Nanometer resolution of three-dimensional motions using video interference microscopy
W. Hemmert, M. S. Mermelstein and D. M. Freeman
Research Laboratory of Electronics
Massachusetts Institute of Technology, Cambridge, MA 02139, USA
e-mail: whemmert @mit.edu; Tel: (617) 258-5889
An interferometric video system for measuring microelectromechanical systems (MEMS) with nanometer resolution is demonstrated. Interferograms are generated by combining light reflected from the target with light reflected from a reference mirror. Motions are determined from sequences of stop-action interferograms obtained with stroboscopic illumination.
The system was used to measure motions of a microfabricated accelerometer. In-plane motions were determined by analysis of brightfield images using gradient methods with subpixel resolution. Results are compared for brightfield images obtained by blocking light from the reference arm of the interferometer and for brightfield images reconstructed from interferograms. Out-of-plane motions are determined by analyzing interferograms obtained with different positions of the reference mirror.
Results demonstrate nanometer resolution of in-plane motions and subnanometer resolution of out-of-plane motions.
A major goal of this work is to develop a method for high-resolution motion measurements of MEMS in all three directions of motion. We have previously achieved nanometer in-plane resolution by analyzing images from a video microscope with a gradient method . However, out-of-plane resolution was more than a factor of 5 worse than in-plane resolution. A variety of interferometric systems provide extraordinarily high out-of-plane resolution [2,3,4,5].
In this paper we demonstrate an interferometric video system that combines the subpixel in-plane resolution of Computer Microvision (CMV) with the superior out-of-plane resolution of interferometry. We call this new system Interferometric Computer Microvision (ICMV).
We assembled a microscope (Figure 1) with two objective lenses (Zeiss LD-Epiplan 20´ , 0.4 NA, working distance: 9.8 mm, fitted with 160 mm tube-length correctors) positioned to receive illumination from the split beam of a high-performance LED (Nichia NSPG 500S LED, wavelength at maximum power l = 525 nm, half width D l = 40 nm, with the plastic lens milled away and the remaining surface polished flat). One objective lens was focused on a microfabricated accelerometer and the other was focused on a reference mirror. The same beam splitter used for illumination combined the reflected light from the microfabricated accelerometer and reference mirror, in the spirit of a Michelson interferometer. Interferograms are generated in the coincident back-focal-planes of the objectives. A piezo-electric actuator (NEC AE0505D08) supported the reference mirror to change the phase in the interferogram by sub-wavelength adjustment of the optical path-length difference. Brightfield images were also generated by blocking the optical path to the reference mirror. Both interferograms and bright field images were collected using a CCD array (Kodak Megaplus 1.6i, 1534´ 1024 pixels with 9mm spacing, unity fill ratio, and 10-bit gray-value resolution). The field of view was 690´ 461 m m. A vibration isolation table (Integrated Dynamics Engineering, Woburn MA) supported the apparatus.
Figure 1: Interference microscope. Light pulses from an LED reflect off the beam splitter and illuminate the target, which is then imaged by the CCD camera. Light from the LED also passes through the beam splitter, reflects off a reference mirror (positioned axially by a piezo-electric actuator) and beam splitter, reaches the camera, and interferes with light from the target. A shutter is used to block light in the reference path to obtain brightfield images.
To assess both in-plane and out-of-plane sensitivity of ICMV, we measured motions of a microfabricated accelerometer (MCNC, Research Triangle Park, NC) while stimulating only one comb drive with a DC-biased sinusoid (5 V peak-to-peak plus 50 V bias); the shuttle was grounded. The LED was driven with a 1/8 duty cycle pulse-train synchronized with one of eight equally spaced phases of the excitation signal to stroboscopically "freeze" the MEMS motion. An arbitrary waveform generator (AWG2005, Sony/Tektronix Corp., Tokyo, Japan) produced the stimulus, strobe signal, and command voltage for the piezo-electric actuator.
