 Hz. In both cases, the phase first approaches the stable start-up phase 
before the amplitude increases, and then the amplitude increases, leading to the oscillations that would eventually converge to the operating point, indicated by the square on the bottom response curve.
Another way to analyze the behavior at small amplitudes is to linearize (3), i.e. replacing all occurring nonlinear dependencies of a with a linear approximation. The detailed derivation is also given in the supplementary materials. For in-phase excitation, at small amplitudes Eq. (3) becomes the well-known Mathieu Equation
 |
13 |
and 
indicate the offset and the slope of the comb drive signal at 
, given by
 |
14 |
Mathieu’s Equation can be analyzed by Floquet theory27, and the typical analysis also shows the mentioned frequency range where the in-phase excitation can lead to parametric oscillations. In contrast, for anti-phase excitation with 50% duty cycle Eq. (3) simplifies to a linear harmonic oscillator
 |
15 |
Experimental results and discussion
Start-up with anti-phase
Figure 4 shows measurement results, including the mirror amplitude during start-up with anti-phase and in-phase excitation with a frequency of 4970 Hz. Using conventional in-phase excitation results in a delay of around 460 ms before the mirror amplitude reaches the first peak. Furthermore, the start-up time is highly dependent on external factors such as vibration and temperature. Even in similar conditions, the start-up time is not consistent, as shown by the yellow shadow, indicating the standard deviation of 600 measured start-up trajectories. The start-up time of in-phase excitation in simulation is highly dependent on the initial angle. With an initial amplitude of amplitude of 
degree the measured start-up time was similar to the one obtained by the simulation. Anti-phase excitation, on the other hand, does not suffer from those issues, because torque is immediately generated, regardless of the mirror angle. This also achieves a very predictable start-up time. The derived model shows good agreement with the measured trajectory. Figure 5 shows the measured start-up speed, defined by the amplitude divided by the time to the first overshoot of the evolution of the amplitude. It is possible to see that anti-phase excitation achieves high start-up speeds for most operating point, except the ones close to the bifurcation points of the response curve. The startup-speed of in-phase excitation is significantly lower, needing more time and even achieving smaller amplitudes. For both excitation schemes higher voltages achieve higher start-up speeds, since the injected energy is higher. Depending on the selected operating point, an improvement by a factor reaching from 8 to 50 is possible.
Fig. 4.
Measured and simulated start-up of in-phase and anti-phase excitation with various duty cycles. In-phase excitation generally requires more time until the mirror begins to oscillate, while anti-phase excitation achieves a very fast and reliable initial oscillation, regardless of the duty cycle.
Fig. 5.
Measured start-up speed for different frequencies and excitation schemes with 50% duty cycle, measured by the amplitude divided by the time to the first overshoot of the trajectory.
Verification of the slow flow model
The slow-flow model described in () is verified through measurements of equilibrium points, which define the response curve of the MEMS mirror. The equilibrium points are computed in MATLAB with the command fsolve by solving the nonlinear equation for 
for a given amplitude or phase. It is important to note that, when solving for the equilibrium points of (), one variable of the set 
must be provided, while the nonlinear solver determines the remaining two variables.
The dynamic characteristics and response curves are measured by a custom-made testbench28. The testbench uses a 1-dimensional position-sensitive detector with a dedicated aligning procedure for accurate angle measurements. Figure 6 shows the measured response curves with an actuation voltage of 40 V for 4 different excitation conditions: In-phase excitation with 50% duty cycle, and anti-phase excitation with 25%, 45%, and 50% duty cycle. All the response curves show a top and a bottom response, characteristic for the Duffing equation21. The response curves display bifurcations, the most prominent beeing the jump from the bottom to the top response curve. Additional bifurcations arise due to the parametric excitation16. With in-phase excitation, the MEMS mirror only starts in a very specific frequency range15,16,20. In contrast, with anti-phase excitation, the response curves extend over the full frequency range, i.e. 
never represents an equilibrium point. However, the geometry of the comb drive of this mirror still causes a fast increase in amplitude at low frequencies, including a small unstable amplitude range in between. Therefore, in this region, additional bifurcations, i.e. amplitude jumps can be observed. All such jumps are indicated by arrows in Fig. 6.
