一種垂直梳齒驅動V型梁微鏡設計*

李曉瑩，孫瑞康，燕 斌，喬大勇(西北工業大學微納米系統實驗室，西安710072)

Scanning mirrors are widely used in diverse applications such as barcode readers［1］，laser printers［2］，confocal microscopes［3］，fiber optic network components［4］，laser projection displays［5］，etc．However，traditional macroscale mirrors have limited the performance significantly because of their large volume，high power consumption and slow response．In recent years，MEMS technology enables the creation of light weight，miniaturized，low energy and fast scanning speed micromirrors．And their prominent advantages include low costs，high integration and exce-llent performances over conventional scanning mirrors．

Generally，the MEMS-based scanning micromirrors may be categorized into three classes according to their functionality．The first class works at only two states:ON and OFF．This kind of micromirrors modulates the light beam positioning digitally．Typically，Texas Instruments has made the DMD scanner for digital projection displays［6］．The second class can produce a steady-state deflection，and its tilted angle varies continuously．Under the control signal，the mirror plate can stay any position needed in the range of torsional angle．Such devices are often used in the field of photonic communication networks，e．g．WDM swithes［7］and variable optical attetnuaors(VOA)［8］．The third class is operated in resonant mode，which cannot be used as controllable positioning mirrors to achieve static deflections．However，resonant scanning micromirrors can yield large scan angles with relatively small excitations due to the mec-hatnical oscillation magnifying effect at resonant freque-ncy．Resonant micromirrors have been demonstrated successfully in the applications of wearable displays［9］and spectroscopy［10］．

There are several various actuation methods to excite these micromirrors:electromagnetic actuation，electrostatic actuation，piezoelectric actuation and thermal actuation．Electrostatic excitation is more compatible with IC process and having faster response speed compared with others．In this paper，an electrostatic comb-driven scanning micromirror with V-shaped suspension beams was designed and modeled，which is to be used in spectrum scanning．This micromirror can work in analog mode and resonant mode．The static and dynamic characteristics were analyzed with finite element methods combined with theory calculation，respectively．

1 Operation Principle

The schematic of the scanning micromirror is shown in Fig．1(a)．Fig．1(b)gives the details of one movable finger and two fixed fingers．The scanning micromirror is designed based on single SOI wafer process，and the device is composed of polysilicon layer，insulated layer，device layer，buried oxide layer and handlelayer(the last two layers are not shown in the diagram for simplicity)．Harald Schenk presented a micromirror configuration with coplane comb drives，and the oscillation is started by asymmetries induced during fabrication［11］．However，this way is not convenient and sometimes it may be difficult to start．Here the polysilicon layer separated from device layer is a starting electrode which can provide an effective initial deflection before oscillation．

Fig．1 Schematic of the scanning micromiror and a pair of comb fingers

In Fig．1(a)，there are six electrodes through which the micromirror can be operated flexibly and multifunctionally．When the electrostatic torque between V2and V3pulls down the left side of the micromirror，a voltage can be applied to V4and V5to obtain the other side upward．Of course，the micromirror may also turn around towards the oppisite direction by applying signals to V1，V4and V3，V6．So the driving voltage is reduced by dual side actuation in static mode．With the static deflection，the micromirror achieves oscillation easily under rectangular shaped wave or sinusoidal wave excition，the frequency of which is double of the mirror plate inherent frequency．More details about dynamically excitation was given in reference［12］．

2 Modeling and Analysis

2．1 Geometric Modeling

Fig．1(a)reveals the configuration of the micromirror，except the rectangular shaped torsional bars．Here in our design，we use V-shaped spring beams instead of the traditional rectangular ones，as illustrated in Fig．2．

Fig．2 Schematic of V-shaped spring beams of the micromirror

Dimensions including mirror plate length(Lm)，mirror plate width(Wm)，finger length(Lf)，finger width(Wf)，overlapped finger length(Lo)，finger gap(g)，and thickness of polysilicon(t1)，isolation(t2)，device layer(t3)are totally listed in Table 1．Parameters of the V-shaped spring beams are set as below:Ls=400 μm，Ws=5 μm，b=20 μm，t=30 μm and θ=6．8°．

Table 1 Partial geometric dimensions of the micromirror

2．2 Static Characteristic Analysis

Under the control of DC voltages，the vertically comb-driven actuators will generate an electrostatic torque induced by the change of electric field energy．This torque makes the mirror plate tilted until the electrostatic torque equals to the restoring torque of torsional beams．The two torques can be expressed by(1)and(2)

