Abstract:
A MEMS apparatus for scanning an optical beam comprises a mirror operative to perform a rotational motion to a maximum rotation angle around a mirror rotation axis, and a bouncing mechanism operative to provide a bouncing event and to reverse the rotational motion. The bouncing event provides the mirror with a piecewise linear response to actuation by intrinsically nonlinear electrostatic forces. The bouncing mechanism includes an element chosen to impart an overall nonlinear stiffness to the system and is selected from the group of elements consisting of a bouncer and a pre-curved nonlinear stiffness element.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS  
       [0001]     The present invention claims priority from U.S. Provisional Patent Applications Nos. 60/494,353 filed 12 Aug., 2003, 60/550,850 filed 8 Mar. 2004, and 60/xxx,xxx filed XX May, 2004, the contents of which are incorporated herein by reference. 
     
    
     FIELD OF THE INVENTION  
       [0002]     The present invention relates generally to scanning micro-mirrors implemented in Micro Electro Mechanical Systems (MEMS) or Micro Opto Electro Mechanical Systems (MOEMS), and more particularly to tilting micro-mirrors used for scanning.  
       BACKGROUND OF THE INVENTION  
       [0003]     A tilting micro-mirror is a central element in many MEMS or MOEMS devices. When used for scanning, its elements and operation principle are shown in  FIG. 1 . A tilting micro-mirror (or simply “mirror”)  100  comprises a generally flat plate (e.g. made of silicon) that has a reflecting surface  104 . Plate  102  is held suspended by two torsional hinges  106 ′ and  106 ″ aligned along a common torsion (and tilt) axis  108 . The two hinges render mirror  100  operative to tilt clockwise and counterclockwise around axis  108  in a given range of angles (typically ±5 degrees). A laser beam  110  that impinges on the reflecting surface of the mirror is redirected by the mirror to a scanned area  112 . The tilting mirror is actuated by an actuation moment  120  that can be provided by well-known MEMS actuation systems.  
         [0004]      FIG. 2  shows the required time dependence of the rotational (or “tilt”) angle of a scanning mirror, i.e. the required shapes of a scanning path  202  to be followed by the reflected laser beam on scanned area  112 . A triangular signal is needed for the forward-backward scanning, as shown in box  206 , while a sawtooth signal  208  is needed for one directional scanning, as shown in box  210 . 1&gt;2&gt;3&gt;4&gt;5&gt;6 represent scans in the time domain. In this kind of applications, the necessity to create an image free of spatial and temporal distortions imposes specific requirements on the scanning micro-mirror motion. These include long term frequency stability and constant angular velocity (for small rotations), see J. H. Lee et al,  Sensors and Actuators A-Physical  96 (2-3) pp. 223-230, 2002. A mirror of this type has been implemented recently in a virtual retinal display, see T. M. Lippert et al,  “Overview of Light Beam Scanning Technology for Automotive Projection Displays” , available at Microvision Inc.®, http://www.mvis.com/pdfs/sid_auto.pdf.  
         [0005]     Tilt mirrors are also used in optical switches and variable optical attenuators implemented in communication systems, and in light processing devices used in projection technology. A large variety of designs and operational modes have been reported, depending on the requirements imposed by the specific application. For example, in optical communication applications, the requirements of long term positioning accuracy combined with high optical quality and low thermal sensitivity are the most challenging. In contrast, micro-mirrors used in projection devices for the digital processing of visible light must fulfill requirements of high reflectivity, short switching time and high reliability, while positioning issues are usually less crucial.  
         [0006]     While linear motions are highly desirable in all micro-mirros, it is difficult to provide it. The difficulty is mainly the consequence of the intrinsic nonlinearity and high level of uncertainty of operational forces developed by MEMS actuators. A large variety of micro-device actuation principles and methods are known. These include electrostatic, magnetic, thermal, piezo, laser and flow-induced actuation, as well as actuation based on shape memory alloys. Electrostatic actuation and magnetic actuation remain the most widely used methods. The main advantage of magnetic actuation is the linear relationship between the input signal (electric current) and the actuation force. However, the price paid is usually a high power consumption resulting in high heat generation, intricacy of the design and relatively complicated fabrication processes. In addition, the scaling laws of magnetic actuators are less favorable that those of electrostatic actuators.  
         [0007]     The required typical size of a micro-mirror for scanning application in a retinal display (from hundreds of microns up to a millimeter) and the required operation frequencies (tens of KHz) make electrostatic actuation attractive for this use. In addition, advantages of electrostatic actuation include simple, well-established processes used for the fabrication of electrostatic devices, low power consumption, and developed modeling tools and large variety of design concepts reported in literature. However, the central difficulty of electrostatic actuation is the intrinsic nonlinearity of electrostatic forces. In the case of a scanning mirror, this results in a nonlinear dependence of the actuating torsion moment on the tilting angle and a nonlinear (quadratic) dependence on operational voltage. Moreover, the nonlinearity of electrostatic forces combined with the linearity of elastic restoring mechanical forces lead also to pull-in instability, which limits the operational range of the device.  
