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The ability to actively shift the primary resonance of a 2D scanning micromirror allows the user to set the scanning direction, set the scanning frequency, and lift otherwise degenerate modes in a symmetrically designed system. In most cases, resonant scanning micromirrors require frequency stability in order to perform imaging and projection functions properly. This paper suggests a method to tune the tip and tilt resonant frequencies in real time while actively suppressing or allowing degeneracy of the two modes in a symmetric electrothermal micromirror. We show resonant frequency tuning with a range of degeneracy separation of 470 Hz or by approximately ±15% and controllable coupling.
© 2017 Optical Society of America
1. Introduction
Microelectromechanical systems (MEMS) have enabled the miniaturization of numerous optical technologies such as projection systems [1–5] and medical imaging components [6–8]. Projection and imaging systems work by scanning a micromirror in the form of a raster, using a fast axis and perpendicular slow axis simultaneously to form a uniform projection area, or by using a Lissajous scan. The Lissajous scan is performed by exciting a bi-directional mirror at resonance along two perpendicular axes.
Lissajous scanners are more prominent in the imaging community as such systems can provide high resolution images. However, the required computations in a Lissajous imaging system are greater than their raster counterparts and, thus, depend upon advanced software and hardware techniques [9]. The effectiveness of 2D MEMS in biomedical imaging has been demonstrated in numerous research findings [10–12], where micromirrors are at the core of the technology. MEMS have also been used as a means to deflect fiber optics in order to scan an image. Park et al. [13] mounted fiber optics to a MEMS and tuned the scanning MEMS by adding or removing mass until the Lissajous scan covered the majority of the scan area. Similarly, Seo et al. used an electrothermal hot-cold arm MEMS to scan an attached fiber optic at two perpendicular resonance frequencies [14]. Another method of forming in-situ the necessary pattern is by using internal mechanical structures to switch from one axis to the perpendicular axis [15]. In all cases in which a Lissajous pattern is desired, the mechanical frequencies must be separated and tunable in order to optimize the uniformity of the scanning area. Display and imaging applications could benefit from a continuously tunable resonant mirror where the oscillation of the mirror is no longer confined to a static frequency band and where coupling between mechanical resonance modes can be precisely controlled. For actuators, such as those in scanning micromirror and fiber systems, the tunability is useful in providing linearity and either coupling or decoupling mechanical modes in order to suppress or enhance the mechanical response. In many cases, in-situ tuning is employed by providing capacitive forces to alter the effective spring constant of a resonator [16]. Mechanical restriction by applying force to specific springs using electrothermal actuators may be used to tune the resonant behavior of a MEMS device [17,18]. Other techniques involve using electrothermal actuators to alter the relative positions of comb drive actuators [19], effectively changing the capacitive forces between the static comb digits and those connected to the mirror.
This paper focuses on the characterization of an electrothermal micromirror described by Morrison et al. [20] under small amplitude harmonic drive signals. At low drive amplitudes, the system behaves as a linear harmonic oscillator. An in-depth investigation into the shifting resonance peaks demonstrates that the mechanics of the system can be predictably tuned using independently actuated thermal bimorphs. We show a method to provide a wide range of frequencies that can be specifically matched or decoupled, lifting and permitting degeneracy in more than one dimension. This level of flexibility provides an enormous amount of control over the direction of optical deflection and sensitivity of the scanning system. The predictive tunability opens up an entirely new domain for controlling and optimizing a scanning micromirror system.
2. Device and actuation principle
The mirror is fabricated using the PolyMUMPs process [21] by MEMSCAP consisting of one immobile polysilicon layer, two mechanical polysilicon layers, and a gold layer. In this process the polysilicon layers may all be joined or separated by etching a sacrificial oxide layer prior to the subsequent polysilicon deposition. By joining the gold layer to a fixed-free polysilicon beam, a stress gradient due to the fabrication process provides a bending moment that can be leveraged to form an electrothermal actuator. A central mirror is placed on a platform levitated above the substrate by four such thermal bimorphs. Four serpentine springs couple the platform mechanically to the bimorphs. The platform constitutes the mirror itself with the gold layer acting as a reflective surface. The design is derived from reference [20] where segmented polysilicon and gold actuators a allow a variable focus actuation atop the mirror platform. However, for this study the mirror need not be varifocal, thus the platform and segments are merged structurally with the platform below to provide a relatively flat reflective surface.
