Abstract:
Devices with tilting plates, in particular tilting mirror plates, having improved dynamic flatness and shock resistance and methods for forming and operating such devices. A mirror device includes a minor plate operative to perform a tilt motion around a mirror tilt axis and an elastic foundation which provides distributed support to the minor plate. In an embodiment, the minor plate and the elastic foundation are formed in a single silicon-on-insulator (SOI) wafer. In another embodiment, the minor plate and the elastic foundation are formed in separate SOI wafers. The elastic foundation may include spiral or serpentine or any other appropriate shaped springs distributed uniformly or non-uniformly between the minor plate and a base.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application is related to and hereby claims the priority benefit of U.S. Provisional Patent Application No. 61/539,491 filed Sep. 27, 2011. 
     
    
     FIELD 
       [0002]    Embodiments disclosed herein relate generally to devices which include tilting plates and more particularly to devices which include tilting micro-minor plates and implemented in Micro Electro Mechanical Systems (MEMS) or Micro Opto Electro Mechanical Systems (MOEMS). 
       BACKGROUND 
       [0003]    Tilting plate devices, for example tilting minors implemented in MEMS or MOEMS devices are known. Such devices may exemplarily comprise rectangular, circular or elliptical plates suspended on elastic torsion members of rectangular or square cross section.  FIG. 1  shows an exemplary tilting mirror  100  having a plate  102  suspended through two elastic torsion members  104  of length L A  on anchors  106 . Plate  102  is rectangular with a thickness h, width B and length 2R and is tiltable around the torsion members (or around a “torsion axis” or “tilt axis” defined by these members) through a tilt angle θ.  FIG. 2  shows initial and deformed configurations of the minor and schematics of the inertial pressure acting on the minor plate. In use, plate  102  undergoes dynamic deformation which can be separated into two components: (1) a rigid body tilting of the plate associated with angle θ, (configuration  202 ), and (2) dynamic deviation (“dynamic deflection”) from the planarity of the plate appearing as a result of inertial forces  204  engendered by time-dependent accelerations. The forces are indicated by arrows with varying length. The dynamic deflections are expressed by differences between the planar tilted plate configuration  202  and an actual deformed configuration  206 . The accelerations are maximal at maximal tilting angles θ(t) (where t is time) when the angular velocity {dot over (θ)}(t) is close to zero but the acceleration is maximal. The overdot ( )=d/dt denotes derivative with respect to time. Inertial force  204  acting on plate  102  is a distributed force (i.e. force per unit area). For a thin plate and a harmonic motion such that the tilting angle is θ(t)=θ MAX  sin(ωt)), the inertial pressure is given by 
         [0000]      p inertial =ρh{umlaut over (θ)}r=ρhω 2 θr   (1)
 