         [0008]     To overcome these difficulties, different solutions were proposed in prior art. Specifically, a generated square root (of voltage) input signal was used by W Zhang et al,  Applied Physics Letters  82 (1) pp. 130-132, 2003, for the operation of a micro-resonator near the parametric resonance. The use of a vertical comb drive permits the elimination of the actuation moment dependence on the tilting angle and the reduction or even elimination of the square dependence on voltage, see e.g. J H Lee et al,  Sensors and Actuators A - Physical  96 (2-3) pp. 223-230, 2002, H. Wada et al,  Jpn. J. Appl. Phys.  41 (10B) pp. 1169-1171, 2002, and H Schenk et al,  Sensors and Actuators A - Physical  89 (1-2) pp. 104-111, 2001. The necessity to provide a triangular signal which is required for video applications leads to very high actuation voltages or, in the case of magnetic actuation, very large currents. This difficulty is not related to the linearity of the motion and it is a result of high angular accelerations during the inversion of the velocity. To overcome this difficulty, frequency, I. Bucher, in  Proc. of  29 th    Israel Conference on Mechanical Engineering,  May 12-13, 2003, Technion, Haifa, Israel, suggested to represent the required triangular response as a Fourier series of sinusoidal components, and to excite each of these components at the resonance  
         [0009]     The problems with tall such solutions include high complexity, difficulty to provide resonant frequencies with high accuracy, and consequently high sensitivity to fabrication tolerances and extreme difficulty in tuning the resonant frequency.  
         [0010]     There is therefore a widely recognized need for, and it would be highly advantageous to have a scanning micro-mirror that has optimized motion linearity combined with high operational frequency and low actuation voltages.  
       SUMMARY OF THE INVENTION  
       [0011]     The present invention discloses a tilting “bouncing mode” micro-mirror that uses either an additional stiffness element (also referred to throughout this description as “bouncer”) or a pre-curved nonlinear stiffness element to achieve a superior scanning performance. The present invention further discloses a micro-mechanical actuator used to move a member carrying a payload (e.g. a mirror) along an axis in a periodical nonlinear angular trajectory. The present invention further discloses a novel tilting micro-mirror mode of operation that permits the achievement of a piecewise linear response of a micro-mirror device operated by intrinsically nonlinear forces. In one embodiment, the “bouncing mode” of operation according to the present invention includes a contact event between the mirror and an elastic constraint, which takes place each time the mirror reaches a prescribed rotation angle. This contact event is followed by the bouncing of the mirror and by the inversion of the angular motion, hence the name “bouncing mode”. In another embodiment, the “bouncing mode” is achieved by the action of pre-curved nonlinear stiffness elements coupled at one end to an actuator and at another end to the mirror. Illustrative examples of bouncing-mode scanning micro-mirrors actuated electro-statically by both parallel-plate electrodes and planar or vertical comb drives are presented in detail. It is shown that in all embodiments, the response frequency can be tuned through the control of the actuation voltage. This feature allows to compensate for uncertainties in the parameters of micro-fabricated devices and to synchronize the response frequency with a precision sufficient for video applications. The resonant-mode operation and the application of actuation forces during the contact event, when the mirror is close to the electrode (in the bouncer plus parallel plate actuation example), enable operation at extremely low power and voltages and provide a compact and low cost actuator. Additional improvements in linearity may be achieved through the application of a correction voltage during the mirror motion between bouncing events.  
         [0012]     In essence, the present invention uses a bouncer or a nonlinear stiffness element in a similar manner to a nonlinear oscillator, specifically to an impact oscillator, for the shaping of an output signal. While impact oscillators incorporating bouncing are known and intensively studied in the nonlinear dynamics literature, the use of a bouncer or a pre-curved nonlinear stiffness element as disclosed herein and for the purposes set forth herein is unknown.  
         [0013]     Advantageously, in the bouncing mode using bouncers disclosed herein, the actuation forces can be applied either to the mirror (passive barrier) or directly to the bouncer (active barrier). Note that in many MEMS applications, a typical situation exists in which the structure of the micro-device is linear mechanically, and nonlinearity arises only due to actuation forces. For example, in the case of a scanning mirror the stiffness of the torsion axis is constant. As a result, the motion of the mirror in the absence of actuation forces during the time interval between bouncing events is actually a free motion of a linear system. The angular velocity of such a motion depends on the initial conditions defining the initial kinetic and potential energy of the system and on the stiffness of the torsion spring. In the case when the kinetic energy is the dominant part of the total energy of the system, the variation of the velocity during the free motion is minor and a desirable level of the linearity of motion is achievable. This situation is realizable when the stiffness of the torsion spring is small compared with the stiffness of the barrier.  
         [0014]     Other important advantages of the devices and methods disclosed herein include low sensitivity of the frequency and shape of the output signal to the system parameters, and good controllability of frequency. These advantages arise from the fact that the frequency of the mirror is defined by the deformation of the barrier, which for its part depends on the level of the energy supplied during each bouncing event. Viscous losses during the free motion, as well as uncertainties in the system parameters, can be easily compensated by controlling this energy supplement through the actuation voltage. Note that the uncertainty in the resonance frequency of the mirror can be critical, as mentioned in H. Wada et al. above, who reported a discrepancy of 25-30% between calculated and measured values. This ability to tune the natural frequency using the actuation voltage permits long-term stabilization of the response frequency and precise synchronization of the mirror motion with a video input.  
         [0015]     According to the present invention there is provided a tilting micro-mirror device comprising a substrate, a micro-mirror operative to perform a tilt motion around a tilt axis positioned substantially parallel to the substrate, the tilt motion defined by a tilt angle range ending in a positive or negative maximum tilt angle, and at least one pair of bouncers, each bouncer positioned on opposite sides of the tilt axis and operative to couple to the micro-mirror when the mirror reaches the positive or negative maximum tilt angle, the coupling providing an a bouncing event.  
         [0016]     According to the present invention there is provided a tilting micro-mirror device comprising a substrate, a micro-mirror operative to perform a tilt motion to a positive or negative maximum tilt angle around a tilt axis positioned substantially parallel to the substrate, and at least one pair of pre-curved nonlinear stiffness elements having each two ends, each pre-curved element connected at one of its ends to the micro-mirror and at the other of its ends to the substrate, whereby each nonlinear stiffness element is operative to impart the mirror motion an essentially piecewise linear characteristic.  
