WO2008087631A1

WO2008087631A1 - A tilting device with integrated motion transformer and amplifier

Info

Publication number: WO2008087631A1
Application number: PCT/IL2008/000058
Authority: WO
Inventors: Slava Krylov; Assaf Ya'akobovitz; Yossef Shacham
Original assignee: Teraop Displays Ltd.
Priority date: 2007-01-16
Filing date: 2008-01-14
Publication date: 2008-07-24

Abstract

The present invention provides a novel architecture and operational principle of a tilting actuator with integrated motion transformer and amplifier fabricated using single structural layer of silicon on insulator (SOI) wafer. The device incorporates an integrated compliant motion amplifier realized as an eccentric elastic torsion link that transforms small out-of-plane motion of the parallel plate electrostatic transducer into large amplitude angular motion of the tilting element. This generic architecture, combining simple fabrication process with robustness of SOI based devices, is efficient for various applications such as static and resonant operation of various tilting micro devices.

Description

A TILTING DEVICE WITH INTEGRATED MOTION TRANSFORMER AND

AMPLIFIER

FIELD OF THE INVENTION

The present invention relates generally to itiicromachined structures incorporating tilting element and more particularly to tilting elements suitable for use in optical applications.

BACKGROUND OF THE INVENTION

Micromachined structures incorporating tilting elements are a core part of many microdevices . Those include mainly light processing devices, micromirrors for laser display such as those described in "Optical Raster- Scanning Displays Based on Surface-Micromachined Polysilicon mirrors", by M. Hagelin and 0. Solgaard, IEEE J. of Selected Topics in Quantum Electronics, 5 (1999) 67-74, optical communications applications such as those described in "An Optical CrossConnect (OXC) Using Drawbridge Micromirrors", by A. Q. Liu et al, Sens. Actuators A, 97-98 (2002) 227- 238 and "A 5-V Operated MEMS Variable Optical Attenuator by SOI Bulk Micromachining" by K. Isamoto et al, IEEE J. of Selected Topics in Quantum Electronics, 10 (2004) 570-578. Other applications are for example angular rate sensors described among others in "Design, Fabrication and Operation of MEMS Gimbal Gyroscope", K. Maenaka et al, Sens. Actuators A, 121 (2005) β 15, radio frequency (RF) devices and biomedical systems. In these devices, tilting elements are typically suspended using elastic torsion axes or bending flexures while an actuating torque is applied directly to the tilting element. In spite of the fact that a large variety of actuation principles are known in the art (e.g. thermal actuation, piezoelectric actuation, magnetic actuation and even acoustic actuation) , electrostatic actuation remains the most widely used due to its efficiency and favorable scaling laws at microscale as well as relative simplicity of fabrication process and compatibility with integrated circuits environments.

The historically first designs of electrostatic devices were based on the use of driving electrodes placed underneath the tilting element. This close-gap architecture is still widely used mainly due the simplicity of the fabrication process and ability to provide high torque at low actuation voltages. However, the tilting angles achievable in the framework of the close-gap architecture are limited, at least for relatively large devices a few hundreds of micrometers in size, due to necessity to provide small distances between electrodes. Vertical comb drive architecture provides an attractive alternative. It permits the achievement of large tilting angles at reasonable actuation voltages and is widely used in micro devices incorporating tilting elements. However, some of the drawbacks of this architecture are the more intricate fabrication process and the necessity to fabricate a large number (typically from a few tens to a few hundreds) of small elements with the dimensions comparable with the minimal feature size of the process. It is worth noting that even though historically first tilting actuators were fabricated from polysilicon using surface micromachining the use of single crystal silicon, especially combined with silicon on insulator (SOI) technology, has only recently become increasingly popular due to its excellent mechanical properties and high reliability and robustness of SOI devices. On the other hand, the number of structural layers available in the framework of SOI technology is rather limited, which in turn restricts possible design solutions or complicates the fabrication process, especially in the case of vertical comb drive architecture. While titling devices fabricated from a single structural layer, operated dynamically and benefit from resonant amplification are devices that are known in the art, the achievement of large tilting angles under static operation of SOI devices is still challenging. Several reported designs mainly based on the vertical comb drive architecture typically utilize prefabricated bonded substrates and multistage etching with multiple aligned masks and critically timed etch stop.

