Abstract:
A thermomechanical actuation system and method includes an elongated thermomechanical actuator (TMA), which is contoured so that electrical resistance at a mid-portion of the TMA is less than at end portions thereof. A pulse generator is electrically coupled to the TMA, and is configured to supply excitation pulses to the TMA. The excitation pulses are transient, so that each pulse is terminated prior to reaching a steady state amplitude, while having sufficient energy to heat the TMA to its predetermined operational temperature range.

Description:
RELATED APPLICATION 
     This application claims the benefit of U.S. Provisional Application Ser. No. 60/970,158, entitled High Speed Pulsing Technique for Non-uniform Heat Generation Thermal System, filed on Sep. 5, 2007, the contents of which are incorporated herein by reference in their entirety for all purposes. 
    
    
     GOVERNMENT SPONSORSHIP 
     This invention was made with government support under Contract Number DMI-0348242, awarded by the National Science Foundation, and Contract Number I-R21-CA118400-01, awarded by the National Cancer Institute/National Institute of Health. The government has certain rights in the invention. 
    
    
     BACKGROUND 
     1. Technical Field 
     This invention relates to thermomechanical actuators, and more particularly to a system and method which uses pulsed actuation of contoured thermomechanical actuators for enhanced dynamic performance. 
     2. Background Information 
     A wide variety of thermomechanical actuators (TMAs) are known in the art. TMAs make use of Joule heating and thermal expansion of materials to generate displacements. Conventional TMAs, which contain slender, constant cross-section microfabricated beams, are well-known for their relatively large force and stroke outputs. Their bandwidths, however, are limited by the heat diffusion process.  FIGS. 1A &amp; 1B  show common TMA configurations, including constant cross-section beam  30  disposed in parallel beam ( FIG. 1A ) and chevron ( FIG. 1B ) configurations. Due to their force/stroke characteristics and ease-of-fabrication, TMAs are frequently found in a variety of meso-/micro-scale devices and positioning systems, such as disclosed in U.S. patent application Ser. No. 11/037,866 (the &#39;866 application), entitled Multiple Degree of Freedom Micro Electro-Mechanical System Positioner and Actuator, filed on Jan. 18, 2005, and which is fully incorporated herein. For instance, they have been used in in-package active fiber alignment devices, micro-scanners used in endoscopes, and meso-/micro-scale nanopositioners. However, drawbacks associated with conventional straight beam TMAs include relatively high power consumption, low efficiency, and low bandwidth, all of which tend to make it difficult to use these TMAs as the basis of practical and efficient devices. 
     Therefore, there are a number of unresolved issues associated with the use of TMAs, which are addressed by the present invention. 
     SUMMARY 
     In one aspect of the present invention, a thermomechanical actuation system includes an elongated thermomechanical actuator (TMA), which is contoured so that electrical resistance at a mid-portion of the TMA is lower than at end portions thereof. A pulse generator is electrically coupled to the TMA, and is configured to supply excitation pulses to the TMA. The excitation pulses are transient, so that each pulse is terminated prior to reaching a steady state amplitude, while having sufficient energy to heat the TMA to its predetermined operational temperature range. 
     In another aspect of the invention, a method of thermomechanical actuation includes providing an elongated thermomechanical actuator (TMA), having a contour, wherein electrical resistance at a mid-portion of the TMA is lower than at end portions thereof. The method further includes electrically coupling a pulse generator to the TMA, configuring the pulse generator to supply excitation pulses to the TMA, and configuring the excitation pulses to be transient, so that each pulse is terminated prior to reaching a steady state amplitude. The excitation pulses are provided with sufficient energy to heat the TMA to its predetermined operational temperature range. 
     The features and advantages described herein are not all-inclusive and, in particular, many additional features and advantages will be apparent to one of ordinary skill in the art in view of the drawings, specification, and claims. Moreover, is should be noted that the language used in the specification has been principally selected for readability and instructional purposes, and not to limit the scope of the inventive subject matter. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIGS. 1A-1B  are schematic plan views of configurations of straight beam TMAs of the prior art; 
         FIGS. 1C-1E  are schematic plan views of contoured TMAs of the type used in embodiments of the present invention; 
         FIGS. 2A ,  2 B are graphical representations of temperature/heat profiles of contoured and straight TMAs subject to the same power input, respectively; 
         FIG. 3A  is a graphical representation of temperature/heat profiles of a contoured TMA heated with a conventional pulse, and with a transient pulse for similar displacement in accordance with embodiments of the present invention; 
         FIG. 3B  is a graphical representation of conventional and transient pulses; 
         FIGS. 4A ,  4 B are graphical representations of temperature/heat profiles of a contoured TMA heated in accordance with embodiments of the present invention, during heating and cooling; 
         FIGS. 5A ,  5 B are graphical representations of temperature/heat profiles of a contoured TMA heated conventionally, during heating and cooling; 
         FIGS. 6A ,  6 B are graphical representations of heat profiles of a prior art TMA heated conventionally, during heating and cooling; 
         FIGS. 7A ,  7 B are schematic plan and perspective views of a representative application using embodiments of the present invention; 
         FIGS. 8A ,  8 B show transmission ratio surface plots as a function of ⊖ l , ⊖ 2  and the number of parallel TMAs; and 
         FIG. 9  is a schematic representation of various exemplary fabrication steps of an embodiment of the present invention. 
     