         [0017]     According to the present invention there is provided a MEMS apparatus for scanning an optical beam comprising a mirror operative to perform a rotational motion to a maximum rotation angle around a mirror rotation axis, and a bouncing mechanism operative to provide a bouncing event and to reverse the rotational motion, whereby the bouncing event provides the mirror with a piecewise linear response to actuation by intrinsically nonlinear forces.  
         [0018]     According to the present invention there is provided a method for scanning an optical beam using a MEMS mirror comprising the steps of providing a mirror operative to perform a rotational motion to a maximum rotation angle around a mirror rotation axis, and providing a bouncing mechanism operative to facilitate a bouncing event and to reverse the rotational motion, whereby the bouncing event provides the mirror with a piecewise linear response to actuation by intrinsically nonlinear forces. 
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0019]     Reference will be made in detail to preferred embodiments of the invention, examples of which may be illustrated in the accompanying figures. The figures are intended to be illustrative, not limiting. Although the invention is generally described in the context of these preferred embodiments, it should be understood that it is not intended to limit the spirit and scope of the invention to these particular embodiments. The structure, operation, and advantages of the present preferred embodiment of the invention will become further apparent upon consideration of the following description, taken in conjunction with the accompanying figures, wherein:  
         [0020]     FIG. I shows the elements and operation principle of a tilting micro-mirror used for scanning;  
         [0021]      FIG. 2  shows the required time dependence of the rotational angle of a scanning mirror;  
         [0022]      FIG. 3  illustrates schematically a prior art tilting mirror that uses parallel plate electrostatic actuators;  
         [0023]      FIG. 4  shows a concept of a bouncing mode mirror and its operation principle according to the present invention;  
         [0024]      FIG. 5  shows schematically the dependence of the torsion stiffness (moment) on the tilting angle in the device of  FIG. 4 ;  
         [0025]      FIG. 6  illustrates schematically motion stages of a bouncing mode mirror as in  FIG. 4 ;  
         [0026]      FIG. 7   a  illustrates the time history (angle-time dependence) of a bouncing mode mirror motion obtained by a simulation;  
         [0027]      FIG. 7   b  illustrates the dependence of the natural frequency of the mirror on the deformation of the bouncer.  
         [0028]      FIG. 8  illustrates the simulated time history of an actuation moment M A  resulting from a bouncing event.  
         [0029]      FIG. 9  shows schematically yet another embodiment of a bouncing mode micro-mirror device in which the bouncers are connected to the mirror;  
         [0030]      FIG. 10  shows schematically yet another embodiment of a bouncing mode micro-mirror device with a pair of additional softer bouncers connected to the substrate;  
         [0031]      FIG. 11  shows schematically yet another embodiment of a bouncing mode micro-mirror device that comprises an active, independently deformable bouncer;  
         [0032]      FIG. 12   a  shows schematically a pre-curved nonlinear stiffness element that may be advantageously used in the bouncing micro-mirror devices of the present invention;  
         [0033]      FIG. 12   b  illustrates the dependence between the applied force and the elongation of the nonlinear stiffness element of  FIG. 12   a ;  
         [0034]      FIG. 13  shows an embodiment of a parallel plate actuated bouncing mode mirror with bouncers: a) isomeric view; b) top view; c) side view without actuation; d) side view with actuation;  
         [0035]      FIG. 14  shows in (a) top view, and (b) and (c) isomeric views an embodiment of a mirror device with a bouncers driven by parallel plate electrostatic actuators;  
         [0036]      FIG. 15  shows: (a) top view and, (b) isomeric view an embodiment of a rotational parallel plate actuated bouncing mode mirror device with pre-curved nonlinear stiffness drive elements; (c) isomeric view emphasizing a side with section A-A; (d) section A-A before deformation; (e) section A-A after deformation by electrostatic forces.  
         [0037]      FIG. 16  shows in (a) top view and (b) isomeric view an embodiment of a planar comb drive-actuated bouncing mode mirror device with pre-curved nonlinear stiffness elements;  
         [0038]      FIG. 17  shows another embodiment of a bouncing mode mirror with in-plane pre-curved nonlinear stiffness elements: (a) top view of the device, (b) top view of the central section with mirror, and (c) isomeric view of the mirror section;  
         [0039]      FIG. 18  shows various shapes of pre-curved nonlinear stiffness beams;  
         [0040]      FIG. 19  shows a detailed embodiment of a vertical comb drive actuated bouncing mode micro-mirror with bouncers: (a) general view, (b) enlargement;  
         [0041]      FIG. 20  shows in cross section various layers of a double-active-layer silicon-on-insulator (SOI) wafer;  
         [0042]      FIG. 21  shows an exemplary process for the fabrication of a bouncing mode micro-mirror using a double-active-layer SOI wafer. 
     
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS  
       [0043]     The present invention discloses a tilting bouncing mode micro-mirror that uses either an additional stiffness element (“bouncer”) or a pre-curved nonlinear stiffness element to achieve a superior scanning performance. While a bouncer such as a cantilever or beam is an element with linear stiffness, we have found inventively and advantageously that its cooperative action with other elements of the system (specifically a torsion spring) that have a different stiffness yields a combined “nonlinear stiffness element” effect. The bouncing mode uses an actuation operation mode based on a special nonlinear actuation and control principle. The “bouncing-mode” actuator that actuates the mirror is operated in the self-exciting mode and its motion is actually a limit cycle. The actuator moves the mirror in a piecewise linear trajectory. The nonlinear actuation and control principle provides a set of desirable features such as small size and weight, low power and low heat dissipation, high frequency cycle with fine and accurate feedback controllability, quasi-linear trajectory intervals, low acoustical noise and more. A full description of the theory and simulations that describe the nonlinear actuation and control that provide the bouncing mode of the present invention may be found in “Bouncing mode electro-statically actuated scanning micro-mirror for video applications” by V. Krylov and D. I. Barnea, submitted for publication in  Smart Materials and Structures, March  2004, which is incorporated by reference for all purposes set forth herein.  