An SEM image of the mirror and actuation components is shown in Fig. 1(a). Each of the U-beam bimorphs are connected to two bonding pads (out of view in the image) in order to provide an electrical bias along the bimorph length. In this case, the actuators are controlled using joule heating and have a continuous actuation range in which the level of actuation is dependent on the electrical bias. The device is annealed prior to actuation to 175 C to increase the initial residual thin film stresses and enhance the bending moment at room temperature. Current is delivered through the bimorph as a means to generate heat. Due to the low impedance of 2.5 Ω, the device is current biased to reduce the effects of non-zero line resistance. However, a constant current requires more finesse and supervision at high electrical power due to the positive temperature coefficient of resistivity. Care must be taken to avoid thermal runaway since the power is proportional to the square of the resistance.
Fig. 1 (a) SEM image of micromirror derived from [20] and (b) the angular deflection as a function of differential power to the bimorph legs.
It can be shown analytically [22] that the dissipated power is not proportional to the deflection of the tip of each beam. For this design, the power is, however, approximately proportional to the optical deflection angle of the mirror platform as seen in Fig. 1(b). The differential power at a constant offset power for all bimorph legs is plotted with the absolute value of the deflection angle for a full range scan.
3. Resonator frequency response
The frequency response for most MEMS devices can be characterized by approximating the MEMS as a spring-mass system given by:
where m is the effective mass of the device, γ is the dissipation factor, F(t) is the time dependent force, and k is the effective spring constant. For this analysis, the device is considered to be a single resonant structure with multiple modes rather than treating it as multiple resonators coupled through the bimorph springs. In addition, the analysis is performed assuming the conversion from linear coordinates to a torsional spring-mass system. In reality, additional inertial terms will contribute to the overall response but for now they can be ignored.The magnitude of the solution of the spring-mass system is given as:
where the parameters have been redefined such that the resonant frequency is defined by and the quality factor is . The eigenvalue frequencies can be determined by examining each spring (namely the serpentine springs and the bimorph U-beam actuators themselves). It is possible to calculate the compliance matrix for the serpentine springs and U-beams using techniques outlined by reference [23]. For this use case, however, the eigenvalue matrix would require six degrees of mechanical freedom. The interactions between the deflections both rotational and translational are significant. Instead, FEM simulations can take into account the residual stresses within the layers to provide both the vertical lift from the stresses and the resonant modes linearized about the projected position of the mirror.Figure 2 shows the mode shapes of the first three resonance modes simulated using COMSOL Multiphysics®. The gold and polysilicon layers are given an initial in-plane stress value until the calculated vertical deflection reflects experimental measurements nearing 200 μm. The resonant modes are simulated to be 1.24 kHz for the piston mode and approximately 2.02−2.07 kHz for the degenerate tip and tilt modes. These values agree with experiment to within 35% for the first and 40% for the second and third modes. The variation between the two simulated frequencies of the degenerate modes is an artifact of asymmetries in the formulated mesh and of relaxed convergence requirements set in order to return timely solutions. While asymmetries may exist for other reasons in the actual device (such as those incurred by thermal cycling of bimorph actuators), the simulation assumes a perfectly symmetric system. The simulations are provided for a qualitative comparison regarding the approximate resonance frequencies and their corresponding directionality or shape. Because the simulations are meant to be supplemental and qualitative in nature, a 35% to 40% agreement is sufficient. It can be improved with a refined mesh, fine tuning the thermo-mechanical coupling approximations, more precisely matched component geometries, including air damping effects, and iterative nonlinear techniques to better solve multiply coupled device physics with a nonlinear geometry. Any further investigation is beyond the current scope of this paper.
Fig. 2 FEM simulations of a mirror with in-plane residual stresses. The modes are given as a) vertical or piston, b) tip, c) tilt. Note that the direction of the tip and tilt are mirror images.
3.1. Driving scheme
To activate the angular scanning mode, the mirror is actuated using a differential bias. In lieu of a feedback loop to control the power bias, an I–V set of data is used as a look up table for the electrical power consumption under DC actuation. The lookup table is created by performing an initial I–V sweep and fitting the powers to a cubic polynomial. The biases for each power are then extrapolated from the fit shown in Fig. 3.
Fig. 3 Fits to the power as a function of current using quadratic and cubic functions. The quadratic fit does not fall within the error bars.