         [0000]    Here ρ is the density of the plate material, {umlaut over (θ)}=ω 2 θ is the angular acceleration, ω is the frequency of the mirror&#39;s vibrations and r is the distance between the tilting axis and the point where the pressure is calculated. The inertial pressure increases linearly with distance from the tilting axis and causes the dynamic deflection of the plate with respect to configuration  202 . 
         [0004]    Current solutions for the improvement of the dynamic flatness include mainly the incorporation of stiffeners (which is often challenging from the process point of view and manifests only limited efficiency) as well as suspension of the plate using an external frame when the torsion axis is attached not directly to the plate but to the frame. These solutions are not satisfactory. 
       SUMMARY 
       [0005]    In various embodiments there are provided tilting plates in which there is significant reduction or even complete elimination of dynamic deflections caused by inertial pressure. In some embodiments, the tilting plates are mirror plates. The following description is focused on, but by no means limited to such tilting mirrors. Tilting mirrors disclosed herein include an elastic foundation which provides distributed support of the mirror plate (referred to hereinbelow sometimes simply as “mirror” or “plate”), in contrast with the concentrated support common in the art. In some embodiments, the elastic foundation is realized as multiple springs. In some embodiments, the elastic foundation is realized as optimally located springs provided under the mirror plate. The tilting of the mirror results in the reaction (restoring pressure acting in the direction opposite to the minor plate deflection) of the elastic foundation, which increases linearly with the distance from the mirror tilt axis. Since the inertial pressure increases linearly with the distance from the axis as well, the reaction of the foundation compensates the inertial pressure at every point of the mirror plate. 
         [0006]    In an embodiment, there is provided a device comprising a tilting plate operative to perform a tilt motion around a tilt axis and an elastic foundation which provides distributed support to the tilting plate and reduces dynamic deflections of the plate during the tilt motion, thereby improving plate dynamic flatness and shock resistance. 
         [0007]    In an embodiment, the plate is a minor plate. 
         [0008]    In an embodiment, the elastic foundation includes a plurality of springs. 
         [0009]    In an embodiment, the springs are distributed uniformly relative to the plate. 
         [0010]    In an embodiment, the springs are distributed non-uniformly relative to the plate. 
         [0011]    In an embodiment, the non-uniform distribution provides a minimal deviation of the plate from a planar plate shape in a given integral norm. 
         [0012]    In an embodiment, the springs are spiral. 
         [0013]    In an embodiment, the springs have a serpentine shape. 
         [0014]    In an embodiment, the elastic foundation includes a plurality of torsion axes and links. 
         [0015]    In some embodiments, the device is formed in at least one silicon-on-insulator (SOI) substrate. 
         [0016]    In an embodiment formed in at least one SOI substrate, the elastic foundation is formed in the same SOI substrate as the mirror plate. 
         [0017]    In an embodiment formed in two SOI substrates, the elastic foundation and the mirror plates are formed in different SOI substrates. 
         [0018]    In an embodiment, the elastic foundation has a stiffness per unit area p elastic  which balances exactly a plate inertial pressure p inertial  such that p elastic −p inertial· =0. 
         [0019]    In an embodiment, there is provided a method for improving the dynamic flatness and shock resistance of a device which includes a tilting plate, the method comprising the steps of providing an elastic foundation and rotating the plate around a tilt axis while the elastic foundation interacts mechanically with the tilting plate to reduce dynamic deflections of the plate during the tilt motion, thereby improving the dynamic flatness of the tilting plate and improving the device shock resistance. 
         [0020]    In an embodiment of the method, the device is a tilting mirror and the plate is a minor plate. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0021]    Non-limiting embodiments are herein described, by way of example only, with reference to the accompanying drawings, wherein: 
           [0022]      FIG. 1  shows a conventional configuration of a tilting minor plate suspended using elastic torsion members; 
           [0023]      FIG. 2  shows initial and deformed configurations of the mirror and schematics of the inertial pressure acting on the mirror plate; 
           [0024]      FIG. 3  shows schematically an embodiment of a tilting mirror device disclosed herein, in two configurations, (A) un-deformed and (B) deformed; 
           [0025]      FIG. 4  shows schematics of the mirror and of the supporting springs located under the minor: (A) spiral springs and (B) serpentine springs; 
           [0026]      FIGS. 5A-D  show stages in an exemplary fabrication process of springs of the embodiment in  FIG. 4 ; 
           [0027]      FIG. 6  shows schematics of a serpentine spring and clamped-guided beams connected in series; 
           [0028]      FIGS. 7A-D  show schematically stages in the realization of yet another embodiment of the elastic foundation in a tilting mirror disclosed herein; 
           [0029]      FIGS. 8A , B show schematically stages in the realization of yet another embodiments of the elastic foundation in a tilting mirror disclosed herein; 
           [0030]      FIG. 9  shows schematically yet another embodiment of a tilting minor disclosed herein. 
       