         [0044]     The invention makes use of loop-shaping methods based on the utilization of the properties of a nonlinear resonator. The method permits the shaping of an output signal to a required form (for example to a triangular or saw tooth form) from another shape of an input signal (for example sinusoidal or rectangular).  
         [0045]     The motion of a generic nonlinear oscillator, excited for example by a sinusoidal signal, can be described by the equation:  
                 θ   ¨     +     c   ⁢           ⁢     θ   .       +       ∑   k     ⁢       ɛ   k     ⁢     θ   k           =     A   ⁢           ⁢     sin   ⁡     (     ω   ⁢           ⁢   t     )                 (   1   )             
 
 where θ,{dot over (θ)} and {umlaut over (θ)} are respectively the angle and its first and second time derivatives, c and A are constants, ω is the frequency, t is time, and ε k  represents parameters of nonlinearity. Since the equation is nonlinear, the periodic output signal (if is exists) is not sinusoidal and contains higher harmonics:  
                     ⁢     θ   =         ∑   k     ⁢       a   k     ⁢           ⁢     sin   ⁡     (     k   ⁢           ⁢   ω   ⁢           ⁢   t     )           +       b   k     ⁢     cos   ⁡     (     k   ⁢           ⁢   ω   ⁢           ⁢   t     )                     (   2   )             
 
         [0047]     The parameters of nonlinearity ε k  can be chosen in such a way that the parameters defining the shape of the output signal, namely a k ,b k  take the values resulting in the required shape of the output signal. Tuning of the resonant frequency is possible as well, due to the nonlinearity of the system. A transducer of this type has an advantage of compactness, since it is based on a single element.  
         [0048]     The very special properties of the bouncing mode actuator enable a set of applications in electrical and optical wave-shaping, where the actuator&#39;s payload forced movement (by an input signal) is transformed into a new shape (e.g. from sinusoidal to a triangular saw tooth) and frequency. Exemplary applications may include the movement of a micro-mirror so as to achieve a retinal scan-injection of a video display. The movement of the mirror facilitates a relatively simple electronic interface between standard video streams into the signaling that activates the display system. Another application may be in RF devices used, for example, for generation of various signals by a single element.  
         [0049]     The following description uses electrostatic tilting as an exemplary actuation means. The electrostatically-tilted micro-mirror described herein fulfills the requirements of motion linearity, high operational frequency and low actuation voltages imposed by laser display applications. It is understood that the electrostatic actuation described may be replaced by other actuation methods such as thermal, magnetic, or piezoelectric actuations.  
         [0050]      FIG. 3  shows schematically a prior art tilting mirror device  300  that uses parallel plate electrostatic actuators. Device  300  comprises a mirror  302  suspended by a torsion axis  304  at a distance d from two parallel plate electrodes  306   a  and  306   b  located under the mirror on a substrate  308 . Due to the rotation of the mirror around axis  304 , at a rotation angle close to the contact angle with the electrodes a distance d′ between a mirror edge  310  and the electrode e.g.  306   b ) is much smaller that the initial distance d. The incorporation of at least one pair of additional stiffness elements (bouncers) in the form of cantilevers, double edge clamped beams, etc. into this device is shown in  FIG. 14 .  
         [0051]      FIGS. 4-13  describe the invention schematically in conceptual terms.  FIGS. 14-19  describe the invention in specific MEMS geometries and designs, applicable for example to silicon or silicon on insulator (SOI) MEMS processes.  FIGS. 20, 21  illustrate schematically the steps of a silicon MEMS process used to fabricate a preferred embodiment of the bouncing mode micro-mirror device of the present invention.  
         [0052]      FIG. 4  shows schematically the concept of a bouncing mode micro-mirror device  400  and its operation principle according to the present invention. Device  400  comprises in (state a) a mirror  402  suspended on a torsion (tilt) axis  404  above a base plane  406 . Under actuation, the mirror tilts (rotates) around axis  404  through a tilt angle θ. Inventively and in contrast with prior art tilt mirrors, device  400  comprises at least one pair of nonlinear stiffness elements (e.g. elastic “bouncers” made of cantilevers, beams or strips or pre-curved nonlinear stiffness elements shown exemplarily in  FIGS. 12 and 18 )  408 ′ and  408 ″ that come into contact with the mirror when the tilt angle θ is larger that some prescribed contact angle value θ c . The following discussion of  FIGS. 4-11  refers specifically to bouncers. Coming in pairs, the bouncers are preferably (although not necessarily) positioned symmetrically relative to (i.e. on both sides of) tilt axis  404 . When contacted by the mirror (state b), elements  408 ′ and  408 ″ invert the mirror movement because of the increased torsion stiffness, as shown in  FIG. 5 . The nonlinear stiffness element (made of a particular material, e.g. silicon) may be designed and optimized such as to achieve a particular characteristic (bouncing effect) in response to an applied actuation force, using well known rules.  
         [0053]      FIG. 5  shows schematically the dependence of the torsion stiffness (moment) M on tilt angle θ in the device of  FIG. 4 . The torsion stiffness is composed of the stiffness of torsion axis  404  and of the additional stiffness of a bouncer  408 ′ or  408 ″ on the tilting angle θ. When the mirror contacts the bouncer, i.e. when θ&gt;θ c , the stiffness (defined by the slope of the M vs. θ curve) increases significantly. This provides a saturation-type, stiff nonlinearity necessary for the formation of the triangular signal.  
