Keywords: Micromirror, MEMS, electrostatic torque, serpentine spring, restoring torque, coupled field analysis. 2. MICROMIRROR STRUCTURE AND DESIGN

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1 Design and Analysis of the performance of a Torsional Micromirror for an Optical switching system Ajay A. Kardak *, Prasant Kumar Patnaik, T. Srinivas, Navakanta Bhat and A. Selvarajan National MEMS Design Centre, Department of Electrical and Communication Engineering, Indian Institute of Science, Bangalore, KA INDIA 561 ABSTRACT This paper describes the design and analysis of a particular micro-electromechanical-system (MEMS) micromirror device that can be used for an optical switching application. Analytical expressions for electrostatic torque and restoring torque of serpentine springs are derived. The performance of the proposed structure is compared with the existing structures having straight torsional beams. The design of the micromirror along wit h the serpentine springs is optimized for large scan angle. Sufficient details are provided to enable this device to be used as benchmark problem for coupled opto-electromechanical analysis. Keywords: Micromirror, MEMS, electrostatic torque, serpentine spring, restoring torque, coupled field analysis. 1. INTRODUCTION Micro-mechanical optical switches are under intensive development these days for the fast growing market of optical fibre networks and optical measurement systems 1,. MEMS optical switches have many advantages aver other switches in terms of low insertion loss, low cross talk and are independent of polarization, wavelength and bit rate and data format. These devices are constructed by using techniques which are very similar to those used in semiconductor industry for fabricating integrated circuits, i.e., a series of deposition and etching processes. Several such processes exist of which one of the most popular for prototyping systems is multi user MEMS processes (MUMPS). The goal of the designer of a micromirror system is to achieve a specified switching time, within a particular pull-in voltage and rotation angle. The design must also be robust. In other words, it must achieve its design goals in the face of inaccuracies in the physical structure and material properties due to random effects in the fabrication process. To achieve large scan angle with relatively small actuating voltage we introduce serpentine springs instead of straight beams.. MICROMIRROR STRUCTURE AND DESIGN In this paper we propose a new configuration of the mirror which improves its performance. A -D system is proposed, which has -degrees of freedom (DOF) micro mirror arrays working in unison. It consists of gimbal configuration. The rectangular mirror is attached to the gimbal and the gimbal in turn is attached to the frame using serpentine springs. Both the mirror and the gimbal are free to rotate with respect to orthogonal axis. This device is actuated using four electrodes; two are placed under the mirror and the other two under the gimbal (fig: 1). Figure 1: (a) Isometric view of Torsional micromirror. (b): Top view of torsional micromirror. * ajay@mems.ece.iisc.ernet.in; phone: ; Fax: ; mems.ece.iisc.ernet.in - 1 -

2 Mirror and serpentine spring dimensions are as follows: Mirror = x µ Serpentine springs: P = µ, b = µ, t = µ Gap between mirror and bottom electrode is µ The overall length and width dimensions of the silicon substrate are 6 µ x 5µ.. ANALYSIS The mirror can rotate around the y-axis by twisting about the inner gimbal structure or the x-axis by twisting about the outer gimbal. In each case, the electrostatic forces rotate the mirror while restoring torque is generated by twisting of the rectangular beams (torsion springs) which support the mirror structure. Electrostatic and restoring torque equations are derived and equated to obtain the scan angle for a given applied voltage. In order to calculate the angle of rotation of torsional mirror, we need to calculate the electrostatic torque and mechanical restoring torque. Since the mirror bends at an angle to the substrate, the profile of the electric field is complex. For calculating electrostatic torque, we use the notations as shown in fig (). W R x Figure (a): Top view When the plate is rotated inward by an angleθ, the length mirror and the bottom electrode. The length of a an arc at x on mirror is Hence d a = x θ sin θ Electrostatic field at x : v v E = = a ( d sin θ x)θ Electrostatic Torque is given by a = ( R x) θ.. (1) (v/m) () Figure (b): Side view L v w x Te = dx ( d x) θ sin θ v w d Lsin θ = log L + θ sin θ d L sin θ R = d sin θ where d is the initial gap between the () Solving the above equation we get: T e (4) The restoring torque equation of the serpentine springs is calculated on the basis of the bending of the vertical beams when a torque is applied about the axis as shown in fig (). Science the horizontal beams in the figure are very small, their twists are neglected. There are two vertical beams of length p and five of length p. The angular rotation for a vertical beam due to torque T is given by: Ml Tp T (P) θ = = or. (5) EI EI EI - -

