## Abstract

On-line and on-wafer characterizations of mechanical properties of Micro-Electro-Mechanical-System (MEMS) with efficiency are very important to the mass production of MEMS foundry in the near future. However, challenges still remain. In this paper, we present an in-plane vibration characterizing method for MEMS comb using optical Fourier transform (OFT). In the experiment, the intensity distribution at the focal plane was captured to characterize the displacement of the vibrator in the MEMS comb structure. A typical MEMS comb was tested to verify the principle. The shape and the movement of MEMS comb was imitated and tested to calibrate the measurement by using a spatial light modulator (SLM). The relative standard deviations (RSD) of the measured displacements were better than 5%, where the RSD is defined as the ratio of the standard deviation to the mean. It is convinced that the presented method is feasible for on-line and on-wafer characterizations for MEMS with great convenience, high efficiency and low cost.

© 2013 OSA

## 1. Introductions

The comb structure used as an actuator [1, 2] or a sensor [3] is popular in MEMS devices such as micro accelerator [4], micro magnetometer [3] and micro capacitor [5]. Since the comb structure usually works under in-plane vibration, therefore dynamic characterization for the in-plane motion of comb structure is of great importance to the study and the evaluation of MEMS performance. Furthermore, the requirement of high efficiency and low cost for characterizations of MEMS is necessary in the manufacturing line.

Great efforts have been made for characterizing MEMS systems. Christian Rembe et al. developed a stroboscopic microscopy interferometer system (SMIS) in 2001 [6]. Also with the method of combining stroboscopic microscope with image processing, Sylvain Petitgrand and Alain Bosseboeuf implemented simultaneous mapping of in-plane vibrations of MEMS in 2004 [7]. Dung-An Wang et al. proposed a method of in-plane vibration characterization for MEMS using acousto-optic modulated partially incoherent stroboscopic imaging in 2010 [8]. The in-plane vibration measuring approach of these methods can be featured with stroboscopic light and image processing. Another common approach is optical microprobe. By adding gratings into the device under test (DUT), J. M. Dawson et al. realized through-wafer optical micro probing for MEMS high-resolution position feedback in 2003 [9]. Guangya Zhou et al. developed a system for both in-plane and out-of-plane displacement of MEMS with optical micro probing in 2006 [10]. Using knife edge instead of grating, Alain Bosseboeuf et al. also succeeded in measuring in-plane vibration of micro devices with the method of optical micro probing [11]. Ying Zhong et al. developed a diﬀerential laser Doppler system for one-dimensional in-plane motion measurement of MEMS in 2006 [12], which also used optical microprobe.

However, all the above methods need complicated optical or electro-optical components, such as microscopic systems, stroboscopic light sources and precise optical focusing systems. Furthermore, most of methods are time-consuming sub-steps, for example, the precise positioning of DUT or synchronizing the stroboscopic light with vibration, are necessary in the methods mentioned above. Therefore, these methods may lead to risks of cost and schedule if used as dynamic characterization in on-line and on-wafer tests of MEMS high volume production. Since, challenges still remain for on-line and on-wafer characterizations of mechanical properties of MEMS structure, the feasible and high efficiency in-plane vibration characterization is required for MEMS on-line and on-wafer tests.

In this paper, we present a novel approach of in-plane vibration characterization for MEMS comb structure, which is one of the most common structures in MEMS, by using OFT. The proposed method has a great advantage of low cost, less complexity, and shorter elapsed time. A simplified optical system is proposed in this paper, the vibrating frequency and the displacement of the vibrator can be calculated. Compared with the methods mentioned above, this approach is more effective and easy-implemented.

## 2. Principle

The typical comb structure in MEMS is shown in Fig. 1
. It usually consists of a stator and a vibrator. When it works as a sensor, the capacitance between a stator and a vibrator is modulated by the displacement of the vibrator. When it works as an actuator, the electrostatics force between the charged parts will drive the vibrator. In Fig. 1, “*a*” is the width of the teeth, “*b”* is the overlap of the teeth, *“s”* is the offset of the two opposite parts and *“T”* is the spatial period respectively. In driving and sensing situations, the vibrator usually moves asymmetrically. Assuming that the feature along the teeth can be ignored and the number of teeth is very large, it is valid to treat the comb as a periodic function, which is typically a plane periodic pattern. OFT has the great advantage of characterizing the periodic feature of MEMS comb structure. The transmittance “*t” _{0}* of comb structure is described by:

One of the most significant and useful properties of a Fourier transform lens is its ability to perform two-dimensional Fourier transformation. The Fourier transforming operation can be performed in a coherent optical system [13, 14].

