Optical Characterization Technique for MEMS Comb-Drive Resonators Y. SABRY, M. MEDHAT, B. SAADANY, A. SAFWAT AND D. KHALIL
Abstract— A novel optical technique is proposed for the characterization of MEMS resonators. The proposed technique is based on measuring the response of the resonator (resonance frequency and quality factor) optically, which eliminates the electrical parasitic effects. The proposed technique was applied to a comb-drive resonator and the obtained results show good agreement with the standard electrical technique with 2-5% deviation in the resonance frequency and 2-10% in the quality factor. Index Terms— Comb-Drive Resonator, Fabry-Pérot Interferometer, MEMS Resonator, Optical Characterization, Quality Factor, Resonance Frequency
I. SUMMARY
C
omb drive actuators are essential elements in many MEMS applications such as in sensors or telecommunication fields [1-2]. They have been extensively studied in the last years where different actuators designs have been implemented in inertial as well as optical applications [3-4]. However, the characterization of MEMS comb actuators is a field that needs much more elaboration to overcome the parasitic effects, usually masking the intrinsic resonator performance [5]. On the characterization level, two parameters are usually required to extract the comb equivalent circuit, the resonance frequency and the damping ratio (or the quality factor). When measuring these parameters using standard electrical techniques (such as frequency sweeping or time domain response) we are usually suffering from the interference of the parasitic elements on the measurement results. In addition, in electrical characterization we measure the equivalent impedance without direct access to the comb displacement (which is the main output of the actuator). The proposed technique is based on applying a voltage step on the comb-drive resonator and measuring the reflected optical power from a moving mirror driven by the comb (and connected to it) as shown in Fig. 1. In this case, the damped mechanical oscillations of the comb-drive are translated to equivalent damped oscillations in the sensed optical power. The resonance frequency and the quality factor can thus be obtained by fitting the measured optical power to the theoretically estimated one. A MEMS resonator operating in one-port configuration has an equivalent electrical circuit that consists of two branches in parallel [6]. One of them is the series
Y. Sabry, A. Safwat and D. Khalil are with Ain-Shams University, Faculty of engineering, 1El-Sarayt St. Abbassia, 11517, Cairo, Egypt, e-mail:
[email protected] B. Saadany, M. Medhat, and D. Khalil are with the MEMS Division of SiWare Systems, 3 Khalid Ibn Al-Waleed St., Heliopolis, Cairo 11361, Egypt. e-mail:
[email protected].
combination of the motional parameters Rm, Cm and Lm and the other branch represents the gap capacitance Co. Additional series and parallel components can also be added to represent the electrical parasitic effects. Assuming the time constant of the electrical parasitic elements in conjunction with the capacitance Co is relatively small with respect to the mechanical time constant, an applied unit step voltage on the resonator, results in a displacement given by: (1) x (t ) = x1 + x2e − tζωs sin(ωd t + cos −1 ζ ) where x1 and x2 are constants and 0 < ζ < 1 is the
ωd
damping ratio,
ωs
is the angular damped frequency and
is the angular natural frequency given by:
ζ = 2Rm
Cm 2 , ω d = ω s 1 − ζ and ω s = Lm
1 Lm C m
This is the typical response of a second order system from which system dynamics parameters can be extracted as shown in Fig. 3 (b). As the comb is driving the mirror in front of the fiber (see Fig. 1), the fiber-mirror combination can be considered as a low finesse Fiber Fabry-Pérot Interferometer (FFPI) whose reflected power is approximated by [7]:
P ( x ) = PDC ( x ) + PAC ( x ) cos(
4π x
λ
)
(2)
where λ is the laser wavelength, and PDC and PAC are both slowly varying functions of x with respect to the sinusoidal term. If x(t ) = xo + Δx(t ) such that Δx