Fig. 6.
Measured and simulated response curves for anti-phase driving and in-phase driving. For in-phase driving, the mirror amplitude can only increase from zero in a very specific range, marked with a double-ended arrow and a dotted purple line. A more extensive verification is included in the supplementary materials.
During anti-phase excitation, the top response curve can be reached by a frequency up-sweep from at low frequencies for a duty cycle of 45%, and 50%. However, for a duty cycle of 25%, and for in-phase excitation, this is not possible. In these cases, the top response curve can only be accessed by first operating on the bottom response curve, and reducing the driving frequency until the amplitude jump to the top response curve occurs.
The behavior of the phase during anti-phase excitation with high duty cycles, e.g. 45% or 50% shows two possible solutions for the phase, where both of them have a nearly identical amplitude, resulting in two different top response curves. For a duty cycle of 50%, both top response curves reach the same peak amplitude. For 45% duty cycle, one top response terminates sooner than the other, and the mirror phase jumps to the value of the other top response curve, as indicated by the arrows in Fig. 6. This behavior is analyzed further in the next Section.
Energy analysis of anti-phase and in-phase excitation
To analyze the behavior of the anti-phase excitation in detail, six different operating points are considered, as shown in Fig. 7(a). Operating points 1 and 2 (op.1 + op.2) represent two stable equilibrium points when the MEMS mirror is driven with 40 V and 50% duty cycle, at a frequency of 5 kHz. Similarly, operating point 3 and 4 (op.3 + op.4) correspond to the stable equilibrium of an actuation signal with 45% duty cycle. At a duty cycle of 25% only one operating point (op.5) exists. The last operating point (op.6) lies on the response curve of in-phase excitation with a duty cycle of 50%. Since for all operating points the mirror is oscillating at 5 kHz, all operating points have a similar amplitude, but the phase is different.
Fig. 7.
(a) Simulated amplitude-phase plot of anti-phase and in-phase excitation with 40 V peak voltage of driving signals. (b) Energy Plot for an amplitude of 2.37 degrees, for an arbitrary phase 
. The operating points are given by the intersection of the dissipated energy 
with the injected energy 
. (c) Power generated by the left comb drive 
(red solid line) and the right comb drives 
(blue solid line) with driving signal and angle of the mirror. The time-integral over the power, indicated by the colored areas, represents the injected energy by the driving signals with the phase of the corresponding operating point.
Figure 7(b) shows the driving signals and the mirror trajectories in time-domain for all operating points. Furthermore, the highlighted areas represent the energy injected by the comb-drive torque
 |
16 |
where 
is the total power generated by the comb-drive, given by the sum of the left and right side. The power is defined by torque of the left and right side 
and 
, respectively, and angular velocity
 |
17 |
The energy injected within one period is
 |
18 |
The behavior of this energy function for an arbitrary phase 
is shown in Fig. 7(b). Furthermore, the energy dissipated by the MEMS mirror within one period is given by
 |
19 |
with 
being the energy at the beginning of the period, given by the kinetic energy of the normalized oscillator as
 |
20 |
determined solely by the amplitude a and frequency 
. An equilibrium point is reached if the injected energy 
equals the dissipated energy 
, which is only the case for very specific phases 
. For some amplitudes, there are multiple stable solutions, while for others there is only one. With increasing amplitude a, the dissipated energy 
grows too. This results in a relative upwards shift of the dotted line representing the dissipated energy in Fig. 7(b). In that case, the operating point 3 (op.3) will be the first to vanish, and the system jumps to op.4. Furthermore, from Fig. 7(b) it can be concluded, that the maximum reachable amplitude of anti-phase excitation is significantly smaller than the one of in-phase excitation, in agreement with the results demonstrated in Fig. 7(a) and Fig. 6. Furthermore, the amplitude reachable with 50% anti-phase excitation is lower than the amplitude of anti-phase excitation at lower duty cycles. An intuitive explanation for this can also be found in Fig. 7(c). With a duty cycle of 50%, a shift in the phase does only slightly influence the injected energy, i.e. the colored area. On the other hand, looking at anti-phase excitation with 25% duty cycle (op.5), a change in phase drastically increases the injected energy, since the driving signal can be shifted such that the comb drive does not extract energy of the system. Compared to in-phase excitation, the reachable amplitude is still lower, since the left and right comb drive are used alternately, cutting in half the maximum injected energy per period.