    
    
     DETAILED DESCRIPTION 
       [0031]    Returning now to the drawings,  FIG. 3  illustrates schematically an embodiment of a tilting device (exemplarily a tilting minor)  300  disclosed herein, in two configurations: (A) un-deformed and (B) deformed. Mirror  300  includes in addition to a plate  302  which tilts around two torsion members (or tilt axis)  304 , a substrate  306  and an elastic foundation  308 . The substrate may be for example a handle layer of a silicon-on-insulator (SOI) wafer. In device  300 , the elastic foundation is formed using a plurality of springs  310 . 
         [0032]    In use,  FIG. 3C , the foundation provides distributed elastic forces acting on the minor plate in a way which prevents dynamic deflections. The deflection of every point of the plate (considered to be a rigid body) is linearly proportional to the distance from the tilting axis. The reaction of the foundation at each point (under the assumption of the small tilting angles), which can be viewed as a pressure of the elastic foundation with units N/m 2  acting on the plate is 
         [0000]      p elastic =kθr   (2)
 
         [0000]    where k [N/m 3 ] is the stiffness of the elastic foundation. Consequently, the total pressure acting at each point of the plate is 
         [0000]        p   inertial   −p   elastic   =ρhω   2   θr−kθr =(ρ hω   2   −k )θ r    (3)
 
         [0000]    Therefore, by choosing the stiffness of the foundation per unit area to be k=ρhω 2  it is possible to eliminate the dynamic deflection of the plate (i.e. cause p elastic −p inertial  to be zero). 
         [0033]    A simple physical explanation of this phenomenon is as follows: during free or forced resonant vibrations, the inertial forces, which are a harmonic function a time, should be dynamically equilibrated by the elastic forces. In the case of the simplest single degree of freedom mass-spring system, this “dynamic equilibrium” is satisfied at all times in the point representing the mass. In the case of a system with distributed parameters (continuous elastic system) such as a micro-minor plate, the inertial forces are applied to the plate whereas most of the elastic restoring forces are concentrated in the vicinity of the tilt axis. To compensate for the lack of the elastic forces within the area of the minor plate, these elastic restoring forces are produced by the dynamic bending of the plate. By distributing the elastic support forces over the plate area, the necessity for the mirror plate to bend (to balance the inertial forces as explained above) is eliminated. 
         [0034]    Consider now the dynamics of the tilting motion of the mirror. The equation of the free vibrations of the plate is 
         [0000]        I{umlaut over (θ)}+Kθ= 0   (4)
 
         [0000]    The inertial mass moment I of a rectangular plate with thickness h, width B and length 2R is I=ρhB(2R) 3 /12=2ρhBR 3 /3. The overall equivalent torsional stiffness K of the elastic foundation is calculated by integrating the moment dM created by the reaction of the elastic foundation (acting on an elementary area in the shape of a strip of the area Bdr oriented in parallel to the tilting axis) around the axis of the minor 
         [0000]    
       
         
           
             
               
                 
                   
                     dM 
                     = 
                     
                       k 
                        
                       
                           
                       
                        
                       θ 
                        
                       
                           
                       
                        
                       
                         r 
                         2 
                       
                        
                       B 
                        
                       
                           
                       
                        
                       dr 
                     
                   
                    
                   
                     
 
                   
                    
                   
                     M 
                     = 
                     
                       
                         
                           ∫ 
                           
                             - 
                             R 
                           
                           R 
                         
                          
                         
                            
                           M 
                         
                       
                       = 
                       
                         
                           
                             
                               k 
                                
                               
                                   
                               
                                
                               θ 
                                
                               
                                   
                               
                                
                               
                                 r 
                                 3 
                               
                                
                               B 
                             
                             3 
                           
                            
                           
                             | 
                             
                               - 
                               R 
                             
                             R 
                           
                         
                         = 
                         
                           
                             
                               2 
                                
                               k 
                                
                               
                                   
                               
                                
                               θ 
                                
                               
                                   
                               
                                
                               
                                 R 
                                 3 
                               
                                
                               B 
                             
                             3 
                           
                           = 
                           
                             K 
                              
                             
                                 
                             
                              
