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Optics Express

Optics Express

  • Editor: Andrew M. Weiner
  • Vol. 21, Iss. 4 — Feb. 25, 2013
  • pp: 5063–5070
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Optical Fourier transform based in-plane vibration characterization for MEMS comb structure

Yongfeng Gao, Liangcai Cao, Zheng You, Jiahao Zhao, Zichen Zhang, and Jianzhong Yang  »View Author Affiliations


Optics Express, Vol. 21, Issue 4, pp. 5063-5070 (2013)
http://dx.doi.org/10.1364/OE.21.005063


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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

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
Fig. 1 Simplified comb structure (a): structure model, (b): cross-section view. (a: width of the teeth; b: overlap of the teeth; s: offset of the two opposite parts; T: the spatial period).
. 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:

t0(x,y)=[(Rec(xT)Rec(xa)+Rec(xsa))comb(xT)]Rec(yb)
(1)

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

13. J. W. Goodman, Introduction to Fourier Optics(third edition) (Robert & Company Publishers, 2005), Chap. 5.

, 14

14. G. J. Zhang and S. H. Ye, “Online measurement of the sizes of standard wire sieves using an optical Fourier transform,” Opt. Eng. 39(4), 1098–1102 (2000). [CrossRef]

].

As shown in Fig. 2
Fig. 2 Performing the Fourier transform operation with a positive lens.
, 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 t0(x,y). Then, the field distribution Uf across the back focal plane of the lens is given by [13

13. J. W. Goodman, Introduction to Fourier Optics(third edition) (Robert & Company Publishers, 2005), Chap. 5.

]:

Uf(x,y)=Aexp[jk2f(xf2+yf2)]jλft0(x,y)exp[j2πλf(xxf+yyf)]dxdy
(2)

Where λ is the wavelength of the incident plane wave and f is the focal length. Thus the intensity distribution If is given by [13

13. J. W. Goodman, Introduction to Fourier Optics(third edition) (Robert & Company Publishers, 2005), Chap. 5.

]:

If(x,y)=A2λ2f2|t0(x,y)exp[-j2πλf(xxf+yyf)]dxdy|2
(3)

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

{If(fx,fy)=A2λ2f2Ix,f(fx)Iy,f(fy),Ix,f(fx)=|[Tsinc(πTfx)asinc(πafx)(1+exp(j2πsfx))]comb(fxT)|2,Iy,f(fy)=|bsinc(πbfy)|2
(4)

wherefx=xf/(λf),fy=yf/(λf).

In Eq. (4), Ix,f (fx) 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 axis is given by:

Δfx=1/T,i.e.,Δxf=λf/T
(5)

It indicates that the spatial period T of the MEMS comb can also be measured by the interval of the spectral lines. To simplify, fy is set to zero, where Iy,f (fy) reaches its maximum b2. Non-zero value of Ix,f (fx) can only be considered when xf = n Δxf (n ∈N), which is called the nth order of the spectrum pattern. Substituting n = 1 and 2 into Eq. (4), we can get:

I1=If|n=1=4A2b2T2λ2f2sinc2(πaT)cos2(πsT)=I1maxcos2(πfsT)
(6)
I2=If|n=2=4A2b2T2λ2f2sinc2(2πaT)cos2(2πsT)=I2maxcos2(2πsT)
(7)

In Eqs. (6) and (7), a, b, λ, A, f and T are constant, while only s is variable. That means the intensity I1 of the first diffraction order and the intensity I2 of the second diffraction order are modulated by the offset s. Therefore, the offset s can be obtained by measuring I1 and I2.

In Fig. 3
Fig. 3 Theoretical intensity distribution of the spectral plane when s = 0.25T (a) and s = 0.5T (b) (c) the semi-linear relationship between I1 and square root of I2.
, 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 I1max and I2max are required when s from I1 or I2 needs to be calculated independently. However, the situations of I1max and I2max may be impossible to reach directly because the available displacement may be limited. So it is essential to find the values of I1max and I2max by some indirect means.