Smooth transitioning between anti-phase and in-phase excitation
In-phase excitation can reach a larger oscillation amplitude, while anti-phase excitation provides a faster and more reliable startup. To use both benefits, it is necessary to transition between the two excitation schemes during operation without disturbing the mirror movement, which is possible with the response curves shown in Fig. 6. To do so, a transition point where the amplitude of both response curves is similar needs to be selected. At the switching event, the phase difference between the two operating points is applied to the driving signal.
Figure 8 shows this with an example, switching from op.4 to op.6 shown in Fig. 7(a). Without the phase adjustment of the driving signal, the mirror amplitude and frequency show oscillations of up to 
, and 
Hz, respectively, after the switching event performed at 
. If the phase is adjusted according to the model, the amplitude and frequency of the MEMS mirror remains nearly unaltered after the switching, only changing the amplitude by 
, which corresponds to a reduction by 92.5%. The smooth transitioning between excitation schemes allows the combination of benefits of different driving signals, for example the combination of anti-phase excitation to quickly obtain an initial oscillation, and the subsequent switching to in-phase excitation with a DAsPLL for optimal closed-loop startup26,29.
Fig. 8.
Smooth transitioning between the anti-phase excitation and in-phase excitation. The mirror operation is switched from op.4 to op.6, indicated in Fig. 7(a). (a) The mirror angle and the driving signals of the left and right sides when switched with the correct phase adjustment. (b) Evolution of amplitude and mirror frequency, for one measurement performed without any phase adjustment, and one measurement with correct phase adjustment. In the latter case, the amplitude and frequency of the mirror remain nearly unaltered.
In summary, the demonstrated results provide a model-based description why anti-phase excitation allows a faster start-up of the MEMS mirror. Furthermore, the derived model allows the calculation of the equilibrium points, enabling a smooth transition between different excitation schemes.
Conclusion
This paper proposes a faster start-up procedure for resonant MEMS mirrors by separating the out-of-plane comb drive between the left and right sides and utilizing anti-phase excitation, i.e. operating the left and right side of the comb drive with a 180-degree phase shift. An accurate model, based on four complex Fourier series - one for the angular derivative of the comb drive and another for the driving signal - is derived. Measurements of the response curves show excellent agreement with the theoretical model. The behavior of anti-phase excitation is analyzed by considering the injected and dissipated energy of the MEMS mirror during one oscillation period, leading to an intuitive understanding of the observed behavior, reachable amplitudes, and possible operating points.
The model also enables a comparison of start-up times between conventional in-phase driving and the proposed anti-phase driving. Simulation results of both start-up behaviors are presented and validated through experiments, showing good agreement between measurements and simulations also for transient behavior. Additionally, the model allows the calculation of phase differences between different driving signals, making it possible to switch between excitation schemes during operation without causing any distortion of the mirror motion.
Because of the generality of the Fourier series approximations of the comb drives and the driving signals, the proposed analysis can be extended to more complex comb drive structures and driving signals. This generality may also allow various application domains such as RF MEMS oscillators30 and capacitive micromachined ultrasonic transducers31.