                             θ 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The equivalent torsion stiffness of the elastic foundation is K=2kBR 3 /3. The natural frequency of the tilting motion is 
         [0000]    
       
         
           
             
               
                 
                   ω 
                   = 
                   
                     
                       
                         K 
                         I 
                       
                     
                     = 
                     
                       
                         
                           
                             
                               2 
                                
                               
                                 kBR 
                                 3 
                               
                             
                             3 
                           
                           × 
                           
                             3 
                             
                               2 
                                
                               
                                   
                               
                                
                               ρ 
                                
                               
                                   
                               
                                
                               h 
                                
                               
                                   
                               
                                
                               
                                 BR 
                                 3 
                               
                             
                           
                         
                       
                       = 
                       
                         
                           
                             
                               k 
                               
                                 ρ 
                                  
                                 
                                     
                                 
                                  
                                 h 
                               
                             
                           
                           → 
                           k 
                         
                         = 
                         
                           ρ 
                            
                           
                               
                           
                            
                           h 
                            
                           
                               
                           
                            
                           
                             ω 
                             2 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0000]    We see therefore that the stiffness of the elastic foundation which satisfies the resonant condition also automatically satisfies the condition of zero total pressure (and consequently zero dynamic deflection) at all times. Note that in the case of a thin plate, the frequency of the tilting motion is identical also to the frequency of the up and down piston motion of the minor. However, for a small but finite thickness of the plate, additional rotational inertia leads to a certain decrease of the tilting frequency. As a result, the tilting frequency is a lowest frequency of the system, as desired. 
         [0035]    In addition to improving the dynamic flatness, the distributed support also improves significantly the robustness of the minor and its sustainability against mechanical shock. In a conventional configuration such as that of mirror  100 , the inertial force due to the shock acts mainly on the minor plate (the largest mass of the system), while the stiffness is provided by the torsion members. That is, the result of the inertial force and the result of the support elastic force are spatially separated and are applied to the minor plate at two different points. In the case of the distributed elastic support, the inertial force originating in the shock and acting on each infinitesimal element of the mirror is equilibrated by the distributed support elastic force acting on the same element. As a result, no bending moments arise in the mirror plate and the minor plate is not deformed under the shock. 
         [0000]    Realization of an Elastic Foundation with a Uniform Constant Stiffness 
         [0036]    The distributed elastic foundation can be realized by several approaches and, in MEMS or MOEMS, within the framework of the limitations of existing micro-fabrication processes. One possible realization (embodiment) is illustrated in  FIGS. 4A , B. A mirror plate  402  is attached to anchors  406  by two low torsional stiffness (long and thin) members  404 . An array of springs  408  is provided under the mirror plate. Exemplarily, each spring may have a traditional spiral shape,  FIG. 4A , or a serpentine shape,  FIG. 4B . The springs can be fabricated from the thin device layer of a SOI wafer with a tensile layer (e.g. of chromium or silicon nitride) deposited on top of each spring, using processes known in the art. If the number of springs is large enough, they can be viewed effectively as a uniform elastic foundation with a constant stiffness per unit area. 
         [0037]    Note that the distribution of the springs (or of any other type of elastic foundation member that fulfills the function of providing improving minor dynamic flatness and shock resistance) may be uniform or non-uniform. A non-uniform distribution may be chosen in such a way that the dynamic deflection of the plate is minimal in a certain norm. As an example of the flatness criterion, one can use the following expression 
         [0000]    
       
         
           
             
               min 
               A 
             
              
             
               
                 ∫ 
                 A 
               
                
               
                 
                   
                     ( 
                     
                       w 
                       - 
                       
                         r 
                          
                         
                             
                         
                          
                         θ 
                       
                     
                     ) 
                   
                   2 
                 
                  
                 
                    
                   A 
                 
               
             
           
         
       
     