By considering Eqs. (6) and (7), we can get:

I2=I2max(2I1I1max-1)2I2=I2max|2I1I1max-1|
(8)

In the Eq. (8), it indicates that the square root of I2 has a semi-linear relationship with I1, which gives a great help of finding the values of I1max and I2max. In Fig. 3(c), it shows the curve of the semi-linear relationship between I1 and the square root of I2. Therefore, the values of I1max and I2max can be acquired by the linear extrapolation.

3. Experimental configurations

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

15. L. P. Zhao, N. Bai, X. Li, L. S. Ong, Z. P. Fang, and A. K. Asundi, “Efficient implementation of a spatial light modulator as a diffractive optical microlens array in a digital Shack-Hartmann wavefront sensor,” Appl. Opt. 45(1), 90–94 (2006). [CrossRef] [PubMed]

, 16

16. A. Vyas, M. B. Roopashree, and B. R. Prasad, “Digital long focal length lenslet array using spatial light modulator,” in Proceedings of the international conference on optics and photonics. (Chandigarh, India, 30 Oct. −1 Nov. 2009)

]. In this measurement, the movements can be precisely controlled by shifting the pattern from one pixel to another. In Fig. 5
Fig. 5 Photograph of the spectrum pattern.
, 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
Fig. 6 CCD image and intensity distribution of the spectrum of the MEMS DUT (a) and the SLM calibration (b).
, it shows that the image of spatial spectrum patterns and intensity distribution. The intensity I1 and I2 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 I1 and square root of I2 are illustrated by Fig. 7
Fig. 7 Experimental results of semi-linear relationship between I1 and square root of I2.
. As a result of digitalization, I1 and I2 have no units in Fig. 7. It is indicated that I1 and square root of I2 fit the theoretical semi-linear relationship.

The value of I1max was calculated by linear fitting from Fig. 7. Substituting I1max into Eq. (6), the displacement of vibrator, or the offset s, was calculated and plotted in Fig. 8
Fig. 8 Measured results compared with the theoretical plot.
. 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 I1 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 I1 was lower than it should be and the corresponding displacement was shift to the symmetric point (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

Table 1. Average Values and RSDs of the Measured Displacements

table-icon
View This Table
. 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

3. M. J. Thompson and D. A. Horsley, “Resonant MEMS magnetometer with capacitive read-out,” in IEEE Sensors 2009 Conference. (Christchurch, New Zealand, Oct. 25–28, 2009), 992–995.

5

5. Y. Zhu, M. R. Yuce, and S. O. R. Moheimani, “A low-loss MEMS tunable capacitor with movable dielectric,” in IEEE Sensors 2009 Conference. (Christchurch, New Zealand, Oct. 25–28, 2009), 651–654.

], in other words, it always above or below T/2, therefore, the proposed method can be quantitative. I1max and I2max are required before measuring displacement. As a result, the calibration procedure is essential as a pre-measuring step.

5. Conclusion

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.

References and links

1.

R. Legtenberg, A. W. Groeneveld, and M. Elwenspoek, “Comb-drive actuators for large displacements,” J. Micromech. Microeng. 6(3), 320–329 (1996). [CrossRef]

2.

S. Kang, H. C. Kim, and K. Chun, “A low-loss, single-pole, four-throw RF MEMS switch driven by a double stop comb drive,” J. Micromech. Microeng. 19(3), 035011 (2009). [CrossRef]

3.

M. J. Thompson and D. A. Horsley, “Resonant MEMS magnetometer with capacitive read-out,” in IEEE Sensors 2009 Conference. (Christchurch, New Zealand, Oct. 25–28, 2009), 992–995.

4.

H. Chen, M. Chen, W. J. Zhao, and L. M. Xu, “Equivalent electrical modeling and simulation of MEMS comb accelerometer,” in 2010 International Conference on Measuring Technology and Mechatronics Automation. (Changsha City, China, 13–14 March 2010), 116–119.

5.