         [0054]      FIG. 6  illustrates schematically motion stages (as expressed by a curve  602  showing a tilt angle θ vs. time t) of a bouncing mode mirror as in the device of  FIG. 3 . Curve  602  comprises  3  stages: a first fast stage between point A and B defined by a release time T REL , a second slow stage between points B and C defined by a time T SLOW  and a third fast load stage between points C and D defined by a time T LOAD . During the slow stage (in which the mirror is not in contact with the bouncer), the mirror rotates with an almost constant angular velocity that is determined by the bouncer and the energy provided by the actuator. The strain energy stored in the deformed bouncer is transformed into the kinetic energy of the mirror. During the fast stages (i.e. when θ&gt;±θ c ) the mirror is in contact with the bouncer and the kinetic energy of the mirror is transformed back into the strain energy of the bouncer. Since the stiffness of the bouncer is much larger that the stiffness of the torsion axis, the additional angle θ MAX -θ c  that corresponds to the deformation of the bouncer is much smaller that the total tilting angle of the mirror θ MAX .  
         [0055]     In summary, the operational mode of the bouncing mode mirror, as illustrated in  FIGS. 4-6  and as explained in more detail in the Krylov and Barnea reference above, incorporates a contact event between the mirror and an elastic constraint followed by a bouncing event and an inversion of motion. As shown in  FIG. 5 , the stiffness of the system is angle dependent.  
         [0056]      FIG. 7   a  illustrates the time history (tilt angle θ vs. time t dependence) of a bouncing mode mirror motion obtained by a simulation. Details may be found in the Krylov and Barnea reference. The simulation uses a piecewise constant-in-time voltage actuation signal marked by (dotted line) V and applied to the electrodes, and the result (full line) is a triangular angle-time dependence. The angular velocity is practically constant.  FIG. 7   b  illustrates on the left the dependence of the natural frequency of the mirror ω on the deformation of the bouncer. Starting from some value, the frequency is a linear function of the bouncer deformation. This property is very useful for the control of the mirror frequency and permits the tuning of the mirror natural frequency in a very large range. As shown by the “Response frequency vs. voltage” graph in the right box in  FIG. 7   b,  the response resonant frequency may be tuned by voltage control. The tuning of the resonant frequency through the control of the applied voltage permits the synchronization of the response of a micro-fabricated device with a video signal.  
         [0057]     The “quality” of the response shown in  FIG. 7   a,  namely the linearity of the rotation angle in time, is defined by the ratio M B /M TA  between the stiffness of the elastic bouncer M B  and of the torsion axis of the mirror M TA . An increase in this ratio can be achieved by the hardening of the bouncer or alternatively by the softening of the torsion axis. The natural frequency of the mirror is defined mainly by the deformation of the elastic bouncer and consequently by the actuation voltage, and is practically independent of the stiffness of the torsion axis M TA . This suggests that M TA  has to be reduced to a minimum. This reduction can be achieved by the design means and is limited only by the requirements of the minimal out-of-plane stiffness of the axis.  
         [0058]      FIG. 8  illustrates the time history of the resulting actuation moment M A  The proximity of the mirror edge to the electrodes results in the increase in the actuation moment. Attractive electrostatic forces large enough are used to deform the bouncer and invert the velocity. The elastic energy stored in the bouncer is transformed into a repelling force acting on the mirror. The combination of additional stiffness element and parallel plate electrodes working in a proximity mode permit the achievement of very large repelling forces in the electrostatic actuator.  
         [0059]      FIG. 9  shows schematically yet another embodiment of a bouncing mode micro-mirror device  900  according to the present invention. In  FIG. 9 , least one pair of bouncers  902 ′ and  902 ″ is attached to a mirror  904  instead of to a substrate  906 . Bouncers  902  are now mobile, in contrast to bouncers  408  ( FIG. 4 ), which are attached to the substrate and immobile. The advantage of this configuration is a simpler fabrication process, since bouncers  902  can be fabricated in the same layer as the mirror (e.g. an active Si layer in a SOI substrate). As in  FIG. 4 , (a) shows the mirror before it touches the substrate through the bouncer, and (b) shows the mirror in contact with the substrate through the bouncer.  
         [0060]      FIG. 10  shows in (a) schematically yet another embodiment of a bouncing mode micro-mirror device  1000 . The configuration is similar to that of  FIG. 4 , except that the device comprises at least one additional pair of “softer” (relative to the first pair) bouncers  1002 ′ and  1002 ″. More pairs of bouncers with varying softness are of course included in the definition of “at least one additional pair”. Bouncers  1002 ′ and  1002 ″ are operative to manage the contact velocity in order to improve reliability and soften the impact. A stiffness-angle dependence plot similar to that in  FIG. 4  is shown in (b) for this configuration. One can see two added sections  1004  and  1006  with slopes intermediate to the slope of the “slow” section and that of the two “fast” sections. These sections show that the impact velocity is lower.  
         [0061]      FIG. 11  shows schematically yet another embodiment of a bouncing mode micro-mirror device  1100  that comprises at least two active, independently deformable bouncers  1102 . An active bouncer  1102  is deformed not by the mirror but by an additional force P A , which is applied directly to the bouncer (stage a, when θ&lt;θ c ), independently of the mirror motion. This permits to store a larger energy in the bouncer in order to provide non-symmetric saw tooth signals and to reduce the impact velocity to zero, since the kinetic energy of the mirror is not required anymore for the bouncing deformation. In stage b when θ&gt;θ c , the bouncer previously deformed by the force returns its energy to the mirror.  