One of the approaches that enables the achieving of larger tilting angles in electrostatic actuators is the incorporation of motion amplifiers. In these devices, small linear or tilting motion of an electrostatic transducer is transformed into large angular motion of a titling element. The devices fabricated from polysilicon using surface micromachining, typically actuated by linear comb drive or scratch drive actuators and incorporating hinges, are characterized by good performance but often suffer from reliability problems. Also, few designs of SOI made devices were reported as well. A SOI device with an arm type angular-to-angular motion transformer operated using vertical comb drive actuator was reported by J. A. Pelesko and D. H. Bernstein, in "Modeling of MEMS and NEMS", Boca Raton, FL, Chapman & Hall/CRC, 2003, and a SOI motion amplifier transforming linear in-plane motion of a comb drive actuator into tilting motion of a micromirror was reported by V. Milanovic et al in "Laterally Actuated Torsional Micromirror for Large Static Deflection", IEEE Photonics Technology Letters, 15 (2003) 245-247. A SOI device incorporating polysilicon elements and an arm type angular to angular motion transformer and amplifier is also reported in "A Novel Electrostatic Micromirror for Large Deflections in MEMS Applications", by J. Singh et al., Thin Solid Films, 504 (2006), 64-68. A kinematically excited resonant SOI device incorporating a piezoelectric film and based on the transformation of an out-of-plane motion of the transducer into an angular motion of a micro mirror has been described by Filhol, et al in "Resonant Micro-Mirror Excited by a Thin-Film Piezoelectric actuator for fast optical beam scanning", Sens. Actuators A, 123 (2005), 483-489. A device with magnetically actuated micromirror and magnetic motion measurement was described by Z.Cui et al in their "High Sensitive Magnetically Actuated Micromirror for Magnetic Field Measurement", Sens. Actuators A, 138 (2007) 145- 150. Though large tilting angles were presented in most of those works, they are typically characterized by intricate fabrication process.

In general, the efficiency of the motion amplification combined with the parallel plate electrostatic transducer is based on the nonlinear dependence of the electrostatic force on the distance between electrodes. Due to this nonlinearity, the gain in force achieved by the decrease in the distance between the electrodes is relatively larger than the reduction of the achievable displacement of the electrodes and the motion amplification is beneficial.

The disclosure of the references mentioned throughout the present specification are hereby incorporated herein by reference in their entireties and for all purposes. SUMMARY OF THE INVENTION

It is the object of the present invention to provide a method and device for actuation of tilting that allow motion transformation and amplification using SOI-based technology.

It is another object of the present invention to provide a tilting actuator realized from a single structural layer of a SOI wafer and incorporating an integrated compliant motion transformer and amplifier.

It is yet another object of the present invention to provide a method whereby small out-of-plane deflection of an energetically efficient large area parallel plate electrostatic transducer is transformed into a large angular motion of a titling element.

It is another object of the present invention to provide a method for combining a simple fabrication process with robustness of SOI based devices, whereby obtaining devices that can be efficiently used for static and resonant operation of titling micro devices including micro mirrors, optical switches and attenuators and micro manipulators .

Other objects of the invention 'will become apparent as the description of the invention proceeds. According to a first embodiment of the present invention there is provided a tilting element comprising: a rotating plate operative to rotate about one of its axes; an electrode movable by applying electrical voltage thereto and characterized by having at least one of its surfaces located at a plane parallel to that of at least one of the surfaces of said rotating plate, at rest position, i.e. when no electrical voltage is applied, and wherein the rotating plate and the movable electrode are connected to each other by a join, preferably an elastic join.

According to a preferred embodiment of the invention, the tilting element is made of a single layer of silicon on insulator ("SOI") .

In accordance with another embodiment of the invention, the tilting element is further characterized in that small out-of-plane deflections of the movable electrode are transformable into substantially large angular motions of the rotating plate.

By still another embodiment of the invention, the join is attached to the rotating plate at an offset relative to that one of the rotating plate's axes. According to yet another embodiment of the present invention the rotating plate is attached to the moveable electrode by an elastic torsion join, and is preferably attached to a substrate by a pair of elastic torsion axes . In accordance with another preferred embodiment of the present invention the tilting element is further characterized in that substantially large tilting angles are achievable under , static operation of the tilting element . These and other embodiments and objects of the present invention will become apparent in conjunction with the description and claims that follow.

BRIEF DESCRIPTION OF THE DRAWINGS For a more complete understanding of the present invention, reference is made to the following detailed description taken in conjunction with the accompanying drawings wherein: FIG. 1 presents a schematic view of the operational principle of a deivce constructed in accordance with an embodiment of the present invention;

FIGs. 2 illustrates a schematic view of the device: FIG. 2A is an issometric view;

FIG. 2B is a top view, and

FIG. 2C is a deformed cross section of the device; FIG. 3 illustrates lumped model parameters of the device;

FIG. 4 presents a unit cell area with and without release holes;

FIGs. 5 present equilibrium curves for different DOF' s of the device calculated while using the "force control" approach:

FIG. 5A - the mechanical tilting angle of the plate, FIG. 5B - the actuator deflection,

FIG. 5C - the plate deflection, and

FIG. 5D - the mechanical tilting angle of the actuator;

FIGs 6. present equilibrium curves for different DOF' s of the device calculated using the "displacement control" approach:

FIG. 6A - the mechanical tilting angle of the plate, FIG. 6B - the actuator deflection, FIG. βC - the plate deflection, 'and FIG. 6D mechanical tilting angle of the actuator, where the dashed lines correspond to unstable branches of the equilibrium curve;

FIGs 7. present static response by way of illustrating the dependence of the mechanical tilting angle on various parameters : FIG. 7A illustrates the dependency of the motion amplification on the deflection of the actuator, and

FIG. 7B the pull-in angle for different offset values;

FIG. 8 - presents a natural mode of the device obtained by the Finite Elements ("FE") method;

FIGs. 9 - present time history response for certain set of conditions, γ_dc =v_∞ =62 V , Q-factor of 10 and excitation frequency of 3.2 kHz:

FIG. 9A shows the mechanical tilting angle of the plate,

FIG. 9B shows the deflection of the actuator,

FIG. 9C shows the deflection of the plate, and

FIG. 9D shows the mechanical tilting angle of the actuator. One observes that the response is not symmetric.

