Linearization of a two-axis MEMS scanner driven by vertical comb-drive actuators
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J. Micromech. Microeng. 18 (2008) 015015 (8pp) doi:10.1088/0960-1317/18/1/015015
Linearization of a two-axis MEMS
scanner driven by vertical comb-drive
actuators
Jui-che Tsai
1, Li-Cheng Lu
1, Wei-Chi Hsu
1, Chia-Wei Sun
2and Ming C Wu
31Graduate Institute of Photonics and Optoelectronics and Department of Electrical Engineering, National Taiwan University, Taipei 10617, Taiwan
2Medical Electronics and Device Technology Center, Industrial Technology Research Institute, Hsinchu, Taiwan
3Department of Electrical Engineering and Computer Sciences and Berkeley Sensor and Actuator Center (BSAC), University of California, Berkeley, CA 94720-1774, USA
E-mail:jctsai@cc.ee.ntu.edu.tw
Received 2 August 2007, in final form 24 October 2007 Published 3 December 2007
Online atstacks.iop.org/JMM/18/015015
Abstract
A driving scheme using a pair of differential voltages (Vx, Vy) over a bias voltage is proposed
to linearize the dc characteristic (angle versus voltage) of a two-axis MEMS scanner. The micromirror has a gimbal-less structure and is driven by vertical comb-drive actuators in conjunction with a leverage mechanism. At an optimal bias voltage of 53 V, a linear optical scan range of±3.2◦is achieved experimentally in both the x and y directions with the differential voltages ranging from−10 V to + 10 V.
(Some figures in this article are in colour only in the electronic version)
1. Introduction
MEMS (micro-electro-mechanical systems) scanners have been widely adopted in a variety of photonics-related research fields or products. In telecommunications, they are the key enabling components for two-dimensional (2D) [1] and three-dimensional (3D) [2, 3] optical cross-connects (OXCs), dynamic gain equalizers [4, 5], wavelength add-drop multiplexers (WADMs) [6,7] and wavelength-selective switches (WSSs) [8–10]. They offer low optical insertion loss and crosstalk, independent of polarization and wavelength, as well as optical transparency for bit rate and data format. In adaptive optics, tip-tilt-piston micromirror arrays [11] and deformable mirrors [12] compensate wavefront distortions introduced by the medium and help achieve sharper images approaching the diffraction limit, which is particularly vital for space observation. MEMS optical scanners are also widely used in projection displays such as digital light processing [13] and laser scanning displays [14], endoscopic imaging [15] and confocal microscopy [16].
Electrostatic actuation is one of the most popular driving mechanisms for MEMS scanning mirrors [17]. Typically,
capacitive structures are embedded to generate the required electrostatic torque for mirror rotation. The electrostatic torque, explicitly as a function of V2, does not linearly vary
with the actuation voltage. This results in strong distortion of the scan pattern when driving with linearly ramped voltages. In some cases, nonlinearity in the capacitance gradient with respect to rotation angle (θ ) further aggravates the distortion; for instance, the capacitance of a parallel-plate scanner exhibits 1/θ dependence.
Several approaches have been proposed to eliminate the distortion, resulting in scan patterns with high linearity. Chiou
et al divided the bottom electrode into multiple segments to
fulfil the linearization of a one-axis parallel-plate micromirror [18]. Hiroshi et al demonstrated the linearization of a two-axis parallel-plate MEMS scanner by using a differential-voltage driving scheme, which was relatively simple and could be realized in open-loop operation [19]. The two polarities of the differential voltage Vdiffwere superimposed on
a bias voltage Vbias.The resulting sum voltages, (Vbias+ Vdiff)
and (Vbias− Vdiff), were applied on the opposite electrodes.
This induced electrostatic torques in opposite directions and proportional to (Vbias+ Vdiff)2and (Vbias− Vdiff)2, respectively,
J. Micromech. Microeng. 18 (2008) 015015 J-c Tsai et al
Fixed Comb Fixed Comb Movable Comb
No overlap between the movable and fixed combs on this side
Vertical Comb-Drive Scanner
Electrode Electrode
Parallel-Plate Scanner Electric
field
Figure 1. Comparison between parallel-plate and vertical
comb-drive scanners.
on the mirror. The square terms V2
biasand Vdiff2 were cancelled
out and only the term Vbias × Vdiff remains in the net
electrostatic torque, which is hence proportional to the control differential voltage Vdiff under a constant bias. Moreover,
PID control (by Zhao et al [20]) and neural network methods (by Zhao et al [21]) have been respectively incorporated with the differential-voltage driving scheme to further enhance the scanner linearity.