The quadratic fit yields residuals as large as 2.5 mW which shifts the deflection angle, increases measurement errors, and increases the likelihood of damaging the mirror by biasing beyond the damage threshold. The cubic fit requirement is indicative of the non-ohmic response of the bimorph legs and remains within the measurement error bounds shown in the figure.
A current biased system set explicitly to obtain a specific dissipated power will only work in static configurations. The thermal time constant of the device ultimately governs the relationship between the measured amplitude and average power at a given frequency. The non-Ohmic response of the bimorph U-beam requires that the minimum DC electrical power on all four bimorphs needs to be shifted close to 5 mW, around which current biased oscillations are near symmetric in power amplitudes. The amplitude of oscillations for frequency sweeps are small compared to the offset required to ensure a relatively linear temperature oscillation. Low current amplitudes allow us to approximate the oscillations of the temperature to be small enough to ignore the resistance oscillations. For drive frequencies much greater than the thermal time constant of the mirror, the resistance oscillations will not be significant as the U-beams will not have sufficient time to heat and cool. However, the average resistance will increase as the average temperature changes based on the root-mean-square of the power amplitude. The power scheme is implemented as is shown in Fig. 4 although the actual value of Pamp drops as the period of oscillation falls below the thermal time constant. For this design, the thermal time constant is approximately 2 ms and frequencies above 80 Hz yield significantly smaller deflections than would a DC drive. The drop in amplitude can be modeled using a one pole, one zero transfer function with one zero similar to what is described in [24].
Figure 5 depicts a frequency sweep of the device fitted to the two pole transfer function. The fit to the mechanical response consistently yields z0 ≈ 250 Hz and p̃0 ≈ 80 Hz. Although the transfer function has a large gradient at low frequencies, at frequencies above approximately 1 kHz the vertical shift is negligible. Between 1.3 kHz and 1.7 kHz the mechanical deflection oscillation amplitude decreases by less than 3%. Thus, at frequencies much greater than the thermal bandwidth, the transfer function is approximated to be independent of frequency. Because the mechanical oscillations do not decrease significantly in amplitude, the measurements at frequencies beyond 1.3 kHz all have approximately the same temperature amplitude of oscillation.Fig. 4 Two legs are biased at an offset power and a differential power amplitude while the opposite legs are biased by a separate constant tuning power. The inset shows the effect of increasing the tuning power on the spring shape.
The purpose of the differential drive and two DC power offsets (namely Poffset and Ptuning) is to sweep through the offset and tuning powers while providing a constant amplitude power. Systematically sweeping through the power biases based on the I–V curve will yield insight into the range of frequencies at which the device will resonate. In addition, the form of the resonant mode will be interpreted to determine how variations in the initial deformation impact the resonance frequencies.
By varying the offset biases along each axis, the amount of strain energy in the springs and temperature of the springs and bimorphs will differ appreciably. The deformation of and temperature change in the serpentine springs and attached bimorphs will shift both the resonant frequencies and the mode shapes because the effective spring constants will not be symmetric along the two axes. However, this presents a secondary shift in the vertical position of the mirror. The range corresponding to the offset powers discussed in this paper is approximately 100 μm. The optical design surrounding the scanning system needs to take into account this range or maintain a subset of tunable parameters in order to minimize vertical displacements.
3.2. Electronics and readout system
A position sensitive detector (PSD) is used to measure the frequency response of the micromirror. A collimated laser diode is directed such that the angle of incidence on the mirror is at 45° and the reflection from the mirror is directly incident on the PSD. The 20 mm by 20 mm square detector is composed of two sensors perpendicularly aligned to one another. Each of the perpendicular sensors outputs two voltages, one corresponding to the total light measured and another corresponding to the position of the beam along that axis. By monitoring all four outputs, the position of the beam on the sensor in two dimensions can be tracked in real time.
Typically, the distance from the mirror to the PSD is 5 − 7 mm to maximize the measurable optical reflection angle. An overview of the measurement flow is outline in Fig. 6. The input forces (electrical powers) are as described in Section 3.1 where all inputs are current sources by design. The differential drive output is maintained exactly out of phase at the source.
Fig. 6 The frequency response is measured using a differential current bias output corresponding to a specified DC power amplitude and a tuning power offset to the bimorph legs driving the opposite axis. The reflection from the mirror is recorded by reading the output of a PSD over multiple periods and converting the cartesian coordinate into two angles.