FIGs. 10 present mechanical peak to peak tilting angle of the plate for Q - factor of 10. In these figures the time dependency of the ac component of the voltages applied is demonstrated at:

FIG. 1OA the resonant frequency {ω=ω_{) , and

FIG. 1OB a half of the resonant frequency (U)=CSo₁);

FIGs. 11 present mechanical peak to peak tilting angle of the plate for Q-factor of 10 at all directions, versus the normalized angular frequency of excitation, where

FIG. HA is presented for κ_Λ =50F, V_ac=5Q>V and FIG. HB for FIGs. 12 illustrates main stages of a fabrication process carried out in accordance with an embodiment of the present invention;

FIG. 12A - Initial SOI wafer; FIG. 12B - Front side lithography, DRIE of the device Si layer and RIE of the silicon dioxide (BOX) layer;

FIG. 12C - Back side lithography and RIE of the backside silicon dioxide (hard mask) ;

FIG. 12D Front side protection using silicon dioxide layer;

FIG. 12E - Back side DRIE of the handle; and FIG. 12F - Release of the chip using hydrofluoric (HF) acid;

FIG. 13 presents SEM micrograh of devices fabricated in accordance with the present invention;

FIG. 14A illustrates the experimental setup used;

FIG. 14B illustrates extraction of optical tilting angle, θ_opt '

FIG. 14C presents several screen shots of the scanned laser beam corresponding to different values of excitation frequency;

FIG. 15 presents experimental resonant curve (dots) of the device with integrated motion amplifier kinematically excited using an external piezoelectric transducer; FIG. 16A shows a schematic representation of the pure kinematically excited device;

FIG. 16B presents SEM micrograph of the pure kinematically excited device;

FIG. 17 presents experimental resonant curves for the pure kinematically excited device actuated using an external piezoelectric transducer for applied voltage of 5 V (open circles) and 10 V (blackened circles);

FIG. 18 presents experimental resonant curves (dots) of the device with integrated motion amplifier corresponding to the actuation voltages of F_Λ = 20F K₄₀ =SOK applied to the parallel-plate actuator; and

FIG. 19 presents experimental resonant curves of the device with integrated motion amplifier for actuation voltages of

applied to the parallel-plate actuator.

DETAILED DESCRIPTION OF THE INVENTION

The present invention will be understood and appreciated more fully from the following detailed examples taken in conjunction with the drawings.

Let us consider now Fig. 1 which presents an example of a schematic illustration of the underlying operational principle of a device according to an embodiment of the present invention. A tilting element (referred to hereinafter as the "plate") is operative by rotating around an axis fixed in space, and a movable electrode (referred to hereinafter as the "actuator") of a parallel plate electrostatic transducer, are connected by a rigid join. The join is attached to the tilting element at some offset e measured from the axis. In the case that this offset is comparable with the stroke of the parallel-plate electrode, small out-of-plane displacement of the actuator w_AC is transformed into a large tilting of the plate with an angleθ = arcsm(w_AC /e)∞w_AC /e , as demonstrated in Fig. 1. It should be noted, that since the angle between the join and the axis and the plate is changed during the deformation, these elements should preferably be attached to the tiling element by hinges. Although

I hinges can be realized in the framework of the polysilicon based process, they often characterized by inferior reliability. The device exemplified is shown schematically in FIG. 2. It comprises the tilting element attached to the substrate by a pair of elastic torsion axes and to the parallel plate electrostatic transducer (the actuator) by an elastic torsion join. The actuator is attached to the substrate by elastic bending flexures. Due to the offset, e between the axis and the join, a small out-of-plane z- direction motion of the actuator is transformed into a large angle tilting motion of the plate (as may be seen for example in the deformed cross section illustrated in FIG. 2C) The pair "torsion axis-torsion join" serves therefore simultaneously as an integrated suspension, motion transformer, and motion amplifier.