The differential driving scheme performs best on the occasion that the capacitance derivative with respect to the rotation angle (∂C/∂θ ) remains constant, which unfortunately is not the case for parallel-plate scanners. On the other hand, vertical comb-drive actuators, whose ∂C/∂θ can be considered quasi-constant, seemingly prevail in this regard. However, the approach with differential-voltage driving is typically not suitable for rotating mirrors powered by vertical comb-drive actuators. This is due to the fact that, at the side that travels upward during rotation, the movable comb parts away from the fixed comb and their overlap vanishes at large angles. The consequence is that an opposite torque fails to be established, contrary to the case of parallel-plate scanners. This can be explained in figure1.
Recently, we have proposed a novel two-axis analog micromirror powered by vertical comb-drive actuators in conjunction with leverage mechanism [22] (figure2). Large mechanical rotation angles (±6.7◦ for both axes at 75 V)
Lever-mirror joint Lever Mirror Vertical combdrive actuators x y Torsion spring (lever rotation axis/ fulcrum)
Fixed comb Movable comb
Anchor to the substrate
Figure 2. Schematic of the two-axis MEMS scanner driven by vertical comb-drive actuators in conjunction with a leverage mechanism [22].
Figure 3. Simulation model. The unit for the numbers in this figure
is µm.
Figure 4. Lever characteristic: angle versus voltage, vertical
displacement of the lever-mirror joint versus voltage.
and high resonant frequency (5.9 kHz before metallization) were achieved experimentally. The device was manufactured through a five-layer surface-micromachining process offered by Sandia National Laboratory. It can also be replicated along the x direction to form a high fill-factor 1D mirror array [22].
(a) (b)
(c) (d )
(e)
Figure 5. Distribution of Ef,totalunder different bias voltages: (a) Vbias= 20 V, (b) Vbias= 30 V, (c) Vbias= 40 V, (d) Vbias= 50 V and (e) Vbias= 53 V.
The mirror is equipped with four lever-comb-drive pairs and the design architecture ensures that all the four electrodes contribute toward the mirror rotation under any circumstance. Therefore, a differential driving scheme becomes possible for a comb-drive-driven mirror constructed in this manner. However, one has to note that the fundamental principle
of differential driving of such a comb-drive-lever mirror is different from that of a parallel-plate scanner. As mentioned above, in the differential driving scheme of a parallel-plate mirror, electrostatic torques in the opposite directions are generated to cancel out the quadratic nonlinear terms. In our device, however, each comb-drive-powered lever possesses
J. Micromech. Microeng. 18 (2008) 015015 J-c Tsai et al
(a) (b)
(c) (d )
(e)
Figure 6. x–y plane projections of Ef,totalfor (a) Vbias= 20 V, (b) Vbias= 30 V, (c) Vbias= 40 V, (d) Vbias= 50 V and (e) Vbias= 53 V, where the values greater than 0.1 are discarded. The red square is the largest square scan pattern that has its sides aligned parallel/orthogonally to the axes and is enclosed by the projection image.
control over the motion of one mirror corner. By measuring the angle–voltage characteristic of each lever, we are able to determine the optimal bias point to achieve the maximum scan pattern that exhibits high linearity using open-loop operation.
2. Simulation determining the optimal bias point
2.1. Model
At most four independent voltages can be allowed to operate our comb-drive-lever mirror. The voltages applied on the four
Figure 7. Schematic of the experimental setup.
levers are denoted as V1, V2, V3 and V4, respectively. We
further express them as
V1= Vbias+ v1
V2= Vbias+ v2
V3= Vbias+ v3
V4= Vbias+ v4.
(1)
In figure3, we define the coordinate of the mirror center as (0, 0, zbias), where zbiasrepresents the lift of the mirror center
and is determined by the bias voltage Vbias. The coordinates
of the four lever-mirror joints are (50 µm, 50 µm, zbias+ z1),
(−50 µm, 50 µm, zbias + z2), (−50 µm, −50 µm, zbias+
z3), and (50 µm, −50 µm, zbias+ z4), where z1, z2, z3and
z4are determined by v1, v2, v3and v4, respectively. f is the
distance between the mirror and the observation plane. For an incident light in the direction of (0, 0,−1) to be reflected to a point (x, y) on the observation plane, the MEMS mirror has to be rotated so that its normal vector becomes (x/2, y/2, f ) as depicted in figure3. For each targeted point on the observation plane, a combination of (z1, z2, z3, z4) and a corresponding (v1,
v2, v3, v4) set can be solved. The step-by-step procedure is
as follows:
(1) Experimentally measure the angle versus voltage curve of the lever and, hence, the vertical displacement of the lever-mirror joint versus voltage.