Where symbols Te，Tsdenote electrostatic torque and spring torque of torsion beams．C，φ，V，Kφdenote total capacitance，angle，applied voltage，and torsion beam stiffness respectively．According to the euqation Te=Ts，the relation between driving voltage and torsional angle can be derived．

In order to obtain the relationship between the voltage and the static rotational angle，we must firstly evaluate the torsional stiffness Kφand the change rate of capacitance with respect to φ．It’s very complicated to calculate the V-shaped beam torsional stiffness and the differential theoretically due to the presence of the strong fringe fields．So a Finite Element Method tool ANSYS was ultilized to extract the parameters．

Fig．3 is the transverse schematic view of the micromirror．Total capacitance is composed of the left capacitance and the right capacitance．In ANSYS environment，with the element solid122，one pair of left capacitance and right capacitance was extracted using the‘CMATRIX’command respectively for different rotational angle．Then at different rotational angle，the sum of left capacitance and right capacitance was multiplied by 2Nfto gain the total capacitance value，where Nfis the number of movable fingers on single side．In our design，Nfis set to be 63．

Fig．3 Transverse view of the micromirror in static mode

The relationship between the capacitance and the rotational angle has been fitted with a five-orders polynomial expression:

Where piare the coefficients reported in Table 2．

Table 2 Coefficients of the capacitance versus angle relationship

From Polynomial(4)，the derivative ?C/?φ can be computed easily．The capacitance versus static rotational angle and the derivative of capacitance with respect to angle φ curves are illustrated in Fig．4．The curves show that the capacitance changes little when the rotational angle reaches about 1．4 degrees．

Fig．4 Capacitance versus rotational angle and differential curves

The torsional stiffness of V-shaped beams is another important parameter needed to be known before static angle versus excitation voltage relationship is characterized．It was extracted in accordance with procedures below:the micromirror with V-shaped suspension beams has been modeled geometrically in a cylindrical coordinate system(Fig．5)，and the rotational axis is along the x direction．Two opppsite forces(Fzand-Fz)have been applied to the two sides of the mirror plate to generate a couple of torques．Variable forces result different tangential displacement around x-axis．the torque and rotational angle can be expressed as(5)and(6)．

Where Lmis the mirror plate length illustrated in Fig．1(a)，and Ztis the tangential displacement．

Fig．5 Meshed model with applied forces

Through applying forces ranging from 0 to 5 μN on nodes，a few angle datas have been obtained．Then Fitting the curve linearly with Matlab Package developed by Mathworks Corporation(Fig．6)，torque versus angle relationship is detailed in(7)．

Where K1and K0have been known to be 1．9837 μN·m/rad and 0．

According to Equation(3)and the simulation results above，the static response of tilted angle with respect to dc bias voltage has been shown in Fig．7．Further increase of the angular range can be achieved by increasing the upper and lower comb fingers’thickness or vertical offset．Decrease of driving voltage can be obtained by increasing the comb finger numbers，finger overlap areas or reducing the finger gap spacing，whereas which has the limitations of being more susceptible to electrostatic instability［12］．

Fig．6 Torque versus rotational angle relationship curve

Fig．7 Static response of angle versus dc bias voltage

2．3 Dynamic Characteristic Analysis

2．3．1 Modal Analysis and Comparison of different beams

For a 1 － D scanning micromirror，generally the first five oscillation modes include torsional motion around spring axis，horizontal motion in mirror plane，rolling motion in mirror plane，vertical motion perpendicular to mirror plane and rocking around the axis perpendicular to spring beams，as shown in Fig．8．Among these modes，torsional oscillation mode should be primary with the lowest frequency．Furthermore，the other modes frequency had better to be large enough and to be separated sufficiently with the first mode frequency．Otherwise the micromirror may work on unstable state and tend to be ruined easily，e．g．the rolling motion in mirror plane may destroy the comb fingers since the moving fingers and fixed fingers are possible to hit each other in this mode．So factors affecting the oscillation modes such as mirror plate size and torsional beams should be considered cautiously．The mirror plate size has been chosen to be 1 000 μm × 1 000 μm，limited by the effective light beam area in our spectroscopy application．Therefore we just need to consider the spring beams．