Y. Zhu, M. R. Yuce, and S. O. R. Moheimani, “A low-loss MEMS tunable capacitor with movable dielectric,” in IEEE Sensors 2009 Conference. (Christchurch, New Zealand, Oct. 25–28, 2009), 651–654.

6.

C. Rembe, L. Muller, R. S. Muller, and R. T. Howe, “Full three-dimensional motion characterization of a gimballed electrostatic microactuator,” in Proc. IEEE Int. Rel. Symp. (Orlando, FL, Apr. 30,2001),91–98.

7.

S. Petitgrand and A. Bosseboeuf, “Simultaneous mapping of out-of-plane and in-plane vibrations of MEMS with (sub) nanometer resolution,” J. Micromech. Microeng. 14(9), S97–S101 (2004). [CrossRef]

8.

D. A. Wang, F. W. Sheu, and Y. S. Chiu, “In-plane vibration characterization of microelectromechanical systems using acousto-optic modulated partially incoherent stroboscopic imaging,” Opt. Lasers Eng. 49(7), 954–961 (2011). [CrossRef]

9.

J. M. Dawson, L. Wang, P. Famouri, and L. A. Hornak, “Grating-enhanced through-wafer optical microprobe for microelectromechanical system high-resolution optical position feedback,” Opt. Lett. 28(14), 1263–1265 (2003). [CrossRef] [PubMed]

10.

G. Y. Zhou and F. S. Chau, “Grating-assisted optical microprobing of in-plane and out-of-plane displacements of microelectromechanical devices,” J. Microelectromech. Syst. 15(2), 388–395 (2006). [CrossRef]

11.

A. Bosseboeuf, C. Bréluzeau, F. Parrain, P. Coste, J. Gilles, S. Megherbi, and X. Roux, “In-plane vibration measurement of micro devices by the knife-edge technique in reflection mode,” Proc. SPIE 6345, 63451D, 63451D-8 (2006). [CrossRef]

12.

Y. Zhong, G. X. Zhang, C. L. Leng, and T. Zhang, “A differential laser Doppler system for one-dimensional in-plane motion measurement of MEMS,” Measurement 40(6), 623–627 (2007). [CrossRef]

13.

J. W. Goodman, Introduction to Fourier Optics(third edition) (Robert & Company Publishers, 2005), Chap. 5.

14.

G. J. Zhang and S. H. Ye, “Online measurement of the sizes of standard wire sieves using an optical Fourier transform,” Opt. Eng. 39(4), 1098–1102 (2000). [CrossRef]

15.

L. P. Zhao, N. Bai, X. Li, L. S. Ong, Z. P. Fang, and A. K. Asundi, “Efficient implementation of a spatial light modulator as a diffractive optical microlens array in a digital Shack-Hartmann wavefront sensor,” Appl. Opt. 45(1), 90–94 (2006). [CrossRef] [PubMed]

16.

A. Vyas, M. B. Roopashree, and B. R. Prasad, “Digital long focal length lenslet array using spatial light modulator,” in Proceedings of the international conference on optics and photonics. (Chandigarh, India, 30 Oct. −1 Nov. 2009)

OCIS Codes
(070.0070) Fourier optics and signal processing : Fourier optics and signal processing
(120.0120) Instrumentation, measurement, and metrology : Instrumentation, measurement, and metrology

ToC Category:
Instrumentation, Measurement, and Metrology

History
Original Manuscript: December 27, 2012
Revised Manuscript: February 8, 2013
Manuscript Accepted: February 8, 2013
Published: February 21, 2013

Citation
Yongfeng Gao, Liangcai Cao, Zheng You, Jiahao Zhao, Zichen Zhang, and Jianzhong Yang, "Optical Fourier transform based in-plane vibration characterization for MEMS comb structure," Opt. Express 21, 5063-5070 (2013)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-4-5063