The device of the present example has been fabricated using silicon on insulator ("SOI") wafer consisting an upper silicon layer (device layer) , silicon dioxide insulating layer and a bottom silicon layer (substrate) . The fabrication process is described in details further below. The plate, the actuator, the axes and the flexures are fabricated from the device layer of the SOI wafer and their thickness is defined by the thickness of this layer. The initial distance between the actuator and the substrate is determined by the thickness of the sacrificial silicon dioxide layer. The cavity has been provided in the substrate to allow large angular motion of the plate, and the release holes in the actuator, for technological reasons enabling sacrificial release of the large area actuator from the silicon dioxide layer. Although various actuation methods can be implemented while using the suggested motion amplification structure, the present device is operated electrostatically. A voltage difference is applied between the actuator, which operates as a movable electrode, and the substrate, which operates as a stationary electrode and is electrically insulated from the actuator by a silicon dioxide layer. The large surface area and small distance between the stationary and moveable electrodes makes the electrostatic actuation very efficient. It should be noted that due to the parallel plate architecture of the transducer, the device does not incorporate elements with small critical dimensions (e.g., comb fingers in comb drive actuators) or hinges that often suffer from friction related reliability problems.

The Model

In order to analyze main features of the static and dynamic behavior of the device and evaluate the design parameters, a lumped model of the device has been constructed. In the model, the device is considered as an assembly of a rigid actuator and plate and deformable elastic mass less axis, join and flexures. The four degrees of freedom (DOF¹ s) model accounts for out-of- plane and tilting degrees of freedom of the plate and the actuator. The tilting of the plate and of the actuator around y -axis (as shown in FIG. 2A) are denoted by θ and φ , respectively, and deflection of the plate and actuator in the out-of-plane z-direction are denoted bys w_P and w_AC , as shown in FIG. 3.

The motion equations were derived while using variational principle, by substituting expressions for kinetic, strain and electrostatic energy into Euler- Lagrange equations. Since the lateral dimensions of the actuator are much larger than the distance between the actuator and the substrate, the equations were simplified by assuming that the tilting angle of the actuator, φ, is relatively small. It should be noted however that this DOF is not negligible and should be included into the model since for certain combination of parameters electrostatic (pull-in) instability of the actuator is associated with its tilting which results in undesired behavior.

Under the above assumptions, the kinetic energy associated with the angular velocity and linear velocity of the plate and actuator, is given by the expression

T = -I_pθ² +-m_ACw_A ² _C +-m_Pw_P ² +-I_ACφ² (1)

where I_p , I_AC , m_p and m_AC are mass moment of inertia of the plate and actuator and mass of the plate and the actuator, respectively. Hereinafter the use of an over dot, ( ), denotes derivative with respect to time, t.

The potential energy consists of strain energy associated with the torsion and bending of the axis and of the join and bending of the suspension flexures, as well as of the electrostatic co-energy. '

U=±C_L(θ-φ)² +-C^e² +^(W₁, +X^sInOf

+ ^ ^BL I ^WAC + (^XL - XCAC )<P ~ Wp - *L ^{sin θ}f

+-25_S [w_AC + (x_SR -x_CAC)φf +-25_S [w_AC -{x_CAC -x_SL)ψt

2φ {g₀ +w_AC -φL_out /2) 2φ {(g. +w^ -φLJl) )

( 2 ) In this equation, (2), C_4x and B_M are torsion and bending stiffness of a pair of axes, similarly C_x and B_L are torsion and bending stiffness of a pair of links and B_s is the bending stiffness of a pair of suspension flexures (shown in FIG. 3) . In addition, L_ouτ is the outer length of the actuator while g₀ is the initial gap existing between the actuator and the substrate; x_CAC , ^X _AX ' ^X _L ^are the x -coordinates of the actuator's center of mass, neutral axis of the axes and neutral axis of the joins, respectively. The locations of the neutral axis of the suspension flexures that correspond to the right and left side of the actuator are denoted by x_SR and x_SL respectively, as shown in FIG. 3. The term for Rayleigh's dissipating function, resulting mainly from air damping is given as follows:

^D=₂ ^C^^{2 +} ₂ ^C^^{C +} ₂ ^Ci™^{p2 +}I^C^^{2 (3)}

where C₁, C₂, C₃ and C₄ are the damping coefficients associated with the plate's tilting, deflection of the actuator, deflection of the plate and tilting of the actuator.

As explained before, the equations of motion were derived by using variational principle. Substituting for the expressions for the kinetic, Eq. (1) and potential Eq. (2) energy as well as Rayleigh's dissipation function Eq. (3) into Euler-Lagrange equations results in a system of four coupled nonlinear ordinary differential equations. The nonlinearity arises due to large titling angles of the plate as well as nonlinear configuration- dependent electrostatic force. In order to simply the description, it has been assumed that the tilting angle of the plate is relatively small. In this case, as encountered in many MEMS devices, the elastic restoring forces and inertia forces are linear in terms of state variables whereas all nonlinearity is due to the nonlinear electrostatic force, and the equations of motion may be written in a matrix form.