Motion estimates from interferograms
For every stimulus condition (excitation amplitude, frequency, and stroboscopic phase), a sequence of interferograms is collected for different axial positions of the reference mirror. Every pixel in the sequence collects an intensity that varies approximately sinusoidally with the optical phase of the reference mirror. The amplitude of that sinusoidal variation is proportional to the intensity in a brightfield image suitable for in-plane analysis using Computer Microvision [1,6]. The phase of the sinusoidal intensity variation encodes the relative out-of-plane position of the surface at the target (i.e., the microfabricated accelerometer). By estimating both the amplitude and phase of the intensities of each pixel observed in a sequence of positions of the reference mirror, we extract sufficient information for full 3D-motion estimation.
Data were collected in two ways. In the first method, interferograms were collected for 11 values of the control voltage (45 to 50 V in 0.5 V increments) of the piezo-electric actuator that supported the reference mirror. This oversampled data set was analyzed to determine the best-fitting magnitudes, phases, periods, and offsets for each pixel by minimizing thec2 error of the fit. Analysis of the parametric dependence of c2 error also provided a basis for error analysis. In the second method, the increment in control voltage DV needed to move the reference mirror so as to advance the phase of the interference by one-fourth cycle was experimentally determined as follows. A large number of interferograms were collected for closely spaced control voltages, and the summed absolute values of differences between each image and a reference image at the reference control voltage were well fit by a rectified sinusoid. DV was taken to be half the difference from the reference control voltage to the adjacent peak. For each stimulus condition, we collected interferograms for three different control voltages separated by DV. The phase of the interference at each pixel was calculated as the angle whose tangent is (2i2-i1-i3)/(i1-i3), where i1, i2, and i3 represent pixel brightnesses measured for each control voltage.
Displacements between successive stimulus phases were used to reconstruct time waveforms of motion. The magnitude and phase of the fundamental component of that waveform were computed with an FFT. All magnitudes are reported as peak-to-peak values.
For data set 1, only the piezo command voltages were taken as known and no assumptions were made about the resulting optical phases of the reference mirror. Because the mirror is rigid, its relative position can be represented by six degrees of freedom, but since translations and rotation in the plane of the mirror yield no observable effect, only three parameters are needed to characterize its optical phase everywhere in the field of view. Interferograms for a sequence of piezo-voltages (with stimulus conditions held constant) contain many potentially independent observations of sinusoidal intensity variations, one at each pixel location. Minimizing thec2 error of a fit with three sinusoid parameters independent at each pixel (amplitude, phase, and offset) while constraining the sinusoid's sampling phases to agree with the three-parameter mirror position model is a straightforward but computationally intensive problem. With eleven frames for as many piezo-voltages and 1,570,816 pixels per frame, the minimization is also fantastically overconstrained.
The first efforts with this minimization suggested that the mirror's optical phase at every pixel could be reasonably approximated as a linear function of the drive voltage. However, the sensitivity (proportionality constant between voltage and displacement) "drifted" from one stimulus condition to the next (as much as a few percent during a two-hour period). Therefore, four-parameter (amplitude, sensitivity, phase, and offset) sinusoid fits were performed, reducing the required computation from a frame-wide problem to an easier collection of many single-pixel problems. Although this deviation from the more accurate physical model with three degrees of freedom in the mirror position potentially reduces the independence of errors from pixel to pixel for a given stimulus condition, the simplified analysis is sufficient to demonstrate the in-plane and out-of-plane capability of the technique.
The fits minimizedc2 normalized for each pixel intensity value by its standard deviation. The standard deviation was taken to be the square root of the intensity value. Brightfield images were synthesized from the fit amplitudes at each pixel and exported to unmodified Computer Microvision in-plane analysis algorithms [1,6]. The fit phases translated to a relative out-of-plane surface height with a full cycle of phase corresponding to a half-wavelength of height. The fit sensitivities betrayed the long-term instabilities and angular dependencies of the piezo response and were not used. The fit offsets were also discarded. Height estimate standard deviations were derived from contours in c2 space around the minimum .