(2) Set a bias voltage Vbias for all levers, which then
determines zbias according to the curve obtained in
step 1.
(3) Define the scan range and an array of targeted (x, y) points on the observation plane.
(4) Find the mirror normal vector n for the light to be reflected to a certain point (x, y).
(5) For each n, find z1, z2, z3and z4. Then the solution for
(v1, v2, v3, v4) can be obtained using the curve in step 1
along with Vbiasand zbias. (v1, v2, v3, v4) is considered as
the control voltage set.
Table 1. Spans of the square patterns in figure6: scan field size on the observation plane and the corresponding optical scan angle.
Bias Scan field size on the Optical scan angle in the voltage (V) observation plane (µm2) x and y directions (◦)
20 200× 200 ±0.14
30 530× 530 ±0.38
40 1800× 1800 ±1.3
50 1670× 1670 ±1.2
53 5900× 5900 ±4.22
Generally fewer independent voltages are preferred as the control scheme can then be simplified. For a two-axis scanner, at least two independent voltages, typically denoted as Vxand
Vy, are required. Our purpose is therefore to reduce (v1, v2,
v3, v4) to (Vx, Vy), i.e. to express v1, v2, v3and v4 as linear
combinations of Vxand Vyas follows:
v1= 12· (Vx+ Vy)
v2= 12· (−Vx+ Vy)
v3= 12· (−Vx− Vy)
v4= 12· (Vx− Vy).
(2)
At any given bias voltage, Vxand Vycan be solved only
for the solutions in which v1= −v3and v2= −v4. Therefore,
we define the error functions as
Ef1= |v2| − |v4| |v2| + |v4| , Ef2= |v1| − |v3| |v1| + |v3| , (3) and Ef,total= Ef1+ Ef2. (4)
Ef1and Ef2, each of which can actually be viewed as a ratio
of the voltage difference to the average voltage, represent the degrees of V2− V4and V1− V3discrepancies, respectively.
Ideally Ef,total has to be 0 for exactly solving Vx and Vy;
however, we hereby define a tolerance 0.1 such that Vx
and Vy are said to be ‘quasi-solved’ for Ef,total<0.1 (i.e.,
10%). Vx and Vyare then experimentally superimposed on
J. Micromech. Microeng. 18 (2008) 015015 J-c Tsai et al (a) (b) (c) (d ) (e) bias bias bias
Figure 8. Scan fields measured by the PSD under different bias voltages: (a) Vbias= 20 V, (b) Vbias= 30 V, (c) Vbias= 40 V, (d) Vbias= 50 V and (e) Vbias= 53 V.
that a differential driving scheme can be achieved for two-axis rotation.
The coordinate axes in figures 2 and 3 are defined in a way that the x and y axes are parallel and orthogonal to
the mirror array direction, respectively. This is mainly for the convenience of application and is adopted throughout the paper. However, it is worth noting that the coordinate axes can also be set along the mirror diagonals, i.e. 45◦rotation of the
Figure 9. Effect of lens aberration on the measured scan pattern. current coordinate system. In this case the lever-mirror joints are situated right on the axes. There is then no need to use the linear combinations of Vxand Vyin the differential driving
scheme, i.e. they can be superimposed on the bias voltage separately.
2.2. Simulation results
Figure 4 is the measured individual lever characteristic. Simulations are done under various bias voltages. The distance between the mirror and the observation plane is set as 4 cm, which is the focal length of the achromatic doublet lens used in the experiment. The experimental setup will be explained in the next section. The distributions of Ef,totalunder different
bias voltages, 20 V, 30 V, 40 V, 50 V and 53 V, are plotted in figure 5. The x–y plane projections of Ef,total are shown
in figure6, where the values greater than 0.1 are discarded. For each bias voltage, the largest square scan pattern that has its sides aligned parallel/orthogonally to the axes and is enclosed by the projection image is drawn with red solid lines. It is worth mentioning that at any given bias voltage some target points on the observation plane eventually yield solutions in which the resultant zbias+ zi(i= 1, 2, 3 or 4) either
goes beyond the maximum displacement in figure4 or falls below 0. Although these solutions have no physical meaning, the voltages (vi) and error-function values are nevertheless
mathematically calculated by means of extrapolation on the curve in figure 4. After defining the square scan patterns in figure 6, the solutions within the squares are rechecked to ensure that they do not fall within the aforementioned physically meaningless regime. Table1summarizes the spans of the square patterns in terms of the scan field size on the observation plane and the corresponding optical scan angle. It is concluded that the maximum scan range achievable by two independent control voltages occurs at Vbias= 53 V. The
optical scan range starts to decrease beyond this point. At
Vbias = 60 V (not shown in the figures), it becomes ±0.6◦.