Fig．8 Five different oscillation modes for a 1 － D scanning micromirror

In this section，we compare three types of torsional beams:rectangular shape，dual rectangular shape and V shape in aspects of mode frequencies and torsional stiffness ratio，which have be seen in Fig．9 From Fig．9(a)～(c)，it can be seen that primarily b varies from zero to a certain value，secondly θ varies from zero to some value．Of course，during the change，Lsand Wsare staying at constants．Here Lsequals to 400μm，and Wsequals to 5 μm．

Fig．9 Three types of torsional spring beams

Fig．10 shows the first five mode characteristics for the three types of torsional spring beams with b and θ changing．And the inserted curves tell us the frequency ratio of the 2nd order mode to the 1st order mode．For dual rectangular-shaped beams，the frequency ratio falls down with b increasing．Comparing to the V-shaped beams’frequency ratio，which rises first and then decreases with θ gradually increasing，the latter is more reliable to make the mirror functioning correctly．In the vicinity of 0．10 rad，frequency ratio peak is obtained．Actually in our model，θ=0．12 rad was chosen as already mentioned in section 2．1．

Fig．10 Modal comparison of different spring beams

In addition，the vibration mode of the same order is not always keeping the same when parameters change．The modes are all listed in Table 3 for the three types of beams．And also the V-shaped beams with θ=0．12 rad have identical vibration modes with that θ=0．10 rad．The mode eigenfrequency can be described as

Where Kφand Ktare the torsional stiffness and translational stiffness of spring beams，and Iφis the moment of inertia．M is the mirror plate mass．

Table 3 Vibration modes for the three types of beams

The 1st torsional mode stiffness Kφhas been extrcted with FEM method and linearly fitting in Matlab．For rectangular mirror plate，the moment of inertia and the mass for torsional mode are given as

The vertical mode stiffness Ktis expressed as

In Equation(9)～(11)，ρ is the material density，tmis the mirror plate thickness regarded as the device layer thickness t3here，E is the modulus of elasticity，tsis the spring beam thickness which is the same to tm，and the other signs like Wm，Lm，Ws，Ls，Nf，θ have already been illustrated in previous paragraphs．

The simulated resonance frequencies of the first five modes and the theoretically ones of the first two modes for V-shaped beams with θ=0．12 rad are reported in Table 4．the FEM simulation results agree well with the theoretically calculated ones for the 1st order mode，whereas the deviation is large for the 2nd order．And the reason may be that Ktexpression is not so accurate．

Table 4 FEM simulated frequencies of the first five modes and theoretically calculated frequencies for the micromirror with V-shaped beams with θ =0．12 rad．Lm=Wm=1000 μm，Ws=5 μm，Ls=400 μm，Wf=4 μm，Lf=100 μm，tm=ts=t3=30 μm，Nf=63，ρ =2．33 ×10 －15kg/μm3，E=130 ×103Mpa

2．3．2 Stress Analysis

Torsional stress impacts the deflection angle and the durability of a micromirror tremendously．It is very critical to forecast the maximum stress at variable torsional angles，aiming to avoid fracture happening．Fig．11(a)shows the stress distribution contour clearly when torsional angle reaches 0．05rad．And obviously the maximum stress concentrates at the fixed ends of the torsional beams．Fig．11(b)gives the deflection displacements of the mirror plate and the close-up of the distortion torsional beams，which indicates that the distortion consists of bending and torsional deflection effect．

Fig．12 reports that the stress is linearly proportional to the torsional angle．The slope of the curve，i．e．the stress per angle is computed to be 71 MPa approximately，higher than 58 MPa that Alexander Wolter et al．got［13］．Because of the torsional motion，the V-shaped beams have both bending deflection and twisting deflection simultaneously．According to the measured fracture stress(2．0 ～ 2．5 GPa)by Wolter［13］，the maximum mechanical deflection amplitude of our micromirror with V-shaped beams could be 28 degrees．

Fig．11 (a)FEM simulated torsional stress distributed contour plot and(b)the deflection displacement color map of the mirror

Fig．12 Stress versus torsional angle curve，showing approximately linear characteristic

3 Conclusions

With FEM simulation and numerical analysis，a dual-mode excited micromirror was designed．The static torsional angle versus applied driving dc voltage relationship was derivated through capacitance extraction．The dynamic characteristics including modal analysis，torsional stress distribution and dynamic deformation were studied．The frequency ratios show that V-shaped torsion beams make the first mode far from the higher modes and enable a stable operation in torsional mode．The designed micromirror will be utilized to scan spectrum and certainly can also be used in many other applications．

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