    
    
     DETAILED DESCRIPTION 
     In the following detailed description, reference is made to the accompanying drawings that form a part hereof, and in which is shown by way of illustration, specific embodiments in which the invention may be practiced. These embodiments are described in sufficient detail to enable those skilled in the art to practice the invention, and it is to be understood that other embodiments may be utilized. It is also to be understood that structural, procedural and system changes may be made without departing from the spirit and scope of the present invention. The following detailed description is, therefore, not to be taken in a limiting sense, and the scope of the present invention is defined by the appended claims and their equivalents. For clarity of exposition, like features shown in the accompanying drawings are indicated with like reference numerals and similar features as shown in alternate embodiments in the drawings are indicated with similar reference numerals. 
     Where used in this disclosure, the term “axial” when used in connection with an element described herein, refers to a direction relative to the element, which is substantially parallel to its longitudinal dimension. Similarly, the terms “transverse” or “lateral” refer to a direction other than substantially parallel to the axial direction. The term “transverse cross-section” refers to a cross-section taken along a transverse plane. Moreover, for convenience, the term “TMA” refers to the driving beam of a TMA actuator, i.e., to the Joule heated beam of a particular TMA actuator configuration, with the understanding that such a Joule heated beam may be used in conjunction with other components that may include either constant cross-section beams (of relatively thick or thin transverse cross-section), or contoured beams, to form a TMA actuator such as shown in  FIGS. 1D and 1E . 
     Referring to  FIGS. 1C-1E , embodiments of the present invention include a contoured TMA  34  ( FIG. 1C ), which may be used in nominally any configuration, such as in parallel beam ( FIG. 1D ) or chevron ( FIG. 1E ) arrangements. As discussed in the aforementioned &#39;866 application, such contouring may significantly reduce the drawbacks associated with conventional straight beam TMAs, particularly when used in conjunction with the short pulse actuation of the present invention, as will be discussed in greater detail hereinbelow. 
     It is to be noted that the concept of TMA contouring involves varying the electrical resistance (e.g., by varying the transverse cross-section as shown, and/or by varying the structural composition) of a TMA over its length. Although geometric contouring is shown and discussed herein as a desired approach for use in many applications, nominally any type of contouring may be used to vary the resistance of a TMA along its length. For example, TMAs may be contoured electrically, instead of, or in addition to, the aforementioned geometric approaches. This may be accomplished by adding dopants (e.g., ion implantation) during microfabrication of the beam to vary the structural composition of the TMA along its length. In this manner, the TMA may be fabricated to have higher electrical resistivity at end portions of the beam, relative to the electrical conductivity at a mid portion of the beam, with or without any geometric contour. A solely electrically contoured TMA (i.e., one without a geometric contour), may provide many, if not all, of the advantages provided by the geometrically contoured embodiments, particularly when used in applications such as on-chip chemical reactors that have few moving parts. 
     It has been established by the instant inventors that in many (e.g., quasi-static) applications, embodiments of contoured TMA  34  may simultaneously produce more than twice the force and stroke of an otherwise similar conventional TMA  30  of a constant cross-section. The inventors have found that this phenomenon may be due to the constant cross-section beam  30  being relatively inefficient at transforming thermal strain into mechanical energy, e.g., due to its storage of relatively high amounts of the strain energy. The inventors have further found that by contouring a beam, one may increase the thermal strain and elastic range of the beam while decreasing the strain energy that is stored (particularly when using geometric contouring as discussed above) in the beam during actuation. As a result, more energy is available to do useful work. Table 1 summarizes the performance improvements of a TMA  34  in a chevron configuration (as shown in  FIG. 1E ) relative to an otherwise similar constant cross-section TMA in a chevron configuration ( FIG. 1B ). 
                                                                 TABLE 1                   Review of static performance improvements by contouring.                Constant power       Constant Stroke                    Stroke   Force   Power   Temperature                       4X   2.5X   65%   35%                        
Transient Heating Behavior of a Contoured Beam/TMA
 