Mu+Cύ+Ku = P (4)

In this equation, M =1 [l_p,m_AC,m_p,I_AC}^T is the diagonal

mass matrix,

is the diagonal damping matrix, P={o,F,O,M}^r is the load vector composed of the nonlinear electrostatic force and moment acting on the actuator and n = {θ w_AC w_P φ] is the vector of DOFs, I is the 4x4 unit matrix, I_p , I_AC, m_p and m_AC are the mass moment of inertia of the plate and actuator and the masses of the plate and actuator, respectively. The elements k_tJ of the symmetric stiffness matrix, K, are

Jc_n = C^ +C_L+B_Le² k_n=-B_Le k₁₃=B_Le k_u=-(C_L+B_Le²) k₂₂=B_L+4B_{s (5)}

K₂₄ — B_Le + B_SL_OUT + B_SL_OUT k33 ^{= B}AX^+BL

Ki =C_L+ B_Le² + 0.5B_SL_OUT + 0.5B_SL_OUT

where the offset, e, is defined as B = X₁-X_4x (as may be seen in FIG. 3) . It should also be noted that the release holes provided in the actuator for technological reasons affect both the density of the actuator' s material as well as the electrostatic force acting thereon. In this analysis the influence of release holes was approximated by using effective averaged values, while defining a ratio between the unit effective area of the actuator without holes and the unit area with holes (as presented in FIG. 4)

where d_H and L_H are distance between the holes and length and width of one hole respectively, and use in calculations the effective density/? = /?r of the actuator with release holes.

The electrostatic force and moment were calculated from energy considerations by differentiating of the electrostatic co-energy with respect to the deflection and tilting DOF' s of the actuator viewed as an inclined planar capacitor for the case of a single DOF, for the case of two DOF) . Under the assumption that the tilting angle of the actuator is small, the force and moment are given by the expressions

( 6) M _S^_/( w,_r, φ, _TVΛ) = — ^ε _ύV²W_o Q_υuj_τi ⁾\ \ _Λ J_n ( g ^& ₀o + w_A A_Cc —+ φL_o O_uU_τT —/ 2 L₀UTJg₀ + W Ac)

^{V AC}'^{Ψ }} 2φ \ φ {g_Q +w_AC -<pL_OUT /2_, {go +^wAc) -(^Louτ<P/2)

(7)

L_OUT ' ^_OUT ' L₁ and W₁ are the outer length and width of the actuator and length and width of the opening in the actuator (as illustrated in FIG. 2B) , respectively, ε₀ is the dielectric permittivity of air. In addition, in order to account for the influence of the release holes, the actual voltage V which is applied to the actuator and in a general case can be a function of time, is replaced by the effective value

, where r is defined by Eq. (6) . Note that the approximation used for the evaluation of the electrostatic force and moment, Eqs . (7 and 8), neglects the influence of fringing fields arising due to the presence of the release holes. The influence of the fringing field was evaluated using a three dimensional electrostatic simulation performed using the IntelliSuite™ software. It was found that for the case of d_H =15 μm and L_H =10 μm and the thickness of the device layer of 35 μm (see FIG. 4 and Table 1) the relative contribution of the fringing fields to the electrostatic force is of the order of several percents and Eqs. (7, 8) provide a satisfactory approximation.

Model Results

Calculations were performed using as an example a device associated with the parameters presented in Table

1. In addition, the Young's modulus and Poisson's ratio of Si used in the calculations, were, E = 169 MPa and v =

0.28, respectively and the density was p = 2300 kg/m³. The parameters used in calculations were chosen as they were found to lead to a preferred device behavior. However, it should be noted that a change of parameters and selecting an unsuccessful set of parameters might lead to a different, even undesired behavior of such a device.

Table 1. Geometrical parameters of the device used in calculations

Static Mode Operation

In the case of static actuation, the voltage in Eq.

(6) and (8) consists only of steady component V = V_dc , and the system of four coupled nonlinear ordinary differential equations Eq. (4) is reduced to a system of i four nonlinear algebraic equations

Ku =P (8)

This system of Eq. (9) was solved numerically by the least square method implemented in MATLAB. Since the parallel plate actuator may exhibit pull-in instability, design parameters were chosen such that the desired mode of pull-in, namely tilting of the plate and downward motion of the actuator, was achieved. Equations (8) were first solved using the "force control" approach. The voltage applied to the electrodes and consequently the actuating force, was prescribed and the unknown DOF' s were found as solutions of Eq. (9) . Although this approach reflects the physical behavior of the device in an experiment, only stable equilibrium configurations can be described, see FIGs 5. In order to describe unstable configurations, "displacement control" was used and

Equations (8) were solved for the prescribed w_AC

(actuator deflection) while the remaining DOF' s and actuation voltage were considered as unknowns. The results obtained are shown in FIGs. 6. The device with the parameters listed in Table 1 exhibits pull-in instability at the actuation voltage f 129.6 V . pull-in value of the actuator's deflection is -1.33 μm and the corresponding pull-in value of the tilting angle is 4.6° . It should be noted that even at large angles, the mechanical stresses in the axis and the link did not exceed several tens of MPa.

FIG. 7A indicates that a small displacement of the actuator results in large tilting angle of the plate. Motion amplification ratio θlw_AC is influenced mainly by the stiffness of the axis and the join and is independent of the actuation voltage. It is interesting to note that though intuitively one may expect that smaller offset, e , should result in larger amplification ratio, it was found that for given geometrical parameters of the axis and the join, an optimal value of the offset exists resulting in maximal amplification, as may be seen in FIG. 7B.