3 where bt I = 1 The total angular displacement for torque T is the summation of the rotations of all the seven (five long and two short) beams. Then, we can calculate the angular stiffness k constant as fallows: TP 4T (P) 1TP θ total = + =.. (6) EI EI EI Hence Angular stiffness EI k =.. (7) 1P Science there are two serpentine springs, one on either side, effective stiffness will be By definition Torque Therefore, T = Kθ θei θebt T = =.. (8) 5P 6P K = k Where E is the Young s modulus of silicon, b and t are the width and thickness of the serpentine springs. Equating Eq(4) and Eq(8) we can obtain the angle θ for a given applied voltage. P Figure : Torque applied on a Serpentine spring. 4. SIMULATION RESULTS The performance of the micromirror is based on the simulation done using CoventorWare TM. The fig (4) shows the scan angle variation with the voltage applied to mirror electrode (right). The result is compared with that of the mirror having straight beam springs. It can be inferred that a low actuation voltage is sufficient to get the required scan angle when compared to that of the mirror with straight beam springs. The simulation runs predict a pull-in voltage of.5v and an angle of degree for mirror with serpentine springs where as the pull-in occurs at 6.5v and an angle of 1. 1 degree for mirror with straight beams. Fig (5) shows the capacitance variation with the applied voltage. Fig (6) shows the variation of the stress components with the applied voltage. Here S xx, S yy and S zz give the stress distribution in X, Y and Z directions. S mises gives the Von-Mises stress distribution on the mirror and its springs. We found that S xx, S yy, and S mises are concentrated at the junction of the Torsional spring and the mirror. S zz is concentrated at the junction of serpentine spring and the fixed frame. The maximum Von-Mises stress experienced by the structure is 4.4mpa and this is less than the yield stress 7 1 mpa of the poly silicon. Fig (7) and Fig (8) shows the variation of Natural frequency (1 st and nd modes) of the mirror with the applied voltage for both straight beam and serpentine beam mirror respectively. From Fig (7) we infer that the natural frequency is constant with the increasing voltage where as Fig (8) shows that the value of natural frequency decreases with the increase in applied voltage, which is evident from the fact that as the voltage increases the effective spring constant and hence the natural frequency decreases

4 Scan angle (degree) Voltage (v) Serpentine beam Straight beam Figure 4: Scan angle Vs voltage applied to one of the mirror electrode. Capacitance (pf) Serpentine beam Straight beam Volatge (v) Figure 5: Variation of capacitance with the applied voltage. Stress component (mpa) Volatge (v) Sxx Syy Szz Smises Figure 6: Variation of stress components with applied voltage

5 Mirror with serpentine beams Frequencies (Hz) Voltage (v) Mode1 Mode Figure 7: Variation of natural frequency of serpentine beams mirror with applied voltage. Mirror with straight beams 14 1 Frequencies (Hz) Mode1 Mode Voltage (v) Figure 8: Variation of natural frequency of straight beam mirror with applied voltage. 5. CONCLUSIONS A micromirror structure with gimbal configuration is designed using torsional serpentine springs. The performance of this mirror is studied by applying voltage to one of the mirror electrode (right). The results of the mirror with serpentine beam springs are then compared with that of the mirror with straight beam torsional springs. From the above results, we infer that the mirror with serpentine springs has a better performance than that of the mirror with straight beam. Sufficient details are provided in this paper to enable this device to be used as benchmark problem for coupled opto-electromechanical analysis. ACKNOWLEDGMENT One of the authors would like to thank National Program on Smart Materials (NPSM), as this work was supported by NPSM. He would also thank Mr. Jatinder Singh, Mr. Sankaranand and Mr. Saurabh Nishant for their continual help and encouragement through the work at NMDC lab

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