     Turning now to  FIGS. 2A ,  2 B, transient heating profiles of a contoured TMA  34  ( FIG. 2A ) and a straight TMA  30  ( FIG. 2B ) are respectively shown. In these examples, a constant current of 25 mA is supplied to both beams and the temporal change of the temperature profiles is plotted. The parameters used in the examples are listed in Table 2. While not wishing to be tied to a particular theory, it is believed that the differences between the temperature profiles of these two TMAs are due to inhomogeneous distributions of heat along the contoured TMA  34 . 
     
       
         
               
             
               
               
               
             
               
               
               
               
             
               
               
               
               
             
           
               
                 TABLE 2 
               
             
             
               
                   
               
               
                 Design parameters used in examples of FIGS. 2A and 2B. 
               
             
          
           
               
                   
                 Example A 
                 Example B 
               
               
                   
                   
               
             
          
           
               
                   
                 Type 
                 Contour 
                 Constant cross-section 
               
               
                   
                 L S /L L   
                 1 
                 N/A 
               
               
                   
                 L/2L L   
                 3 
                 N/A 
               
               
                   
                 1/w′ 
                 1/2 
                 1 
               
               
                   
                 w e   
                  8 μm 
                  8 μm 
               
               
                   
                 L 
                 600 μm 
                 600 μm 
               
               
                   
                 b 
                  30 μm 
                  30 μm 
               
             
          
           
               
                   
                 Driving current 
                 25 mA 
               
               
                   
                   
               
             
          
         
       