The results provided by the lumped model were further verified by using the coupled three-dimensional simulation carried out using IntelliSuite™ software. This software enables the solution of coupled electromechanical problem, while the elasticity problem (mechanical domain) is solved using the finite element (FE) method and the electrostatic problem (electrostatic domain) is solved using the boundary elements method. Similarly to the case of the lumped model, in order to consider the influence of the release holes, an effective voltage V was used in simulations instead of the actual voltage. In contrast to the lumped model, where the actuator is considered as a rigid body, the release holes affect the stiffness of the actuator. Since the device has thousands of release holes, the direct evaluation of their influence is computationally intensive. To overcome this hurdle, the effective mechanical properties of the perforated actuator, namely Young' s modulus and Poisson' s ratio, were evaluated using the High Fidelity Generalized

Method of Cells (HFGMC) and were confirmed by using the three-dimensional FE analysis. It was found that for a distance between the holes of d_H=\5μm and holes width and length of L_H =10 μm (see FIG 4) the effective Young's modulus in x and y directions is 112 GPa and the Poisson' s ratio is υ = 0.28. These effective mechanical properties were used in the three-dimensional simulation and a good agreement between the FE and the lumped models was achieved as demonstrated in FIG. 8.

Dynamic Operation

First, the natural frequencies of the device were obtained by using the lumped model and were verified by the FE analysis. Both models predicted that the first eigenmode is the desired mode characterized by the dominant tilting motion of the plate, as shown in FIG. 8. The corresponding natural frequency is 3.2 kHz, see Table 2. It should be noted that some eigenmodes obtained using the FE analysis were missed and were not predicted by the lumped model since it does not incorporate the DOF' s associated with these modes. However, the FE simulation shows that higher modes have natural frequencies which are much higher than the frequency of the first, desired, mode and have minor influence on the device dynamics, at least under resonant excitation.

Table 2. Natural frequencies and modes obtained by the lumped model and FE simulation.

In the case of forced excitation, the system of four nonlinear ordinary differential equations Eq. (4) completed by zero initial conditions was solved numerically using the Runge-Kutta solver implemented in MATLAB. The voltage applied to the system was taken as a combination of a steady component V_dc and time dependent voltage with amplitude V_ac

V = V_dc+V_acsm(ωt) (9)

Under forced resonant excitation ^' by V_dc=V_ac = 62V , quality factor of Q = 10 and excitation frequency equal to the device first natural frequency (3.2 kHz) a mechanical tilting angle of the plate of 9 degree pk-pk was obtained, see FIG. 9. For the quality factor of 100, mechanical angle up to 29 degrees pk- pk was achieved for V_dc=V_ac=40V . Note that the mechanical angle pk-pk refers to the angle between the two extreme positions of the plate.

From Eqs. (6) and (7) one may observe that the actuation force and moment are proportional to the square of the actuation voltage

V² = V_d] + 2V_dcV_ac Si_n(O -~V_a ² _c cos(2ωt) ( 10 )

In this expression, there are two time dependent terms. The first term excites the device at the angular frequency of ω while the second term excites the device at angular frequency of 2ύ) . When the excitation frequency ω is close to the first resonant frequency O)₁ of the device, the dominant resonant term in Eq. (10) is 2V_dcV_ac sinζωj) and the dependence of the actuating forces on the ac and dc voltages is linear. In this case, the relation between the applied dc and ac voltage and the amplitude has a shape of bilinear surface, as shown in FIG. 1OA. On the other hand, when the device is excited at half of its resonant frequency

the dominant term in Eq. (10) is 0.5J^. cos(2ω/) and the amplitude has quadratic dependence on the applied ac voltage, FIG. 1OB. Equation (10) suggests that the device reaches resonance when excited at its resonant frequency or half of its resonant frequency, FIG. 11. By changing the ac and dc components of the applied voltage the ratio between the amplitude of these two peaks can be controlled, FIGs HA and HB.

Fabrication of the device and the experimental setup

Devices of several configurations were fabricated of highly doped single crystal Si using SOI wafer as a starting material with upper device layer of 35 μm, silicon dioxide (BOX) layer of 4 μm, handle layer of 390 μm and 3 μm layer of backside silicon dioxide (see FIG. 12A) . Due to the single layer architecture and the absence of elements of critical dimensions (e.g., comb fingers) the device is characterized by a simple fabrication process requiring two masks and no precise alignment. In addition, no multistage critically timed etching is involved. The main steps of the fabrication process are shown schematically in FIG. 12. The spinning of photoresist ("PR") and front side lithography was followed by deep reactive ion etching ("DRIE") of the device layer and reactive ion etching ("RIE") of the BOX layer (FIG. 12B) . The backside process included a deposition of PR, backside lithography with rough alignment and RIE of the silicon dioxide to form the backside hard mask (FIG. 12C) . At this point, in order to provide an additional mechanical strength to delicate elements of the device layer and prevent an undesired over etch, the front side was protected by 1 μm of silicon dioxide (FIG. 12d) using plasma enhanced chemical vapor deposition ("PECVD") . Is should be noted that three-dimensional FE simulations of the device at different fabrication steps revealed that this protective layer is beneficial and may prevent significant warping, buckling and even breakage of tiny suspension elements arising due to high compressive residual stress within the BOX layer. Next, DRIE of the handle was performed in order to create the opening beneath the tilting plate (FIG. 12E) and in order to separate the dies. It should also be noted that the separation of the dies using DRIE prevents the need for the dicing of the wafer - a process which might damage delicate mechanical elements. Finally, the devices were released using hydrofluoric (HF) acid (FIG. 12F) . In order to avoid attachment between the large area actuator and the substrate during the release, the drying process was performed in a super critical point drying tool. Scanning electron micrographs of the fabricated devices are shown in FIG. 13.