As shown in Fig. 2
, an object illuminated by a normally incident monochromatic plane wave of amplitude A is located at the front focal plane of the lens with an amplitude transmittance *t _{0}*(

*x,y*). Then, the field distribution

*U*across the back focal plane of the lens is given by [13]:

_{f}Where *λ* is the wavelength of the incident plane wave and *f* is the focal length. Thus the intensity distribution *I _{f}* is given by [13]:

Substituting Eq. (1) into Eq. (3), the power spectrum of the 2-D comb is given by:

where${f}_{x}={x}_{f}/\left(\lambda f\right),{f}_{y}={y}_{f}/\left(\lambda f\right)$.

In Eq. (4), *I _{x,f}* (

*f*) with the comb structure can be described as a discrete function, which means the intensity distribution is made of discrete points. Therefore, the interval of discrete points along the

_{x}*x*axis is given by:

It indicates that the spatial period *T* of the MEMS comb can also be measured by the interval of the spectral lines. To simplify, *f _{y}* is set to zero, where

*I*(

_{y,f}*f*) reaches its maximum b

_{y}^{2}. Non-zero value of

*I*(

_{x,f}*f*) can only be considered when

_{x}*x*=

_{f}*n*Δ

*x*(n ∈N), which is called the

_{f}*n*

^{th}order of the spectrum pattern. Substituting

*n*= 1 and 2 into Eq. (4), we can get:

In Eqs. (6) and (7), *a*, b, *λ*, *A, f* and *T* are constant, while only *s* is variable. That means the intensity *I _{1}* of the first diffraction order and the intensity

*I*of the second diffraction order are modulated by the offset

_{2}*s*. Therefore, the offset

*s*can be obtained by measuring

*I*and

_{1}*I*.

_{2}In Fig. 3
, it shows that the theoretical intensity distribution of the spectral plane of zero, the first and the second diffraction orders. In Eqs. (6) and (7), it also indicate that *I _{1max}* and

*I*are required when

_{2max}*s*from

*I*or

_{1}*I*needs to be calculated independently. However, the situations of

_{2}*I*and

_{1max}*I*may be impossible to reach directly because the available displacement may be limited. So it is essential to find the values of

_{2max}*I*and

_{1max}*I*by some indirect means.

_{2max}By considering Eqs. (6) and (7), we can get:

In the Eq. (8), it indicates that the square root of *I _{2}* has a semi-linear relationship with

*I*, which gives a great help of finding the values of

_{1}*I*and

_{1max}*I*. In Fig. 3(c), it shows the curve of the semi-linear relationship between

_{2max}*I*and the square root of

_{1}*I*. Therefore, the values of

_{2}*I*and

_{1max}*I*can be acquired by the linear extrapolation.

_{2max}## 3. Experimental configurations

The schematic diagram of the experimental configuration is illustrated in Fig. 4 . The collimated beam from a He-Ne laser passed through the beam expander and then was casted through a stop and a DUT. In the demonstration, a comb-featured MEMS magnetometer works as the DUT. The light modulated by the DUT passed through the Fourier lens. And then, the spatial Fourier spectrum can be observed at the far field plane, where the intensity distribution of spatial spectrum was recorded by a charge-coupled device (CCD) image sensor or photodiode array. The intensity of the first order and the second order were calculated by summing up the intensity of all the corresponding pixels. The CCD sensor is 8-bit grayscale resolution and covers 720 × 576 pixels. As the absolute value of intensity was not concerned, the digitalized value (without unit) could be direct used in Eqs. (6), (7) and (8).

To calibrate the system, we conducted another calibration experiment in which the MEMS DUT was replaced by a transmitting SLM. Since SLM can modulate the spatial transmittance and phase pattern, by loading the shape pattern of the comb structure on the SLM, the periodical movement of comb structure can be imitated [15, 16]. In this measurement, the movements can be precisely controlled by shifting the pattern from one pixel to another. In Fig. 5
, it shows a photograph of the spectrum pattern of the phase pattern. In our experiment setup, the SLM has the resolution of 1024 × 768 and the pixel pitch is 13μm. *T* was set to 20 pixels and *a* was set to 2 pixels. The offset *s* varied from 3 pixels to 10 pixels. As it has the same magnitude as common MEMS combs in size, therefore it was valid for the SLM to imitate a MEMS comb structure.