     
     In  FIGS. 2A and 2B , the x-axis and y-axis represent the position and temperature along TMAs  34 ,  30 , respectively. In both cases, the two ends of the beams are assumed to be attached to a heat sink maintained at room temperature. Starting from T=300° K (room temperature), each temperature profile is offset by a 0.2 milli-second time interval. By qualitatively comparing examples A and B, it is found that the contoured beam has a slightly larger thermal time constant, which is expected because the heat generated at both ends of the contoured TMA  34  would require some time to reach equilibrium state. 
     It is also found that the average temperature on the contoured beam rises faster than the straight beam, which indicates that for identical input/power, the contoured TMA has a higher forward stroke speed. Quantitative observation and calculations may be done based upon  FIGS. 2A and 2B  by calculating the different temperature profiles. 
     Another property of contoured beam  34  is that its temperature profile goes through three different stages as time progresses. At stage  1 , the temperature on both ends of the contoured beam rises faster than in the central region and forms a concave-up profile (as more heat is generated on both ends), such as shown at t 1 =0.2 ms. At stage  2 , the heat accumulates at the central region, and there is a moment in time where the temperature profile becomes relatively “flat” on a rather wide mid-portion of the contoured beam, such as shown at t 2 =1.0 ms. At stage  3 , the system gradually reaches steady state and forms a uniform bell-shape temperature profile, such as shown at t 3 =3.3 ms and t 4 =4 ms. 
     It is also noted that the heating and cooling temperature profiles are different for a contoured beam  34 . In the cooling process, it has been found that the temperature of beam  34  tends to decrease faster than the similar straight beam  30 , which is somewhat counter-intuitive given the greater overall heat content of the contoured beam  34 , as shown by the area under the curves of  FIGS. 2A ,  2 B. It has been found that this faster cooling is provided by the relatively large temperature gradient at the ends of the beam relative to ground, provided by the contour, as shown by the relatively steep profile at the ends of the beam in  FIG. 2A . Embodiments of the present invention use these unique heating/cooling characteristics in combination with a high speed pulsing approach to provide an improved TMA actuation system. 
     Pulsing Technique 
     Referring now to  FIGS. 3A and 3B , embodiments of the present invention use a short pulse actuation in which the aforementioned non-uniform heat generation characteristics of Joule heated beam  34  are used to re-shape the temperature distribution along the beam to form a relatively flat temperature profile, as shown in  FIG. 3A . This flat temperature profile is provided by using transient pulses  46  to heat the contoured TMA  34  to stage  2  of its three-stage temperature profile. 
       FIG. 3A  shows the temperature profiles  40 ,  42  of a contoured beam  34  respectively actuated with conventional steady state pulses  44  ( FIG. 3B ) that heat TMA  34  to its steady state “stage  3 ” profile, and transient, short pulses  46  ( FIG. 3B ) that heat TMA  34  to its transient “stage  2 ” profile. 
     As shown in  FIG. 3B , pulses  44  are referred to as steady state pulses since they tend to reach a plateau or horizontal peak prior to dropping. As mentioned hereinabove, these pulses  44  drive TMAs to a quasi-static state, i.e., steady state, for either a TMA  30  (as shown in  FIG. 6A ) or a TMA  34  in its “stage  2 ” (as shown at  40 ,  FIG. 3A ), and then let the actuators slowly cool down, returning to their original positions. Pulses  44  are, for example, conventionally used in a 50% duty cycle, such as shown in  FIG. 3B . It is noted that advantages associated with embodiments of the present invention would not be realized by simply applying transient pulses  46  to conventional straight TMAs  30 , since the TMAs  30  do not have three-stage temperature profiles, as discussed hereinabove. 
     Pulses  46  are referred to as transient pulses since they are still rising in amplitude (e.g., in current) and thus have not leveled off or plateaued, at the point at which they are terminated. In other words, the transient pulses are those having a voltage and current combination which, if enabled to reach steady state, would heat at least a portion (e.g., central portion) of the TMA beyond its predetermined operational temperature range. (In various embodiments, the predetermined operational temperature range is determined based on design requirements/constraints. However, a temperature closer to the failure temperature will typically increase the overall thermal efficiency. The failure temperature is defined as the temperature at which the modulus of the material (e.g. silicon), will change, e.g., decrease, which may ultimately cause the material to melt. In particular embodiments, the predetermined operational temperature is about 50-90 percent of the failure temperature of the material. Moreover, in some representative embodiments, the pulses  46  have an amplitude (e.g., voltage or current) which is at least about, for example, 1.25 to 1.5 times the level required to heat the TMA to its predetermined operational temperature range, in the event the pulses were permitted to reach their steady state. 
     It is noted that although square waves are used as short pulses  46  in the previous examples, the short pulse signals are not limited to square waves. For example, the pulse  46  may be combined with other nominally any conventional signal control/conditioning/preshaping techniques to achieve desired dynamic performance. The spirit of the short or transient pulse is that its “average amplitude” and “average pulse width” is respectively larger and smaller than that of the conventional steady state signal. 
     It should be noted that while profile  40  is steady state, the flat temperature profile  42  is a transient state. As mentioned above, this transient state profile  42  (“stage  2 ” of TMA  34 ) is steeper at the ends of the beam, denoting a relatively high temperature gradient. The high gradient enables more heat to be removed from the contoured beam  34  to the heat sink, to reduce overall cooling time. 
     Thus, as compared to operation of a conventional straight TMA  30 , the transient pulse  46  activation of this approach enables a contoured three-stage TMA  34  to heated to its transient “stage  2 ” temperature profile, to provide a high temperature gradient on the two ends of beam  34  to increase the cooling speed, increased maximum displacement, faster displacement, and decreased energy consumption. 
     The following illustrative examples demonstrate certain aspects and embodiments of the present invention, and are not intended to limit the present invention to any one particular embodiment or set of features. 
     EXAMPLES 
     Heating and cooling profiles of exemplary contoured TMAs  34 , using both short and conventional pulses, are described in the following Examples, with reference to  FIGS. 4-6 . 
     A Finite Element Analysis (FEA) program (COMSOL) was used to quantify some of the benefits that may be obtained through the contoured beam and pulsing technique of the present invention. Heating and cooling profiles of a Joule heated contoured beam  34  actuated with a short pulse signal are respectively shown in  FIGS. 4A ,  4 B, while temperature profiles of the same beam actuated with a conventional steady state signal are shown in  FIGS. 5A ,  5 B. As a Control, heating and cooling profiles of a conventional straight Joule heated beam  30  driven by a conventional signal is simulated, with heating and cooling profiles shown in  FIGS. 6A ,  6 B, respectively. In these examples, the conventional straight beam  30  has a constant cross section and the same volume and length as the contoured beams  34 . The design parameters for these beams and pulses are listed in Table 3, and the results are summarized in comparison to the conventional approach, in Table 4. 
     