Supposing a region on the MEMS responded to the electrical stimulus in part with an out-of-plane sinusoidal motion, the unwrapped relative height estimate would also vary sinusoidally as a function of the strobe phase. In this case, because of the electronic synchronization between the strobe signal and the MEMS stimulus (and because the stimulus was DC biased), the frequency is expected to be exactly one cycle of motion per cycle of strobe phase. A region of interest (20 x 20 pixels) was identified on the shuttle near the stimulated comb drive of the microfabricated accelerometer (shown in Figure 2) that was believed to have this type of motion. A three-parameter (amplitude, phase, and offset) sinusoidal fit was performed on the relative height estimates for each frequency at each of the 400 pixel locations in the region of interest. The fits minimizedc2 normalized by height estimate standard deviations. The amplitude and phase of motion was thus measured at each pixel in the region of interest. The offset value was once again discarded. Standard deviations in amplitude and phase were computed for each pixel location separately from contours in the c2 space of the new fits. Combining information across the 400 pixels in a region of interest would reduce the standard deviations by a factor of 20 if the measurements from different pixels were truly independent. Although some noise sources (such as photonic and electronic shot-noise) are uncorrelated from one pixel to the next, many potentially important noise-sources could affect all pixels in the region of interest with the same bias (notably, mechanical system vibration and LED power fluctuations). Therefore, we determine error statistics for regions of interest by examining statistics for each pixel without assuming improvement from averaging.
Three-dimensional Computer Microvision
For comparison, the same microfabricated accelerometer was measured with a three-dimensional Computer Microvision system [1,8], using a Zeiss Axioplan II microscope (Zeiss, Oberkochem, Germany, objectives: Zeiss LD Epiplan-20´ with NA of 0.4, working distance: 9.8 mm and LD Epiplan 50´ with NA of 0.6, working distance: 3.5 mm). Out-of-plane motions were resolved by analyzing images at planes of focus above and below the plane of best focus. A piezo-electric focusing system (PiFoc P-721.10, Physik Instrumente, Germany) was used to change the plane of focus.
Motions of a microfabricated accelerometer (MCNC, Research Triangle Park, NC) were stimulated by applying a DC-biased sinusoid (5 V peak-to-peak plus 50 V bias) to only one comb drive. Eleven interferograms were measured at each of 17 frequencies (from 1 kHz to 72 kHz) and 8 stimulus phases for a total of 1496 interferograms, which constitute data set 1. Three interferograms were measured at each of 33 frequencies and 8 stimulus phases for a total of 792 interferograms, which constitute data set 2. Brightfield images in the plane of best focus were also obtained for the latter stimulus conditions (33 frequencies and 8 stimulus phases) for analysis of in-plane motion. Finally, 3D brightfield images were obtained for the latter stimulus conditions using a Zeiss Axioplan II microscope and two objectives: the same LD Epiplan-20´ objective (NA 0.4) used in the interference microscope and an LD Epiplan-50´ objective (NA 0.6). The 20´ data set combined results at 8 stimulus phases and 11 planes of focus with 2 m m spacing for a total of 2904 2D images. The 50´ data set was similar, but the planes of focus were separated by 1 m m. The brightfield images were analyzed using Computer Microvision algorithms [1,6] for comparison with results obtained using the interference microscope.
To estimate the noise floor of the system, measurements were repeated with the stimulus turned off. Displacements were determined in 10 independent trials and the power spectra were averaged.