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References

  1. R. Legtenberg, A. W. Groeneveld, and M. Elwenspoek, “Comb-drive actuators for large displacements,” J. Micromech. Microeng.6(3), 320–329 (1996). [CrossRef]
  2. S. Kang, H. C. Kim, and K. Chun, “A low-loss, single-pole, four-throw RF MEMS switch driven by a double stop comb drive,” J. Micromech. Microeng.19(3), 035011 (2009). [CrossRef]
  3. M. J. Thompson and D. A. Horsley, “Resonant MEMS magnetometer with capacitive read-out,” in IEEE Sensors 2009 Conference. (Christchurch, New Zealand, Oct. 25–28, 2009), 992–995.
  4. H. Chen, M. Chen, W. J. Zhao, and L. M. Xu, “Equivalent electrical modeling and simulation of MEMS comb accelerometer,” in 2010 International Conference on Measuring Technology and Mechatronics Automation. (Changsha City, China, 13–14 March 2010), 116–119.
  5. Y. Zhu, M. R. Yuce, and S. O. R. Moheimani, “A low-loss MEMS tunable capacitor with movable dielectric,” in IEEE Sensors 2009 Conference. (Christchurch, New Zealand, Oct. 25–28, 2009), 651–654.
  6. C. Rembe, L. Muller, R. S. Muller, and R. T. Howe, “Full three-dimensional motion characterization of a gimballed electrostatic microactuator,” in Proc. IEEE Int. Rel. Symp. (Orlando, FL, Apr. 30,2001),91–98.
  7. S. Petitgrand and A. Bosseboeuf, “Simultaneous mapping of out-of-plane and in-plane vibrations of MEMS with (sub) nanometer resolution,” J. Micromech. Microeng.14(9), S97–S101 (2004). [CrossRef]
  8. D. A. Wang, F. W. Sheu, and Y. S. Chiu, “In-plane vibration characterization of microelectromechanical systems using acousto-optic modulated partially incoherent stroboscopic imaging,” Opt. Lasers Eng.49(7), 954–961 (2011). [CrossRef]
  9. J. M. Dawson, L. Wang, P. Famouri, and L. A. Hornak, “Grating-enhanced through-wafer optical microprobe for microelectromechanical system high-resolution optical position feedback,” Opt. Lett.28(14), 1263–1265 (2003). [CrossRef] [PubMed]
  10. G. Y. Zhou and F. S. Chau, “Grating-assisted optical microprobing of in-plane and out-of-plane displacements of microelectromechanical devices,” J. Microelectromech. Syst.15(2), 388–395 (2006). [CrossRef]
  11. A. Bosseboeuf, C. Bréluzeau, F. Parrain, P. Coste, J. Gilles, S. Megherbi, and X. Roux, “In-plane vibration measurement of micro devices by the knife-edge technique in reflection mode,” Proc. SPIE6345, 63451D, 63451D-8 (2006). [CrossRef]
  12. Y. Zhong, G. X. Zhang, C. L. Leng, and T. Zhang, “A differential laser Doppler system for one-dimensional in-plane motion measurement of MEMS,” Measurement40(6), 623–627 (2007). [CrossRef]
  13. J. W. Goodman, Introduction to Fourier Optics(third edition) (Robert & Company Publishers, 2005), Chap. 5.
  14. G. J. Zhang and S. H. Ye, “Online measurement of the sizes of standard wire sieves using an optical Fourier transform,” Opt. Eng.39(4), 1098–1102 (2000). [CrossRef]
  15. L. P. Zhao, N. Bai, X. Li, L. S. Ong, Z. P. Fang, and A. K. Asundi, “Efficient implementation of a spatial light modulator as a diffractive optical microlens array in a digital Shack-Hartmann wavefront sensor,” Appl. Opt.45(1), 90–94 (2006). [CrossRef] [PubMed]
  16. A. Vyas, M. B. Roopashree, and B. R. Prasad, “Digital long focal length lenslet array using spatial light modulator,” in Proceedings of the international conference on optics and photonics. (Chandigarh, India, 30 Oct. −1 Nov. 2009)

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