## 4. Results and discussion

In Fig. 6
, it shows that the image of spatial spectrum patterns and intensity distribution. The intensity *I _{1}* and

*I*were obtained by summing the digitalized intensity of the pixels throughout the corresponding areas together instead of finding the maximum of intensity. The experimental results of the relationship between

_{2}*I*and square root of

_{1}*I*are illustrated by Fig. 7 . As a result of digitalization,

_{2}*I*and

_{1}*I*have no units in Fig. 7. It is indicated that

_{2}*I*and square root of

_{1}*I*fit the theoretical semi-linear relationship.

_{2}The value of *I _{1max}* was calculated by linear fitting from Fig. 7. Substituting

*I*into Eq. (6), the displacement of vibrator, or the offset

_{1max}*s*, was calculated and plotted in Fig. 8 . Most data points fitted the theoretical curve well except the points of 9-pixels real displacement. It is speculated that the point of 9 pixels was close to the symmetric point and that the intensity

*I*was close to zero. Thus the intensities of many pixels in the first order could be less than the minimum CCD resolution, which caused the measured

_{1}*I*was lower than it should be and the corresponding displacement was shift to the symmetric point (

_{1}*s*=

*T/*2). From Fig. 8, it is obvious that the measured results of displacement matched the theoretical results. The average measured displacements and the RSDs were listed in Table 1 . The RSD is defined as the ratio of the standard deviation to the mean here. The observed RSDs were below 5%, which meant that the compensation could be very effective to eliminate the nonlinearity of the average measured displacement.

The proposed method does not need to focus the light beam, which spares the optical focusing system. Based on diffraction theory, the smaller the spatial period of the comb structure is, the larger the interval between two adjacent diffractive orders is. The interval of spectral lines can reach millimeters, so the microscope is also spared. In the stroboscopic system, the stroboscopic light and the vibration of DUT must be synchronized. This step may take precise adjustment and a lot of human efforts. Nevertheless, such trouble can be avoided successfully in the proposed method because the frequency of the movement can be direct measured by acquiring the light intensity and performing FFT. We are convinced that this method has an especial advantage in measuring vibrating frequency. Thus, the presented method takes a great advantage over those mentioned methods in system complexity.

Since optical Fourier transform is not sensitive to the position of DUT, precise positioning of DUT is avoided. It means the measuring system would be more stable and more resistant to the positioning error.

From Eqs. (6) and (7), it is easy to find that the proposed method cannot distinguish the offset from its symmetric position. This problem is caused by the principle of Fourier transform. Generally, this method can only be semi-quantitative with displacement. However, in a lot of situations that “*s”* varies asymmetrically [3–5], in other words, it always above or below *T*/2, therefore, the proposed method can be quantitative. *I _{1max}* and

*I*are required before measuring displacement. As a result, the calibration procedure is essential as a pre-measuring step.

_{2max}## 5. Conclusion

In the paper, a novel method of in-plane vibration characterization for MEMS comb using optical Fourier transform is proposed. The feasibility was demonstrated by testing a comb-featured MEMS magnetometer. The calibration configuration was implemented by using a transmitting spatial light modulator to imitate the periodical movement of MEMS comb structure. The RSDs of the measured displacements were better than 5% in the experiment. It has been demonstrated that our method can work as an approach of characterization for MEMS comb. Further, it implies that our method could be effective for any MEMS device with periodic patterns as it is based on OFT. Without focusing system, microscope and stroboscopic source, the proposed method has a less-complicated optical system with only one Fourier lens, compared with the mentioned methods [6–12]. Furthermore, the synchronization of stroboscopic light with vibration and the precise positioning of MEMS DUT can be avoided in the proposed method. It has the ability of semi-quantitative displacement characterization for MEMS comb structure. Furthermore, the displacement measurement of MEMS comb structure can be completely quantitative in the situation where the movement is limited in half side. It is convinced that the proposed method has an advantage in high efficiency, great convenience and low cost.

## Acknowledgment

This work was financially supported by National Program for Significant Scientific Instruments Development of China (2011YQ030134), National Basic Research Program of China (973 Program) (No. 2012CB934103), National High Technology Research and Development Program of China (863 Program) (No. 2012AA121503) and National Nature Science Foundation of China (51205223, 50905096). Authors also thank Mr. Hao Zhu for the experimental assistance.

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