It should be noted that direct measurements performed using scanning electrons microscope ("SEM") revealed that due to relatively low tolerances of the fabrication process, several structural elements, mainly the link and the axis (see FIG. 13) , have dimensions and even shape different from the nominal values. This deviation of the geometry parameters was taken into consideration and all the model results presented hereinafter correspond to the nominal values listed in Table 1, except the width of the axis and the join, which were replaced by the actual values of d^ =4.6 μm and d_L=4.6μm , respectively. In addition, the part of the substrate having a shape of a narrow holder (the "bridge") and located beneath the part of the titling element, was introduced for technological reasons and was used for the attachment of the internal anchor (FIG. 13) . The influence of the "bridge", which may mechanically limit the titling motion of the plate, was not taken into consideration in the theoretical model presented above.

Actuators of various configurations were operated in ambient air conditions. The experimental setup is presented in FIG. 14A. The motion was recorded by using an optical microscope and a CCD camera. In order to quantify the angular response, an optical deflection technique was implemented. A laser beam pointed to the device was re-directed by the tilting plate to a screen

(FIG. 14B) . A similar setup has been described by Z. Cui et al, in "High Sensitive Magnetically Actuated Micromirror for Magnetic Field Measurement", Sens. Actuators A, 138 (2007) 145-150. The location of the moving laser spot on the screen was video captured and the resulting video data were analyzed using image processing. In the case of static operation by a linearly increasing voltage, the deviation of the laser spot from its initial position was measured (in pixels) using an image processing software and a customized procedure implemented in Matlab. The result was then re-scaled providing the actual position of the spot and the optical tilting angle defined as a total angle scanned by the laser beam (see FIG. 14B) . In the case of resonant operation, the laser spot scanned a line at the screen, FIG. 14C. The length of this line depends on the vibration amplitude of the plate. Since the video recording of the line was synchronized with the linear sweep of the excitation frequency, each video frame

, corresponded to a specific value of the frequency. For each frame, the length of the scanned line was measured using the image processing and the resonant curve was restored in terms of optical tilting angle.

Experimental Results First, the device was actuated statically using only steady dc component of the voltage. The device exhibited the pull-in voltage of 151 V , which is in a good agreement with the theoretically predicted pull-in voltage of 145 V . On the other hand, the measured pull-in optical angle of 3.5°, as well as the maximal optical angle close to 7°, are smaller than the theoretical values (theoretical pull in optical angle is 8° , maximal optical angle is 28°) . This discrepancy could be attributed to the influence of the "bridge" (see FIG. 13) , which limits mechanically the tilting of the plate and reduces the maximally achievable angle. To estimate the influence of the "bridge" on the maximal angle, the ratio between the nominal distance of

4 μm separating the tilting element and the substrate

(the thickness of the silicon dioxide sacrificial layer) in the un-deformed position and the half width of the "bridge" (which is 40 μm) was calculated to be 1:10 which corresponds to the mechanical angle of 0.1 rad. Taking into account that some static deflection of the actuator before pull-in occurs (see FIG. 5C) , thereby reducing the distance between the actuator and the substrate, the maximal static mechanical angle of 3.5° limited by the contact the "bridge" and, consequently, the optical angle of approximately 7° could be explained.

In order to characterize the dynamic behavior of the device it was first excited mechanically, using an external piezoelectric transducer vibrating in the out- of-plane direction. The mechanical kinematical excitation makes it possible to extract the resonant characteristics of the device when it is not affected by the electrostatic force. The results are shown in FIG. 15. The experimental resonant frequency of the device of 3.82 kHz was in excellent agreement with the natural frequency of 3.83 kHz provided by the model and calculated using the actual dimensions measured in SEM. Quality factor of Q = 300 which corresponds to the equivalent damping of £ = 0.002 was extracted using a Lorentzian fit of the experimental resonant curve. It should be noted that for this relatively low damping and in absence of the electrostatic loading, the resonant frequency practically coincides with the natural frequency.