3. Experiments
3.1. Differential driving operation
A schematic of the experimental setup is shown in figure 7. A 632.8 nm He–Ne laser is incident on the 50/50 beam splitter with a 45◦incident angle. The laser beam (1/e2beam
diameter 2w = 0.75 mm) is then focused onto the MEMS
Figure 10. Non-differential driving scheme. The unit for the
numbers in this figure is µm.
scanner by an achromatic doublet lens with 4 cm focal length, resulting in a focus spot size of 2w= 43 µm. The differential voltages, Vxand Vy, are computer programmed and the bias
voltage is provided by a power supply. The MEMS scanner deflects the laser beam, which is then recollimated by the achromatic doublet lens, passes through the beam splitter, and finally is collected by the position-sensing detector (PSD). This arrangement ensures the scan pattern be independent of the distance between the PSD and the lens.
The differential voltages are varied with an increment of 1 V and are limited within±10 V by the data acquisition (DAQ) card. Figure8 shows the scan fields experimentally measured by the PSD under different biases in the differential-driving scheme. The scan range for Vbias= 53 V is limited
to±3.2◦ (in both the x and y directions) by the maximum voltage magnitude which can be provided by the DAQ card. It is expected that the value predicted by the simulation can be reached if sufficient driving voltage is supplied. We note that the required voltages to reach a certain angle in the experiment are larger than those in the simulation. This may result from the resistance at the lever-mirror joints, which are elastic and compliant, but actually not completely free.
The larger scan pattern for Vbias= 53 V exhibits distortion
with a barrel shape, mainly caused by the lens aberration. The effect of lens aberration can be explained with the model in figure9constructed using ZEMAX, a commercial ray tracing software. A point source on the left focal plane emerging optical rays can be treated as the MEMS mirror steering the laser beam, where each ray is viewed as the reflected light beam from the mirror poised at a certain tilt angle. A 0.45 cm× 0.45 cm square aperture, corresponding to the mirror scan pattern for Vbias= 53 V, is placed right on the left of the lens.
The transformed pattern at the PSD, 15 cm on the right of the lens, turns out to be a barrel shape due to the aberration of the achromatic doublet lens.
3.2. Non-differential driving operation
For comparison, a non-differential driving scheme with two independent voltages, Vxand Vy, is implemented. The voltage
combination shown in figure10deflects the laser beam toward the 3rd quadrant. In our experiment, Vxand Vyboth vary from
J. Micromech. Microeng. 18 (2008) 015015 J-c Tsai et al
Figure 11. Scan field measured by the PSD under the
non-differential driving scheme.
using power amplifiers for the voltages output by the computer. Similar voltage combinations can be used for beam deflection toward the 1st, 2nd and 4th quadrants. The complete scan pattern is shown in figure11. It can be seen that the pattern is profoundly distorted even with a small optical scan range of <±0.3◦. The slight asymmetry is due to optical system misalignment and the non-uniformity among levers.
4. Conclusions
We have implemented a differential driving scheme to linearize the dc characteristic of a two-axis MEMS scanner. The gimbal-less micromirror is driven by vertical comb-drive actuators in conjunction with leverage mechanism. A simulation model is developed to determine the maximum scan range. At an optimal bias voltage of 53 V, a linear optical scan range of ±3.2◦is achieved experimentally in both the x and y directions
with the differential voltages varying within the range of ±10 V. It is expected that the value predicted by the simulation (±4.22◦) can be approached if sufficient driving voltage is
supplied.
Acknowledgments
This work was supported by the National Science Council of Taiwan under grants NSC 95-2221-E-002-053 and NSC 96-2221-E-002-198-MY2, Excellent Research Projects of National Taiwan University, 95R0062-AE00-06, and DARPA/SPAWAR under contract N66001-00-C-8088.
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