         [0062]      FIG. 12  shows schematically a conceptual pre-curved nonlinear stiffness element (e.g. a beam, a string, etc.)  1200  with two ends  1102  and  1104  that may be advantageously used in the bouncing micro-mirror devices of the present invention. A “pre-curved” element in the present invention is an element with at least one section having a radius of curvature that is not infinite. Element  1200  is pre-curved in such a way that in an original (non-stressed) pre-curved state, the distance between its two ends is L 0 . A force P with a starting value of P 1  is applied to ends  1102  and  1104 . Length L 0  increases by an elongation ΔL when the force reaches a value P 2 .  FIG. 12   b  illustrates the dependence between the applied force P and the elongation L of element  1200 . The dependence of the elongation on the force is highly nonlinear and is a function of the initial shape of element  1200 . In order to straighten the element completely, a theoretically infinite force needs to be applied. Due to its high nonlinearity, the element characteristics are close to those of an ideal bouncer. The dimensions of a pre-curved element of the present invention may be designed and optimized to achieve a particular characteristic (bouncing effect) in response to an applied force. Implementations of element  1200  as a nonlinear stiffness element are shown in the systems in  FIGS. 15-17 .  
         [0063]      FIG. 13  shows in (a) isomeric view and (b) top view an embodiment of a parallel plate actuated bouncing mode mirror device  1300  that includes bouncers (bouncers) of the present invention connected to a mirror. This embodiment implements the concept shown in  FIG. 9 . Device  1300  comprises a mirror  1302  situated in an XY plane and connected by two torsion bars  1304 ′ and  1304 ″ to a substrate  1306 . Mirror  1302  is rendered electrically conductive (e.g. by a deposited metallization) so that it can be pulled into the −Z direction by one of two electrodes  1308 ′ and  1308 ″ located in a lower XY plane (below and separate from the mirror plane). Device  1300  further comprises at least one pair of spring beams  1310 ′ and  1310 ″ that are fixedly connected to (and in fact preferably part of the same layer as) torsion bars  1304 . Beams  1310 ′ and  1310 ″ serve as impact nonlinear stiffness springs. Device  1300  further comprises at least one pair of stoppers  1312 ′ and  1312 ″ that stop the movement of the mirror when the mirror rotates around torsion bars  1304 , by contacting spring beams  1310 . In essence, the bouncers in this embodiment are beams connected at one end to the mirror (or to the torsion bar of the mirror), the other end being free. The beam bends upon contact with a stopper, building up energy that eventually reverses the mirror rotation, bouncing it back. The location of the bouncers (when connected to the torsion bar) may be optimized to give the mirror a high rotation angle. Enlargements of the area of contact between beams  1310  and stoppers  1312  are shown in FIGS.  13 ( c ) and (d).  FIG. 13 ( c ) shows a spring beam that is not in contact with a stopper.  FIG. 13 ( d ) shows a spring beam during contact with stopper  1312 ′. The bending of the beam during contact is shown in a highly exaggerated way, for illustration purposes only.  
         [0064]      FIG. 14  shows an embodiment of a mirror device  1400  with at least one pair of bouncers (spring beams)  1412  connected to a substrate  1406  and driven by parallel plate electrostatic actuators: (a) top view, (b) and (c) isomeric views taken along a section A-A of  FIG. 14 ( a ). This embodiment implements the concept shown in FIGS.  4  or  10 . Only one pair of spring beams is shown in (b) and (c), although it is understood that two or more springs with different stiffness characteristics may be attached (in pairs) to the substrate on each side, as shown exemplarily in  FIG. 14 ( a ). Device  1400  comprises a mirror  1402  in plane x-y connected by at least one pair of torsion beams  1404  to a substrate  1406 , and pulled in −z (into the page) direction by one of two electrodes  1410   a ,  1410   b  located in a lower x-y plane layer. The mirror rotates until it contacts with its edge at least one spring beam  1412  that is clamped to substrate  1406  at the electrode level (plane). When more than one beam is used, each beam may have different elastic properties, for example the same cross section but different length, as shown in  FIG. 14 ( a ). The mirror deflects the springs and bounces back. In other words, in this embodiment the bending beam is fixedly attached at one end to the substrate instead of to the mirror. The deflection of the springs is shown in a highly exaggerated way in (c).  
         [0065]      FIG. 15  shows in (a) top view and in (b) isomeric view an embodiment of a rotational parallel plate actuated bouncing mode mirror device  1500  with pre-curved nonlinear stiffness elements. This embodiment makes use of pre-curved C-shape elements of the type shown in  FIG. 12  and  FIG. 18   a . Device  1500  comprises a mirror  1502 , in this case rectangular but in general of any regular symmetric shape (for example round). Mirror  1502  is situated in an XY plane and connected by two torsion bars  1504 ′ and  1504 ″ aligned along a common torsion axis  1505  and ending each in elevations (or “pads”)  1507  on top of a substrate  1508 . Device  1500  further comprises two pairs of short “offset” beams  1510   a  and  1510   d,  and  1510   b  and  1510   c,  which are located in a lower part of the mirror, at an offset b from the top surface, see section A-A view in  FIG. 15 ( c ). The offset beams are respectively connected by at least one pair (in this case two pairs) of preferably C-shaped spring beams  1512   a  and  1512   d,  and  1512   b  and  1512   c  to substrate  1508 . Device  1500  further comprises two electrodes  1520   a  and  1520   b  located below the mirror. Note that in principle the at least one pair of C-shaped (and more generally “pre-curved”) beams may comprise only beams  1512   a  and  1512   c,    1512   b  and  1512   d,    1512   a  and  1512   d,  or  1512   b  and  1512   c.    FIG. 15 ( d ) and (e).  FIG. 15 ( d ) show side views of section A-A: (d) shows the C-shaped beams before deformation by the electrostatic force, while (e) shows them after deformation.  