       
         
               
             
               
               
               
               
             
               
               
               
             
               
               
               
               
             
               
               
             
           
               
                 TABLE 3 
               
             
             
               
                   
               
               
                 Design parameters used in FIGS. 4A-6B. 
               
             
          
           
               
                   
                 FIGS. 4A, 4B 
                 FIGS. 5A, 5B 
                 FIGS. 6A, 6B 
               
               
                   
                   
               
             
          
           
               
                 Type 
                 Contour beam 
                 Straight beam 
               
               
                 L S /L L   
                 1/10 
                 N/A 
               
               
                 L/2L L   
                 3/2  
                 N/A 
               
               
                 1/w′ = w L /w S   
                 5 
                 1 
               
               
                 L (length of the 
                 600 μm 
                 600 μm 
               
               
                 beam) 
               
               
                 B (height of the 
                  30 μm 
                  30 μm 
               
               
                 beam) 
               
             
          
           
               
                 Signal type 
                 Short pulse 
                 Conventional 
                 Conventional 
               
               
                 Signal value 
                 54 mA 
                 27 mA 
                 35 mA 
               
               
                 Pulse width 
                 0.94 ms 
                 30 ms 
                 30 ms 
               
               
                 Command max. 
                 1000°K 
                 1000°K 
                 1000°K 
               
               
                 temp. 
               
             
          
           
               
                 Note 
                 Time increment = 10 μs 
               
               
                   
               
             
          
         
       
     
     In these Examples, beams  34 ,  30  are fabricated from silicon, with a peak safe operating temperature set to be 1000° K. As shown in  FIG. 4A , a substantially flat temperature profile was achieved at 1000° K. It is noted that almost 80% of the Joule heated contoured beam may be maintained at 1000° K as compared to a single point at 1000° K for both the contoured beam  34  and straight beam  30  when driven with conventional signals as respectively shown in  FIGS. 5A and 6A . 
     As shown in  FIG. 4B , the temperature at both ends of the contoured, short pulsed beam dropped faster than those of the other examples ( FIGS. 5B ,  6 B), due to the aforementioned high temperature gradient. 
     As shown in  FIGS. 6A ,  6 B, the temperature profiles of the constant cross-section beam  30 , driven with a conventional signal, deviate from parabolic shapes. These conventional temperature profiles tend to be inferior to parabolic profiles as they yield a lower average temperature than parabolic profile at a given T MAX  (T MAX =1000° K in this Example). The instant inventors attribute this effect to the fact that the thermal conductivity for silicon is a function of temperature and tends to decrease relatively quickly as the temperature increases. In terms of actual beam elongation or TMA displacement, the straight beam TMA has been shown to yield worse performance than the inventive TMAs because the value of silicon&#39;s coefficient of thermal expansion at 1000° K is two times higher than the value at room temperature. The combined effects of the contoured beam with short pulse signal of the present invention have been shown to generate 70% more extension in beam length (Δ T ) than a straight beam, with a T MAX  on both beams of 1000° K. 
     As shown in Table 4, it is observed that a simultaneous 28% enhancement in stroke, 11% reduction in cooling time, and 70.9% reduction in required actuation energy is provided (on the same contoured beam  34 ) when using the transient short pulse signal. Moreover, the same displacement provided by the conventional beam/signal at T MAX  of 1000° K may be achieved using the inventive beam and pulsing approach at a T MAX  of only 850° K. Also, in this scenario, the time required for cooling and the reduction in power are further enhanced by 23% and 73% respectively. 
                                                                                 TABLE 4                   Summary of performance of examples in FIGS. 4A-6B.                FIGS. 4A, 4B   FIGS. 5A, 5B   FIGS. 6A, 6B                        Type   Contoured   Straight            Signal type   Short pulse   Conventional   Conventional       Rise time   0.853 ms   13.80 ms   6.66 ms       Fall time   3.02 ms   3.40 ms   1.60 ms       Elongation   2.058 μm   1.605 μm   1.214 μm       Stroke enhancement [%]   ↑ 28.2%   1   ↓ 24.4%       Normalized forward speed   ↑ 19.8X   1   ↑ 1.5X       enhancement       Normalized returning speed   ↑ 1.5X   1   ↑ 1.6X       enhancement       Normalized energy consumption [%]   ↓70.9%   1   ↑ 68.0%            Note   Time increment = 10 μs                    
Pulsed Signal Determination
 