Supplementary Information
Author contributions
F.R. derived the results and conducted the experiments. H.Y., D.B., and G.S. guided the research process and analyzed the results. All authors reviewed the manuscript.
Funding
This work has been supported by the federal ministry of innovation, mobility, and infrastructure (BMIMI) of the Republic of Austria under the scope of the PercAR project (FFG project number FO999915248) in the program “Mit Regulierung und Souveränität zur Innovation - Digitale Technologien 2023”,authorized by the Austrian Research Promotion Agency (FFG). The authors acknowledge the TU Wien Bibliothek for financial support through its Open Access Funding Programme.
Data availability
The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.
Declarations
Competing interests
The authors declare no competing interests.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary Information
The online version contains supplementary material available at 10.1038/s41598-026-39623-z.
References
- 1.Holmström, S. T. S., Baran, U. & Urey, H. MEMS laser scanners: A review. J. Microelectromechanical Syst.23, 259–275 (2014). [Google Scholar]
- 2.Wang, D., Watkins, C. & Xie, H. Mems mirrors for lidar: A review. Micromachines11 (2020). [DOI] [PMC free article] [PubMed]
- 3.Yoo, H. W. et al. MEMS-based lidar for autonomous driving. e & i135, 408–415 (2018).
- 4.Hung, A.C.-L., Lai, H.Y.-H., Lin, T.-W., Fu, S.-G. & Lu, M.S.-C. An electrostatically driven 2D micro-scanning mirror with capacitive sensing for projection display. Sens. Actuator A Phys.222, 122–129 (2015). [Google Scholar]
- 5.Noui, L. & Reitterer, J. Trilite addresses challenges with projection display technologies for ar glasses. Information Display37, 12–16 (2021). [Google Scholar]
- 6.Milanovic, V., Kasturi, A. & Hachtel, V. High brightness mems mirror based head-up display (hud) modules with wireless data streaming capability. Proceedings of SPIE - The International Society for Optical Engineering9375 (2015).
- 7.Strathman, M. et al. Mems scanning micromirror for optical coherence tomography. Biomed. Opt. Express6, 211–224 (2015). [DOI] [PMC free article] [PubMed] [Google Scholar]
- 8.Gorecki, C. & Bargiel, S. Mems scanning mirrors for optical coherence tomography. Photonics8 (2021).
- 9.Shahid, W. et al. Modeling and simulation of a parametrically resonant micromirror with duty-cycled excitation. J. Microelectromechanical Syst.23, 1440–1453 (2014). [DOI] [PMC free article] [PubMed] [Google Scholar]
- 10.Ataman, C. & Urey, H. Modeling and characterization of comb-actuated resonant microscanners. J. Micromech. Microeng16, 9–16 (2006). [Google Scholar]
- 11.Brunner, D., Yoo, H. W., Thurner, T. & Schitter, G. Data based modelling and identification of nonlinear SDOF MOEMS mirror. In Piyawattanametha, W., Park, Y.-H. & Zappe, H. (eds.) MOEMS and Miniaturized Systems XVIII, vol. 10931, 1093117. International Society for Optics and Photonics (SPIE, 2019).
- 12.Izawa, T., Sasaki, T. & Hane, K. Scanning micro-mirror with an electrostatic spring for compensation of hard-spring nonlinearity. Micromachines8 (2017). [DOI] [PMC free article] [PubMed]
- 13.Hah, D., Patterson, P., Nguyen, H., Toshiyoshi, H. & Wu, M. Theory and experiments of angular vertical comb-drive actuators for scanning micromirrors. IEEE Journal of Selected Topics in Quantum Electronics10, 505–513 (2004). [Google Scholar]
- 14.Chae, D., Jeong, D.-h., Yun, S.-j., Jo, K.-w. & Lee, J.-H. Quasi-statically actuated mems scanner with concentric vertical comb electrodes. Micro and Nano Systems Letters12 (2024).