For a two dimensional measurement, it is important to measure all four values simultaneously so that each measured point on the x-y plane of the PSD can be used to calculate the angle. If the direction of the deflection is constant at any angle such that the scan is a perfect line, the phase difference between the x- and y- signals is exactly zero. For this measurement only, the AC amplitudes of each independent axis can be measured and combined. Conversely, most of the measurements within this experiment require point by point evaluation in order to determine the magnitude of the angle as a function of time. The right hand side of Fig. 7 shows two parametric plots of harmonic signals given as
The reflection from the mirror, representing the mechanical response, is given by a Lissajous curve in the form of either a line or an oval as shown on the right side of Fig. 7. As each frequency sweep is performed, the output of the PSD is continuously measured by directly reading out the oscilloscope probes into LabVIEW. The X and Y coordinates are converted piece-wise into angles while the relative phase between the two signals, ϕ, records the eccentricity of each rotation. From this, the magnitude of the angle as a function of time is determined and the maximum and minimum angle magnitudes yield the major and minor axis at each frequency.Fig. 7 The normalized angle magnitude is calculated analytically for two possible phase values in order to convey measurement variables. The plot is not a measurement but a representation of how the variables are extracted. The maximum and minimum angles are depicted for and and shown as arrows in the parametric plot inset.
Figure 7 shows two calculated examples of a signal normalized to the unit circle where the phase difference between the X and Y signals is π/3 and π/9. The mode shape frequency dependence is found by measuring the maximum and minimum angle magnitudes and directions while sweeping the frequencies with a function generator. The dominant mode is defined as the mode corresponding to the maximum angle magnitude. The ratio of the X and Y signal amplitudes provides the direction of the dominant mode while the phase provides the quadrant. When ϕ < π/2, the angle is a measure from the positive x-axis and when π/2 < ϕ < π, the angle is measured from the negative x-axis.
4. Separation of degenerate modes
The full range of possible resonant mode shapes is obtained by iterating through offset and tuning powers, Poffset and Ptuning while keeping the peak-to-peak amplitude power constant with Pamp = 0.5 mW. Figure 8 shows the mechanical response for two frequency sweeps near the tip and tilt resonance peaks fitted with the superposition of two Lorentzian functions given as Equation 2. The figure identifies the change in frequency response for two values of ΔP = Poffset − Ptuning. Figure 8(a) is the response for Poffset = 9 mW and Ptuning = 10 mW. The inset is a FEM simulation with approximately the same power on all four bimorph legs, yielding similar shapes and temperatures. Where the two peaks overlap, the eccentricity of the mode shape is minimized and the mode closely resembles a circle. The shape is shown in Fig. 8(c) based on the measured angle, phase, and direction of θmax for five distinct frequencies. The frequencies are also labeled in Fig. 8(a) by the stars and dotted lines using the color of the legend in Fig. 8(c). At 1.44 kHz the phase is nearly ϕ = π/2. As the frequency is swept, both the phase and direction change continuously. The data is displayed for Poffset = 5 mW and Ptuning = 16 mW in Figs. 8(b) and 8(d). The eccentricity at the two resonance peaks is much greater and the resonance frequencies for this measurement when ΔP = −11 mW have separated significantly compared to when ΔP = −1 mW. The parametric plots in (c) and (d) have the same units and a square aspect ratio, though the specific value is unnecessary and changes based on how far the PSD is from the mirror.
Fig. 8 Lorentzian fits to the magnitude of the dominant resonance angle for (a) nearly equal powers (Poffset − Ptuning = −1 mW) and (b) unequal powers (Poffset − Ptuning = −11 mW). The FEM insets show the approximate spring deformations for the two scenarios. Below the Lorentzian fits, the parametric plots in (c) and (d) correspond to the mode shape at key frequencies labeled in (a) and (b) by the stars and dotted lines.
The tunability is continuous because of the analog nature of the tuning and offset powers. The experiment is run to sweep in integer intervals for every combination of Poffset and Ptuning for 5 mW ≤ Poffset ≤ 16 mW and 5 mW ≤ Ptuning ≤ 16 mW. As the powers are swept the point of degeneracy decreases in frequency further demonstrating the effect of serpentine spring tension on the overall response. As the bimorph legs are actuated by nearly the same amount (5 mW ≤ Poffset ≈ Ptuning ≤ 15 mW), the rotation in all four springs decreases and the temperature increases leading to a drop in the degenerate mode frequency from approximately 1.465 kHz to 1.435 kHz where the original resonance (Poffset = Ptuning ≈ 0 mW) is 1.479 kHz when the tension in all four springs is at a maximum and the temperature is at a minimum.