The mechanical excitation using an external piezoelectric transducer was also used for the resonant operation of the device of another configuration, referred as a pure kinematically excited device and presented schematically in FIG. 16A. In contrast to the device with the motion amplifier, the pure kinematically excited device does not incorporate axes connecting the substrate and the tilting element which is connected only to the actuator by a pair of elastic joins. To provide a dynamic coupling necessary for the kinematical excitation, the center of mass of the plate is located with a certain offset with respect to the joins, as shown schematically in FIG. 16A and in SEM micrographs FIG.

16B. The advantage of this architecture is its simplicity

, and ability to reach large tilting angles. However, as a result of absence of the coupling in stiffness between the plate and the actuator the device can be operated only dynamically. The geometrical parameters of the device are identical to those of the device with motion amplifier listed in Table 1. The experimental resonant curves corresponding to two different voltages applied to the piezo transducer are shown in FIG. 17. One may observe that the resonant curve is of nonlinear character mainly due to stiffening of the short torsion joins at large angles.

The results for dynamic electrostatic operation of the device with the integrated motion amplifier and the integrated actuator are shown in FIG. 18 and FIG. 19. The resonant curve in FIG. 18 corresponds to steady and periodic components of the actuation voltages of V_dc=2QV V_ac=50V , respectively. The obtained resonance curve is typical for linear resonator and can be approximated using the Lorentzian fit. The maximum optical angle in this case was 16° . The resonant frequency and the quality factor extracted using the Lorentzian fit were 3.78 kHz and Q = 226, respectively. The resonant curves for the case of higher actuation voltages of F_Λ=20F F_αc=90F are shown ' in FIG. 19, where the dot markers correspond to the excitation frequency swept up; the open cirlce markers correspond to the frequency swept down, and the solid line represents the Lorentzian fit. The shape of the resonant curve suggests that the response of the device has nonlinear character. It is worth noting that in a certain frequency range the increase in the actuation voltage did not result in a significant increase in the tilting angle. Instead, when the excitation frequency was swept upwards (dots) the response increased until a flat region at the value of the peak-to-peak optical angle of 16° was observed. This result is consistent with the maximal static angle of 7°. During the further increase in frequency, the tilting angle remained practically constant followed by a steep jump down in the response at the frequency of 3.815 kHz. When the frequency was swept downwards (open circles in FIG. 19) the different resonant curve was observed. The steep increase in the response (the jump phenomenon) was not registered and hysteresis phenomenon was obtained implying the coexistence of two different stable solutions at the same excitation frequency and indicating that the response is indeed of strongly nonlinear character.

From the dynamic results presented (mechanic excitation and electrostatic excitation at two different voltages) it appears that the quality factor decreases when the displacement increases. The increase of the tilting angle at higher actuation voltages is accompanied by the increase of the amplitude of the vibrations of the actuator, decrease of the distance between the actuator and the substrate and consequently decreases of the quality factor. While the present invention has been particularly described, persons skilled in the art will appreciate that many variations and modifications can be made. Therefore, the invention is < not to be construed as restricted to the particularly described embodiments, rather the scope, spirit and concept of the invention will be more readily understood by reference to the claims which follow.

The present invention has been described using non- limiting detailed descriptions of preferred embodiments thereof that are provided by way of example and are not intended to limit the scope of the invention. It should be understood that features described with respect to one embodiment may be used with other embodiments and that not all embodiments of the invention have all of the features shown in a particular figure. Variations of embodiments described will occur to persons of the art. Furthermore, the terms "comprise," "include," "have" and their conjugates, shall mean, when used in the claims, "including but not necessarily limited to." The scope of the invention is limited only by the following claims:

Claims

1. A tilting element comprising: a rotating plate operative to rotate about one of its axes; an electrode movable by applying electrical voltage thereto and characterized by having at least one of its surfaces located at a plane parallel to that of at least one of the surfaces of said rotating plate, when no electrical voltage is applied, and wherein said rotating plate and said movable electrode are connected to each other by a join.

2. A tilting element according to claim 1, wherein said tilting element is made of a single layer of silicon on insulator (SOI) .

3. A tilting element of claim 1, further characterized in that out-of-plane deflections of said movable electrode are transformable into substantially large angular motions of said rotating plate.

4. A tilting element of claim 1, wherein said join is attached to said rotating plate at an offset relative to said one of its axes.

5. A tilting element of claim 1, wherein said rotating plate is attached to said moveable electrode by an elastic torsion join.

6. A tilting element of claim 1, wherein said rotating plate is attached to a substrate by a pair of elastic torsion axes.

7. A tilting element of claim 1, wherein the actuation of said moveable electrode is carried out by using at least one of: a parallel-plate electrostatic transducer, a vertical comb-drive actuator, a magnetic actuator and a piezoelectric film.

8. A tilting element according to claim 1, further characterized in that substantially large tilting angles are achievable under static operation of said tilting element.

9. A tilting element of claim 1, further comprising a set of bending flexures characterized in that set of bending flexures is operative to provide out-of plane motion of said moveable electrode and to prevent titling instabilities of the moveable electrode.

10. A tilting device comprising a pair of elements, one of which comprising a torsion axis and the two elements are connected with each other by a torsion link, thereby enabling conversion of linear motion into an amplified angular motion.