Although light from the LED has a short coherence length (tens ofmm), by careful alignment of the optical train, high-contrast interference fringes were generated over the entire shuttle (Figure 2). The prominent horizontal banding in the interferogram in Figure 2 resulted from the asymmetry of the electrical drive: only the top comb was electrically stimulated so that both in-plane and out-of-plane motions resulted. There are 6 prominent bright bands across the width of the shuttle (not counting bands on the comb drives). The difference in out-of-plane positions of adjacent bright bands is half the wavelength of the light from the LED. Thus, differences in out-of-plane positions smaller than 100 nm are readily apparent by simply looking at the interference images.
Quantitative measurements of out-of-plane position can be obtained from sequences of interferograms from different axial positions of the reference mirror. Interference fringes go through one cycle of intensity modulation as the reference mirror moves axially by half the wavelength of the illumination (Figure 2, right stack). The phase of the intensity modulation directly codes axial position of the MEMS.
Axial position can also be determined by analysis of a sequence of brightfield images taken at different planes of focus (Figure 2, left panel). However, axial contrast in such 3D brightfield images is much less than that of 3D interferograms. In Figure 2, the brightfield image blurs gradually as the plane of focus sweeps through the 16 m m distance between the bottom and top images in the brightfield stack. By contrast, the interference modulation period is just 262 nm, 64 times smaller than the distance over which the brightfield image blurs. Thus interference microscopy has the potential for much more sensitive axial resolution than brightfield microscopy.
Figure 2: Brightfield (left) and interference (right) image of the microfabricated accelerometer. A 3D brightfield image is obtained by stacking a sequence of brightfield images from multiple planes of focus (2 m m spacing). The left stack of images illustrates a 3D brightfield image of a single anti-stiction dimple enclosed in a 20 ´ 20 pixel analysis region (black rectangle). A 3D interferogram is obtained by stacking a sequence of interference images for different axial positions of the reference mirror separated by 46 nm (see Figure 1). The right stack of images illustrates a 3D interferogram of the same dimple. These images were obtained with the 20´ objective.
Three-dimensional interferograms were obtained at 8 phases of the stimulus frequency to determine out-of-plane motions. Results are shown in the upper panel of Figure 3. Analysis of the intensity variations for 11 positions of the reference mirror and for the 8 phases of motion (88 numbers) for a single pixel was sufficient to determine the peak-to-peak out-of-plane displacement with nm resolution and response phase to a few degrees. When the number of reference positions was reduced to 3, the results were similar: results obtained from the two data sets differed by less than 1 dB in amplitude and 10° in phase (Figure 3, lower panel). Measurements with three positions of the reference mirror provide no redundancy to calculate standard deviations for each pixel. Instead, the noise floor was measured in the same 20´ 20 analysis region with the stimulation switched off; the RMS-value of the motion was 0.56 nm (80 measurements). Thus, both data sets demonstrate out-of-plane displacement resolutions on the order of nanometers.
Out-of-plane motions were measured as a function of frequency to define an out-of-plane frequency response. The results (Figure 4, left panel) are consistent with a highly-damped second-order resonance. Standard deviations for magnitude measurements using data set 1 were determined from thec2 fits to be less than 4 nm, independent of excitation frequency. Standard deviations of the phase estimates were less than 4 degrees for amplitudes bigger than 50 nm. For data set 2, the RMS-value of the noise floor was 0.56 nm. Amplitude and phase values from both data sets overlapped for frequencies up to 40 kHz, at higher frequencies an accumulating phase difference was apparent. This difference was presumably caused by drift in the measurement system (0.116 nm/s) during the acquisition of the 8 phases for each stimulus condition.
Figure 3: Out-of-plane displacements for an electrical stimulus of 5 V peak-to-peak at 20 kHz (bias: 50 V). Positions determined from one pixel of the analysis region shown in Figure 2 and for 11 reference mirror positions (data set 1, upper panel) are shown by I-beams centered on the values that minimized thec2 fit (see Methods). The vertical extent of each I-beam represents ± one standard deviation for the fit. These standard deviations range from 2.6 to 3.2 nm. The dotted line represents the sinusoid that best fits the data across the 8 stimulus phases for the same pixel. The solid line represents the sinusoid that best fits data from all 400 pixels in the region of interest. Positions determined from just 3 positions of the reference mirror (data set 2) are plotted in the lower panel. Symbols show positions averaged over the measurement region, which were well fit by a sinusoidal fit (dashed line). For comparison, the fit from data set 1 is also shown (solid line). Repeated measurements of the same analysis region with the stimulus off varied by 0.56 nm.