- 15.Schenk, H. et al. Large deflection micromechanical scanning mirrors for linear scans and pattern generation. IEEE Journal of Selected Topics in Quantum Electronics6, 715–722 (2000). [Google Scholar]
- 16.Yoo, H. W., Albert, S. & Schitter, G. Fourier series-based analytic model of a resonant mems mirror for general voltage inputs. J. Microelectromechanical Syst.30, 343–359 (2021). [Google Scholar]
- 17.Schenk, H. Ein neuartiger Mikroaktor zur ein-und zweidimensionalen Ablenkung von Licht. Ph.D. thesis (2000).
- 18.Yoo, H. W. et al. Experimental evaluation of vibration influence on a resonant mems scanning system for automotive lidars. IEEE Trans. Ind. Electron.69, 3099–3108 (2022). [Google Scholar]
- 19.Denis, V., Jossic, M., Renault, A., giraud audine, C. & Chomette, B. Identication of nonlinear modes using phase-locked-loop experimental continuation and normal form. Mechanical Systems and Signal Processing106 (2018).
- 20.Rhoads, J. F. et al. Generalized parametric resonance in electrostatically actuated microelectromechanical oscillators. Journal of Sound and Vibration296, 797–829 (2006). [Google Scholar]
- 21.Nayfeh, A. Introduction to Perturbation Techniques (Wiley, 1981).
- 22.Yoo, H. W., Albert, S. & Schitter, G. Accurate analytic model of a parametrically driven resonant mems mirror with a fourier series-based torque approximation. Journal of Microelectromechanical Systems29, 1431–1442 (2020). [Google Scholar]
- 23.Brunner, D., Albert, S., Hennecke, M., Darrer, F. & Schitter, G. Self-sensing control of resonant mems scanner by comb-drive current feedback. Mechatronics78, 102631 (2021). [Google Scholar]
- 24.Hofmann, U., Janes, J. & Quenzer, H.-J. High-q mems resonators for laser beam scanning displays. Micromachines3, 509–528 (2012). [Google Scholar]
- 25.Strasser, A., Stelzer, P., Steger, C. & Druml, N. Towards synchronous mode of multiple independently controlled MEMS mirrors. IFAC-PapersOnLine52, 31–36 (2019). 8th IFAC Symposium on Mechatronic Systems Mechatronics 2019.
- 26.Brunner, D., Yoo, H. W. & Schitter, G. Digital asynchronous phase locked loop for precision control of moems scanning mirror. IFAC-PapersOnLine52, 43–48 (2019). 8th IFAC Symposium on Mechatronic Systems Mechatronics 2019.
- 27.Kovacic, I., Rand, R. & Sah, S. M. Mathieu’s equation and its generalizations: Overview of stability charts and their features (review article). Applied Mechanics Reviews70, 020802 (2018). [Google Scholar]
- 28.Yoo, H. W., Brunner, D., Thurner, T. & Schitter, G. Mems test bench and its uncertainty analysis for evaluation of mems mirrors. IFAC-PapersOnLine52, 49–54 (2019). 8th IFAC Symposium on Mechatronic Systems Mechatronics 2019.
- 29.Roscher, K.-U., Fakesch, U., Schenk, H., Lakner, H. K. & Schlebusch, D. Driver ASIC for synchronized excitation of resonant micromirrors. In Urey, H. (ed.) MOEMS Display and Imaging Systems, vol. 4985, 121 – 130. International Society for Optics and Photonics (SPIE, 2003).
- 30.Zega, V. et al. Numerical modelling of non-linearities in mems resonators. Journal of Microelectromechanical Systems29, 1443–1454 (2020). [Google Scholar]
- 31.Pekař, M. et al. A fluid-coupled transmitting cmut operated in collapse mode: Semi-analytic modeling and experiments. Sensors and Actuators A: Physical267, 474–484 (2017). [Google Scholar]
Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Supplementary Materials
Data Availability Statement
The datasets used and/or analysed during the current study are available from the corresponding author on reasonable request.