         [0066]     When the mirror and one of the electrodes  1520  are charged, the pull on spring beams  1512  (with eccentricity length b) yields a moment causing the rotation of the mirror in the opposite direction around torsion bars  1504  and common axis  1505 . Due to the straightening, the C-spring beams have a stiffening type nonlinear characteristics required in order to produce “bouncing effect”. It is emphasized that the C-shape springs are used for example only, and that other pre-curved nonlinear stiffness elements, for example V-shape, S-shape or Z-shape may be equally useful for the purposes set forth herein.  
         [0067]      FIG. 16  shows in (a) top view and (b) isomeric view an embodiment of a planar comb drive-actuated bouncing mode mirror device  1600  with pre-curved nonlinear stiffness drive elements actuated by planar comb drives. Device  1600  has elements  1602  to  1610  identical with respective elements  1502  to  1510  in  FIG. 15 . These include a mirror  1602 , two torsion bars  1604 ′ and  1604 ″ elevations  1606 ′ and  1606 ″, substrate  1608 , and two pairs of short “offset” beams  1610   a  and  1610   d,  and  1610   b  and  1610   c  located in a lower part of the mirror, at an offset b from the top surface ( FIG. 16   b ). The offset beams are respectively connected by two pairs of preferably C-shaped spring beams  1612   a  and  1612   d,  and  1612   b  and  1612   c  to two planar comb drive rotors  1614   a  and  1614   b  that have a Y-direction degree of freedom by their connection to substrate  1608  through retaining beam springs  1620  on pads  1622 .  
         [0068]     When a comb drive stator  1624   a  with teeth  1620  and a comb drive rotor  1614   a  with teeth  1616  are charged through electrical conductors  1630   a  and  1632   a,  the eccentric pull (with eccentricity length b) yields a counter-clockwise rotation moment of mirror  1602  around torsion bars  1604 . Comb drive stator  1618   b  and comb drive rotor  1614   b  yield similarly a clockwise rotation of the mirror. The C-spring beams have a nonlinear stiffness designed to transform the movement induced by the comb drives into a linear movement of the mirror (bouncing effect). Is is emphasized that the C-shape springs are used for example only, and that other shape non linear stiffness elements, for example V-shape, S-shape or Z-shape may be equally useful for the purposes set forth herein.  
         [0069]      FIG. 17  shows another embodiment of a planar comb driven bouncing mode mirror device  1700  with in-plane pre-curved nonlinear stiffness elements.  FIG. 17   a  shows the entire device,  FIG. 17   b  shows a vector force in the Y direction arising from a pulling force in the X direction.  FIG. 17   c  shows that as a rotation moment around the torsion axis results from the force developed due to the geometry as explained in (b) in combination with the eccentricity b of this force relative to the torsion axis. Device  1700  comprises a mirror  1702  situated in an XY plane and connected by two torsion bars  1704 ′ and  1704 ″ aligned along a common torsion axis in the X direction to a substrate  1710 . The mirror is pulled by drivers  1730   a  and  1730   b  to the X (and −X) direction either through a pair of curved beams  1706   a  and  1706   b  (by forces  1708   a  and  1708   b  respectively) or by drivers  1730   c  and  1730   d  through a pair  1706   c  and  1706   d  curved beams (by forces  1708   c,    1708   d  respectively). When pulled, beams  1706  have a combined action of a nonlinear stiffness element (beam) with a curved-step shape [see  1802   b  in  FIG. 18 ] with an eccentricity distance ‘a’. This eccentricity provides a vertical force vector effect. In addition, the pulling of an opposite pair of curved beams creates a force (e.g.  1708  by) in plane XY that is vertical to the pulling force vector (e.g.  1708 bx). In other words, the Y direction force vector is eccentric in the Z direction by a distance ‘b’ to the rotation axis  1704 , creating a torsional moment around the rotation axis.  
         [0070]     A driver (e.g.  1730   a ) includes a rotor  1732  pulled by a stator  1750 . The rotor has a X-direction degree of freedom as it is fixed to a carrier beam  1734 . The carrier beam is connected to two four-beam flexures  1736  anchored to substrate  1710  at pads  1738 . The rotor and stator are charged by conductors  1742  and  1752  respectively.  
         [0071]      FIG. 18  shows various shapes of nonlinear stiffness beams that may be used in the embodiments of  FIGS. 15, 16 , and  17 . These include (but are not limited to) a C-shaped beam  1802  shown in (a), an S-shaped beam  1804  shown in (b), a V-shaped beam  1806  shown in (c) and a Z-shaped beams  1808  shown in (d). In each such beam, the application of a force (shown by arrows) changes the beam curvature and shape, from a state a′ to a state b′ (shown for simplicity only for the C and S-shaped curves as  1802   a′, a″  and  1804   a′, a″ ).  
         [0072]      FIG. 19  shows an embodiment of a vertical comb drive actuated bouncing mode micro-mirror device  1900 .  FIG. 19   a  shows an isomeric view, while  FIG. 19   b  shows some details of that view. Device  1900  comprises a mirror  1902  connected through torsion springs (axis)  1904  to a layer  1906  in which the mirror is fabricated. The mirror has four arms  1908   a - d,  which together with respective teeth  1910  form comb drive rotors. The device further comprises stator comb drives  1914   a - d.  For simplicity only set d is shown in  FIG. 19   b.  In use, the activation of comb drive rotors  1908   a  and  1908   b  through conductive pads  1925   a  and  1925   b  on respective stators  1914   a  and  1914   b  causes rotation of the mirror around torsion axes  1904 . To reverse direction, comb drive rotors  1908   c  and  1908   d  are activated by conductive pads  1925   c,    1925   d  on stators  1914   c  and  1914   d.  The rotors themselves are charged through a conductive pad  1927 . Device  1900  further comprises at least one pair of bouncing spring beams  1916   a, b  that are connected each to a respective rotor arm  1908   a, b  (and through it to the mirror). Additional pairs of bouncing springs (e.g.  1916   c, d ) may be connected to rotor arms  1908   c, d . When the mirror is rotated around torsion axis  1904 , beams  1916  contact stoppers (not shown, but located for example on the handle layer) at the end of the mirror rotation, causing the bouncing effect. In principle, the operation and bouncing effect in this embodiment are similar to those of the embodiment in  FIG. 13 . The mirror is situated substantially above a release hole  1929 .  