         [0000]    where w is the dynamic deflection of the plate and A is the plate area. 
         [0038]      FIGS. 5A-D  show stages in an exemplary fabrication process of springs of the embodiment in  FIG. 4 . The starting material,  FIG. 5A , includes a first SOI (“mirror”) wafer with a handle layer  506  and a thin device layer  510  separated from the handle by a buried oxide (BOX) layer  508 . The device layer is covered by a highly stressed layer  512  (e.g. made of silicon nitride or chromium). Springs  514  are patterned first using deep reactive ion etching (DRIE), with the etch stop in the BOX layer,  FIG. 5B . Then, a mirror plate  502  is patterned using a device layer  516  of a second SOI wafer. The mirror wafer is then bonded face down to the second SOI wafer with previously patterned springs,  FIG. 5C . Springs  514  are released using hydrofluoric acid (HF). Due to high residual stress in layer  512 , the released springs are bent up, forming a spatial structure which contacts the mirror plate,  FIG. 5D . 
         [0039]    Exemplary parameters of the springs are calculated next. First, we calculate the stiffness of the elastic foundation required to achieve the desired frequency. Consider a first exemplary minor with a square plate of dimensions 1700×1700 μm 2  and thickness h of 30 μm. In this case, B=1700 μm and R=850 μm (see  FIG. 1 ). In accordance with Eqs. (3) and (6), we have k=ρhω 2 =4π 2 ρhf 2  where f is the natural frequency of the mirror. For the parameters above, ρ=2300 kg/m 3  and f=24 kHz, and the required stiffness of the elastic foundation is k=1.6×10 9  N/m 3 . 
         [0040]    Assume for simplicity that the elastic foundation is built from serpentine springs,  FIG. 4B . Each spring can be viewed as several clamped-guided beams connected in series, see  FIG. 6  The width, thickness and length of each segment of a spring are respectively b s , d s  and L s . The thickness of the stressed film is d f . The stiffness k s   (i)  of each segment of the serpentine spring (viewed as a clamped-guided beam) and the stiffness k s  of the entire assembly are respectively 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         k 
                         S 
                         
                           ( 
                           i 
                           ) 
                         
                       
                       = 
                       
                         
                           
                             12 
                              
                             EI 
                           
                           
                             L 
                             S 
                             3 
                           
                         
                         = 
                         
                           
                             
                               
                                 Eb 
                                 S 
                               
                                
                               
                                 ( 
                                 
                                   
                                     d 
                                     S 
                                   
                                   + 
                                   
                                     d 
                                     f 
                                   
                                 
                                 ) 
                               
                             
                             3 
                           
                           
                             L 
                             S 
                             3 
                           
                         
                       
                     
                     ; 
                   
                    
                   
                     
 
                   
                    
                   
                     
                       k 
                       S 
                     
                     = 
                     
                       
                         
                           k 
                           S 
                           
                             ( 
                             i 
                             ) 
                           
                         
                         n 
                       
                       = 
                       
                         
                           
                             
                               Eb 
                               S 
                             
                              
                             
                               ( 
                               
                                 
                                   d 
                                   S 
                                 
                                 + 
                                 
                                   d 
                                   f 
                                 
                               
                               ) 
                             
                           
                           3 
                         
                         
                           nL 
                           S 
                           3 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0000]    We calculate the curvature of each element of the spring due to the residual stress in the intentionally deposited stressed film (e.g., silicon nitride or chromium). In accordance with the well known Stoney formula (written here for the one-dimensional case of a beam rather than a plate), we have 
         [0000]    
       
         
           
             
               
                 
                   
                     ρ 
                     S 
                   
                   = 
                   
                     
                       
                         E 
                         S 
                       
                        
                       
                         d 
                         S 
                         2 
                       
                     
                     
                       6 
                        
                       
                         σ 
                         f 
                       
                        
                       
                         d 
                         f 
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Here ρ s  is the beam radius of curvature, E s  is the beam Young&#39;s modulus, d s  and d f  are respectively the thicknesses of the beam (spring) and of the stressed film, and σ f  is the residual stress in the film. The elevation of the end of each segment of the spring is 
         [0000]    
       