Comparison of out-of-plane performance to Computer Microvision
Similar out-of-plane measurements were also obtained using a previously described  Computer Microvision (CMV) system (right panels of Figure 4). Because out-of-plane resolution using CMV is strongly dependent on numerical aperture (NA), CMV measurements were done with two objectives: a 20´ objective of the same type used for the ICMV measurements, and a higher NA (0.6) 50´ objective. Generally there is good agreement between the two systems for low frequencies, where the out-of-plane motions were relatively large. The average magnitude for 17 frequencies between 1 kHz and 20 kHz was 66, 56, and 58 nm for ICMV (data set 2), CMV 20´ , and CMV 50´ measurements, respectively. However, the variability differed: standard deviations were 2.4, 16, and 5 nm for ICMV, CMV 20´ , and CMV 50´ , respectively. Noise floors were also determined by repeated measurements with the stimulus off. The largest noise floor, 23 nm, resulted for CMV 20´ . Using the higher NA 50´ objective reduced the noise floor by more than a factor of 5, to 4.6 nm. The noise floor for ICMV, 0.67 nm, was nearly 7 times smaller than that for CMV 50´ , and more than 30 times smaller than that for CMV which used the same objective.
Figure 4: Comparison of out-of-plane motions measured with Interferometric and Brightfield Computer Microvision. Magnitudes and phases of out-of-plane motions of a microfabricated accelerometer (analysis region shown in Figure 2) are shown as a function of the frequency of sinusoidal excitation of the upper comb (5V p-p plus 50V DC). The left panel shows results obtained using interference images taken with a 20´ objective (+ís from data set 1, ´ ís from data set 2). A second order resonance curve was fit to data shown by ´ ís (amplitude at DC: 61 nm; resonant frequency: 31.6 kHz; quality of tuning: 1.2) and is shown as a solid line in all three panels for reference. Circles indicate results from brightfield images using the same 20´ objective (center) and a high NA (0.6) 50´ objective (right). Dashed lines indicate the noise floor determined by analyzing images taken when the stimulus was off.
To analyze in-plane motions with ICMV, brightfield images were reconstructed from interferograms collected at eleven reference mirror positions. For comparison, brightfield images were also obtained by blocking light to the reference mirror. A reconstructed image of an anti-stiction dimple is illustrated in Figure 5 along with a brightfield image of the same dimple (upper panels). Reconstructed brightfield images and measured brightfield images were analyzed with a 2D CMV algorithm to estimate in-plane motions (Figure 5, lower panels). Differences were 2% in amplitude and 5° in phase.
Figure 5: Comparison of in-plane motions from brightfield images (left panels) and from brightfield images reconstructed from series of interference images for 11 positions of the reference mirror (data set 1). Top panels show brightfield and reconstructed brightfield images of an anti-stiction dimple (same region of interest shown in Figure 2). Symbols in the plots represent in-plane (vertical) positions of the dimple as a function of phase through one cycle of the 20 kHz stimulus. The fit (solid line) was derived using data from the right panel, but is also shown in the left panel for comparison. The images correspond to the 270° stimulus phase points in the plots.
3D motion measurements
Both in-plane and out-of-plane responses of the accelerometer are shown in Figure 6 as a function of frequency. The in-plane response showed a sharp (Q = 27) resonant peak at 19 kHz. The phase response exhibited a fast roll-off at resonance, which is typical for highly tuned systems. At frequencies sufficiently below resonance, the amplitude response showed a plateau (50 nm). The in-plane response was well approximated by a weakly damped second-order resonant system. The noise floors for both in-plane and out-of-plane motions are on the order of nanometers (in-plane: 3.5 nm, out-of-plane: 0.67 nm).