         [heading-0073]     Fabrication Process  
         [0074]     The micro-mirrors can be manufactured using two SOI wafers bonded or fused together, an SOI wafer bonded or fused to a regular Si wafer, or using a special double-active-layer SOI wafer. An exemplary fabrication process of a bouncing mode tilting micro-mirror using a double-active-layer SOI wafer is shown in  FIG. 20 .  
         [0075]     Inventively, the present invention uses a “double-active-layer” SOI wafer in an accurate fabrication process that does not require wafer bonding. The backside etches are aligned to the initial hard mask etch, as described in  FIG. 21 . The more accurate the processing steps and the alignment between layer features, the higher the precision functionality of the final devices.  
         [0076]      FIG. 20  shows a representation of the initial double-active-layer SOI wafer prior to processing. The figure shows a first active layer  2002 , a first sacrificial layer  2004 , a second active layer  2006 , a second sacrificial layer  2008 , a silicon substrate  2010 , and a third sacrificial layer  2012 . Third sacrificial layer  2012  is used for patterning the areas that will be removed underneath the mirror and actuators. The three semiconductor layers, first active layer  2002 , second active layer  2006 , and substrate  2010  are electrically isolated from one another by layers of sacrificial material (silicon dioxide). First sacrificial layer  2004  is located between the first and second active layers, and second sacrificial layer  2008  is located between the second active layer and the substrate. Each active layer may have a thickness of a few to a few tens of microns.  
         [0077]      FIG. 21  (steps a-y) shows details of a process used for fabricating a bouncing mode electrostatic tilting micro-mirror using the wafer construction shown in  FIG. 20 . The process begins in (a) with deposition of a blank metal layer  2102 . The metal is then patterned in (b) using a photoresist mask  2104  and a wet etch is used in (c) to form a mirror metal  2106  and electrical contact pads  2108  for the electrostatic actuator&#39;s rotor fingers. A low-pressure chemical vapor deposition (LPCVD) silicon dioxide layer  2110  is then deposited over the metal on the topside of the wafer, in (d). This oxide layer is used both to protect the metal and to form a hard mask for the silicon active layers. The oxide layer is then patterned and etched in (e). The wafer is flipped over, and a photoresist layer  2112  is patterned on the backside in (f) and aligned to the pattern in (e). The third sacrificial layer is etched with reactive ion etching (RIE) in (g), and the photoresist is stripped and a new layer  2114  of photoresist is deposited and patterned in (h). In (i), the third sacrificial layer is etched for a second time using the previously patterned photoresist. The substrate is then etched with a deep RIE (DRIE) process to a typical depth of 50 microns in (j).  FIG. 21 ( k ) and (l) show respectively the backside photoresist being stripped, and another layer of photoresist  2116  being applied. Resist  2116  is patterned in (m), and the substrate is etched in (n) using DRIE until the second sacrificial layer is reached. The second sacrificial layer is etched with RIE in (o), and the DRIE is completed when both the second and third sacrificial layers are reached. In  FIG. 21 ( q ), the wafer is flipped over again and optionally placed on a carrier wafer  2130 . Also highlighted are the back etches  2132  formed in step (p, q) that allow for mirror movement. A photoresist layer  2140  is spun on the topside in (r), and shown patterned in (s). The first active layer is then etched with DRIE until the second sacrificial layer is reached in (t). In  FIG. 21 ( u ), the second sacrificial layer is etched with RIE. A DRIE etch is then used to etch about halfway, i.e. typically 25 microns, through the second active layer in (v). At this point photoresist layer  2140  is stripped and a DRIE etch is used to etch until the first and second sacrificial layers are reached in respectively maskless places, as shown in (w). In the final step,  FIG. 21 ( x ), the carrier wafer is removed from the backside of the wafer, the wafer is diced and the sacrificial layers, along with the deposited LPCVD oxide hard mask, are etched in hydrofluoric acid (HF). At this point the rotor fingers  2142  and bouncer spring  2144 , stator fingers  2146 , and mirror  2150  can be more clearly seen. As the HF etches the first sacrificial layer, the second active layer sections  2148  that remained underneath the rotor fingers fall away from the device between the stator fingers.  
         [0078]     In summary, the invention described above facilitates the formation of a triangular output signal for a scanning mirror and other devices by a single element (bouncer or pre-curved nonlinear stiffness element) having a stiffness nonlinearity (in the case of the bouncer a combined nonlinear stiffness with the rest of the system). The dependence of resonant frequency on the signal amplitude (and therefore voltage) permits its tuning in a very large range. In a parallel plate actuated embodiment, the mirror can be very close to an electrode when it contacts the bouncer without exhibiting pull-in, since the stiffness of the bouncer is very high. The proximity of the electrode permits to develop very large forces. The bouncer transfers attractive electrostatic forces to repelling forces, so that the mirror is actually driven by pulses.  
         [0079]     All publications, patents and patent applications mentioned in this specification are herein incorporated in their entirety by reference into the specification, to the same extent as if each individual publication, patent or patent application was specifically and individually indicated to be incorporated herein by reference. In addition, citation or identification of any reference in this application shall not be construed as an admission that such reference is available as prior art to the present invention.