         
           
             
               
                 
                   
                     w 
                     S 
                     
                       ( 
                       i 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ρ 
                         S 
                       
                        
                       
                         [ 
                         
                           1 
                           - 
                           
                             cos 
                              
                             
                               ( 
                               α 
                               ) 
                             
                           
                         
                         ] 
                       
                     
                     = 
                     
                       
                         ρ 
                         S 
                       
                        
                       
                         [ 
                         
                           1 
                           - 
                           
                             cos 
                              
                             
                               ( 
                               
                                 
                                   L 
                                   S 
                                 
                                 
                                   ρ 
                                   S 
                                 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where L s  is the length of the segment, see  FIG. 6 . 
         [0041]    In an exemplary design, assume that each serpentine spring is assembled from n=10 segments, each segment being L s =180 μm long, b s =18 μm wide and d s =8 μm thick. The area of the spring is L s ×n×b s =180×180 μm 2 . The thickness of the film and the stress are respectively d f =2 μm and σ f =1000 MPa. Young&#39;s modulus of the spring material (Si) is 169 GPa. We assume for simplicity that the film has the same Young&#39;s modulus (although the actual modulus of silicon nitride is usually higher than that of Si). In this case we obtain that the stiffness of the spring is k s =52.1 N/m and the stiffness of the elastic foundation is k=k s /Area=52.1 N/m/(180×180 μm 2 =1.6×10 9  N/m 3 , as required. Equation (9) yields the deflection w s  of the end of the spring of w s =nw s   (i) =179 μm. This result implies that the end point of the mirror is able to deflect up to 180 μm without contact with the substrate, which is equivalent to a mechanical angle of 0.21 rad=12.1 degree. 
         [0042]    The calculations presented here provide only a simple estimation and are exemplary. Careful design and optimization of the spring may allow significant improvement of the results. In an alternative embodiment, the design may be such that the elastic foundation provides only a part of the total tilting stiffness, with another part of this stiffness originating in the torsion axis. In yet another embodiment, the design may include an elastic foundation with variable stiffness. The part of the mirror plate closer to the tilting axis deflects much less in the vertical direction and stiffer springs may be used there. The springs located further away from the axis may be weaker, thus allowing larger deflection of the mirror. 
         [0043]    Consider now another (second) exemplary mirror of dimensions 3000×3000×530 μm 3 , where the thickness of the mirror plate is 50 μm. In this case we have B=3000 μm and R=1500 μm (see  FIG. 1 ). Assume that each segment of the serpentine spring is L s =300 μm long, b s =20 μm wide and d s =6 μm thick. The area of the spring is L s ×n×b s =300×200 μm 2 . The thickness of the film and the stress are d f =0.5 μm and σ f =1000 MPa, respectively. For the adopted parameters, ρ=2300 kg/m 3  and f=2.5 kHz, obtain that the required stiffness of the elastic foundation is k=2.8×10 7  N/m 3 . Equation (7) gives for these mirror parameters k=k s /Area=2.6×10 7  N/m 3 , which is close to the required value. Equation (9) yields the deflection of the end of the spring of w s =nw s   (i) =332 μm. This result implies that the end point of the mirror is able to deflect up to 332 μm without contact with the substrate, which is equivalent to the mechanical angle of 0.22 rad=12.7 degree for the 3 mm wide mirror. 
         [0044]    We now analyze the behavior of this mirror in the case of mechanical shock. First note that the curvature of the springs arises due to the release of the tensile residual stress in the silicon nitride layer and the appearance of the compressive stress in the beam itself. The full flattening of the beam will result in the restoration of the original tensile stress in the nitride layer and zero stress in the Si layer. From this perspective, the stress in a spring cannot exceed an initial “as fabricated” value. The full flattening of a spring is accompanied by contact of the spring with the substrate, which prevents the mechanical failure of the structure. 
         [0045]    The balance between the potential and kinetic energy of the mirror plate yields 
         [0000]        mgH= 1/2 mv   2   →v =√{square root over (2 gH )}  (10)
 