Figure 6: Frequency response for in-plane (left) and out-of-plane (right) motion of the anti-stiction dimple. In-plane components (left column) were calculated from brightfield images recorded from the plane of focus. Out-of-plane motions (right) are determined from interferograms at eleven (´ ) and three positions of the reference mirror (+, same data shown in Figure 4). Solid lines are derived from fits to second-order resonances (for out-of-plane derived from the three-reference position measurement only). Dashed lines indicate noise floors determined by repeated measurements while the stimulus was off (in-plane: 3.5 nm; out-of-plane: 0.67 nm).
Summary and Discussion
Results in this paper demonstrate the use of off-the-shelf optical components to construct an Interferometric Computer Microvision (ICMV) system capable of measuring three-dimensional motions with nanometer resolution. ICMV combines the subpixel in-plane resolution of Computer Microvision with the extraordinary out-of plane sensitivity of an interference method to achieve nanometer sensitivity in three orthogonal dimensions.
Out-of-plane resolution in brightfield Computer Microvision strongly depends on numerical aperture: high out-of-plane resolution can only be achieved with high numerical aperture objectives. Out-of-plane resolution of Interferometric Computer Microvision derives from interference and is much less sensitive to the numerical aperture of the objective. Consequently, less expensive objectives with lower numerical aperture can be used.
Field of view
Generally, high numerical aperture can only be achieved for objectives with high magnifications. The higher the magnification, the smaller the field of view, given the same camera. Since ICMV allows the use of lower numerical apertures than CMV, it also affords greater fields of view for comparable out-of-plane sensitivity. Greater field of view will become increasingly important as the complexity and size of MEMS increases.
Generally, high numerical aperture can only be achieved for objectives with short working distances. Since ICMV allows the use of lower numerical apertures than CMV, it can achieve the same out-of-plane sensitivities using objectives with longer working distances. Increasing working-distance simplifies the integration of the measurement system with other experimental apparatus such as electrical probes and vacuum systems.
Size of data set
Out-of-plane measurements with ICMV are intrinsically less noisy than those from CMV. For example, Figure 3 illustrated that nanometer out-of-plane precision could result from analysis of brightnesses of a single pixel measured with 11 mirror positions. By combining 400 pixels from the small region that encloses a single anti-stiction dimple, similar sensitivity was obtained from interferograms from just 3 positions of the reference mirror. Thus 3D measurements could be obtained with as few as 3 interference images plus one brightfield image per stimulus condition, compared to the 10 to 20 images at different planes of focus that are typically used for 3D analysis using CMV. Even greater economies of data are achieved because lower magnification is required for ICMV. To image a given field of view, the 20´ system uses 6.25 times fewer pixels than the 50´ system. These factors suggest more than an order of magnitude reduction in the amount of data required for 3D motion measurements using ICMV instead of CMV.
Although ICMV has many advantages over CMV, there are also disadvantages. For example, precise adjustment of the two optical paths is a difficult and frustrating task. Furthermore, the high out-of-plane sensitivity makes the system very sensitive to unintended motions, such as those caused by table vibrations and drift in the piezo-electric actuator. Many of these disadvantages could be reduced by improving the design of the apparatus.
The authors acknowledge contributions of our colleagues, including Alexander J. Aranyosi, Michael J. Gordon, Stanley S. Hong, and Samson J. Timoner. Werner Hemmert was supported in part by the Alexander von Humboldt Foundation. Michael S. Mermelstein was supported by the Fannie and John Hertz Foundation. Dennis M. Freeman was supported in part by the W. M. Keck Career Development Professorship. This work was supported by a grant from DARPA (F30602-97-2-0106)
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