Contour plots showing each frequency sweep for a given Poffset are shown in Fig. 9. The color represents the magnitude of the total angle, the horizontal scale is the frequency value, and the vertical scale is the tuning power. For low offset powers, the point of degeneracy is at the same tuning power. As the tuning power is shifted further from the offset power, the separation of the modes grows.
Fig. 9 Magnitude of the dominant resonance angle as a function of frequency and tuning power is shown for various offset powers.
In the interest of understanding how these modes shift as a function of the offset and tuning powers, the frequencies obtained by fitting the magnitudes to two Lorentzian peaks are tracked. Furthermore, the direction can be extrapolated from the relative amplitudes of the X and Y signals while the quadrant is determined by the the phase, ϕ, as defined in Fig. 7. A standardization is adopted to index the peaks and is defined by the mode shape. Let us define the primary mode as the mode with when the resonance frequency is below the degeneracy point and the secondary when when the frequency is below the degeneracy point. This defines the modes by the axis (set of springs) of rotation.
Physically, the strain in the springs should be exactly the same for flipped Ptuning and Poffset values but with the polarization flipped about . Figure 10(a) demonstrates the shift in the primary and secondary frequency values against the difference in the offset and tuning powers. It should be noted that the spread in frequency can be utilized to stabilize any variations in the resonant frequency. The offset power would then be used to compensate for any variations in device stiffness due to parameters such as self-annealing and ambient temperature fluctuations. The primary resonance begins at low frequency and increases as the the difference (defined by ΔP = Ptuning − Poffset) increases in value. The secondary resonance is the mirror image about ΔP = −0.7 mW. The offset in the point of symmetry may be due, in part, to self annealing, but the majority of the shift is most likely due to the root-mean-square (rms) temperature at high frequencies. The thermal filter implies the oscillations in temperature due to an input current will be offset by the rms temperature for periods much less than the thermal time constant.
Fig. 10 Primary (green) and secondary (blue) resonance frequencies are plotted in (a) as a function of ΔP and the difference between the primary and secondary resonance frequencies Δf shown as a function of ΔP in (b). The frequency separation can be tuned from 230 Hz to −240 Hz.
The symmetry, once again, in Fig. 10(a) is the motivation for an inspection of the difference between the primary and secondary resonance peaks. The spread in the frequencies can be accounted for by the overall spring constant decreasing as the total power increases. In effect, the total tension and temperature value, and thus the degeneracy point, will shift as the average total power is increased. By subtracting the primary and secondary resonance measurements for each ΔP, the overall reduction in spring tension is removed from the results. Figure 10(b) shows a plot of Δf = fPrimary − fSecondary as a function of ΔP for each of the fitted peaks. A pattern emerges showing the spread of the primary and secondary resonance modes falls perfectly in line regardless of the average total power or degeneracy point for the given power scheme. From this, the separation range can be extrapolated and is tunable from 230 Hz to −240 Hz yielding a full range of 470 Hz. The phase for the primary (green) and secondary (blue) resonance peaks is measured for each ΔP. The result is then plotted against the difference in frequency (shown in Fig. 11). The implications of a continuously changing phase are in the physical interpretation of ϕ. Since the phase ultimately represents the level of coupling between the primary and secondary modes, a set of offset and tuning powers can be chosen to shift the frequencies and simultaneously lift or permit degeneracy.
Fig. 11 Phase measured at the primary (green) and secondary (blue) resonance frequencies plotted against the difference in frequency.
5. Conclusion
This work outlines the frequency response of a damped, driven harmonic oscillator for a sinusoidal input force. The device is modeled using FEM in order to approximate the mechanical resonance frequency mode shape for the piston and tip/tilt degenerate eigenmodes. A series of frequency sweeps shows the separation of degenerate modes is substantial and can be tuned by shifting the relative powers in the bimorph legs. From this set of data, a pattern emerges in both the frequency values and the overall coupling between the two modes. The continuous range of offset and tuning powers provides a means to separate or enhance degeneracy which can be used to change the reflected Lissajous curve from a straight line along any direction to a circle by changing introducing specific power biases for each axis. Further work needs to be performed to understand the implications of the power bias offsets on the amplitude and quality factor as the coupling and mode shapes continuously vary.
Funding
This work was supported primarily by the Engineering Research Centers Program of the National Science Foundation under NSF Cooperative Agreement No. EEC-0812056.
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