         [0000]    where H=1 m is the height that the device is dropped from. The balance between the kinetic energy of the mirror at the end of the drop and the potential energy of the elastic foundation is 
         [0000]      1/2 mv   2 =1/2 k ×Area Mirror   w   Max   2    (11)
 
         [0000]    where w is the maximal deflection due to the drop, k is the stiffness of the elastic foundation ([N/m 3 ]) and v is given by Eq. (10). For the parameters of this mirror, we get w MAX =290 μm. That is, the full flattening of the springs is not reached. Note that this estimation is very conservative, since it disregards the presence of additional springs holding the mirror plate and assumes an ideally “rigid” contact after the drop. The result can be improved by using a variable stiffness elastic foundation in which the stiffness closer to the tilting axis is higher than at the outer part of the minor. In the case of the minor with the higher frequency of 24 kHz the deflection due to the 1 m drop is much smaller and is 28 μm. 
         [0046]      FIGS. 7A-D  show schematically various stages in the realization of yet another embodiment of the elastic foundation in a tilting mirror  700 . In this embodiment, springs  714  are fabricated of polysilicon (covered by a stressed layer  712 ) and are grown on a mirror plate  702  of a first SOI wafer,  FIG. 7A . The springs are released using hydrofluoric acid,  FIG. 7B . An adhesive layer (polymer, low melting temperature metal)  718  is deposited (e.g. using a shadow mask) at the bottom of a cavity  716 , etched into another wafer,  FIG. 7C . Release of the springs is followed by heating of the assembly and bonding of the springs to the bottom of the cavity. An opening  720  is then etched into a handle layer  706  of the first SOI wafer to expose the optical surface of the minor plate,  FIG. 7D . 
         [0047]      FIGS. 8A , B shows stages in the realization of yet another embodiment of the elastic foundation in a tilting mirror  800  disclosed herein. In this embodiment, an elastic foundation is attached to a minor plate  802  and is formed by an array of torsional springs  804  with low torsional stiffness and links  810  arranged in the direction perpendicular to a direction  801  of a tilting axis  804 ,  FIG. 8A . A serpentine-like geometry as in  FIGS. 4 ,  5  can be implemented to reduce the overall torsional stiffness of the springs. An isomeric view of the elastic foundation realized as an assembly of torsional springs and links is shown in  FIG. 8B . 
         [0048]    In yet another embodiment (not shown), a finite number of springs is provided instead of a distributed elastic foundation. The springs are optimally located to provide highest dynamic flatness and appropriate resonant frequency. Each of the springs is attached to a bending flexure or torsional spring to provide a necessary stiffness in the up and down direction. 
         [0049]    In yet another embodiment, shown in  FIG. 9 , a mirror  902  is attached to anchors  906  by two torsion members  904 . A bridge  908  is provided above torsion members  904 . A gap  910  left between the torsion members and the bridge ensures that there is no contact between them during normal operation of the minor. In the case of high inertial out-of-plane forces arising due to shock, the bridge prevents the excessive out-of-plane deformation of the structure and its failure. The bridge may be fabricated from a material such as polysilicon, metal or polymer (e.g. PDMS) using surface micromachining. Combined with a post  912  etched into the handle  914  of the wafer and fabricated using DRIE and located under the mirror or under the torsion members, the bridge provides protection against shock in all possible directions. 
         [0050]    While this disclosure has been described in terms of certain embodiments, alterations and permutations of the embodiments will be apparent to those skilled in the art. Specifically, while the description focused in detail on tilting mirrors, non-minor tilting plates may also advantageously have improved dynamic flatness and shock resistance through the addition and use of elastic foundations disclosed herein, as long as equation 3 is fulfilled (foundation stiffness leading to zero total pressure). The disclosure is to be understood as not limited by the specific embodiments described herein, but only by the scope of the appended claims.