     A systematic approach may be used to determine an optimal short pulse for a given contoured beam at a desired flat (e.g., maximum) temperature. If too much power is supplied in a short time for a contoured beam (e.g., if the transient pulse is permitted to approach steady state), the beam may fail (burn out), e.g., at the ends of the beam before it reaches a flat temperature profile. If too little power is supplied, however, the temperature profile will become flat at a lower than optimal temperature, at which temperature some of the benefits of the present invention may not be achieved. 
     Two parameters, signal (current) amplitude (I P ) and pulse width (t P ), need to be determined for the short pulse. The example of  FIG. 4A  will be used here to demonstrate how I P  and t P  were found. In  FIG. 4A , a desired temperature is set at 1000° K, and the objective is to find the current, I P , and t P  so that the temperature at the center (i.e., at x=0) of beam  34  is substantially equal to the temperature close to the ends (i.e., at x=±L/2) of the contoured beam at time t P . As shown in Equation (03-06), I P  and t P  may be found with the described objective and a proper initial condition (I.C.) and boundary conditions (B.C.). 
     
       
         
           
             
               
                 
                   
                     
                       
                         ∂ 
                         
                           ∂ 
                           x 
                         
                       
                       ⁢ 
                       
                         k 
                         ⁡ 
                         
                           ( 
                           T 
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           ∂ 
                           T 
                         
                         
                           ∂ 
                           x 
                         
                       
                     
                     + 
                     
                       
                         r 
                         ⁡ 
                         
                           ( 
                           x 
                           ) 
                         
                       
                       · 
                       
                         
                           I 
                           P 
                         
                         
                           A 
                           ⁡ 
                           
                             ( 
                             x 
                             ) 
                           
                         
                       
                     
                   
                   = 
                   
                     
                       ρ 
                       · 
                       
                         C 
                         P 
                       
                     
                     ⁢ 
                     
                       
                         ∂ 
                         T 
                       
                       
                         ∂ 
                         t 
                       
                     
                   
                 
               
               
                 
                   ( 
                   03 
                   ) 
                 
               
             
             
               
                 
                   
                     B 
                     . 
                     C 
                     . 
                     
                       T 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             = 
                             L 
                           
                           , 
                           
                             
                               t 
                               0 
                             
                             = 
                             0 
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       T 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             = 
                             0 
                           
                           , 
                           
                             
                               t 
                               0 
                             
                             = 
                             0 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       300 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       K 
                     
                   
                 
               
               
                 
                   ( 
                   04 
                   ) 
                 
               
             
             
               
                 
                   
                     I 
                     . 
                     C 
                     . 
                     
                       T 
                       ⁡ 
                       
                         ( 
                         
                           x 
                           , 
                           
                             
                               t 
                               0 
                             
                             = 
                             0 
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     300 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     K 
                   
                 
               
               
                 
                   ( 
                   05 
                   ) 
                 
               
             
             
               
                 
                   
                     Objective 
                     : 
                     
                       T 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             1 
                           
                           , 
                           
                             t 
                             P 
                           
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       T 
                       ⁡ 
                       
                         ( 
                         
                           
                             x 
                             2 
                           
                           , 
                           
                             t 
                             P 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       1000 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       K 
                     
                   
                 
               
               
                 
                   ( 
                   06 
                   ) 
                 
               
             
           
         
       
     
     In another example, a series of parallel TMAs  34  were used in an endoscopic scanner for a two-photon endomicroscope having a conventional gradient index (GRIN) lens  50 . It is noted that TMAs  34  are particularly suitable for in vivo endoscopic applications due to their combination of relatively high force and low driving voltage. In this example, an active silicon optical bench  52 , shown in  FIGS. 7A ,  7 B, contains flexure bearings  54  and contoured TMAs  34  that control the z motion of the gradient index (GRIN) lens  50  and the angular motion, θx, of the lens  50  for in-plane 2D scanning. 
     As shown in  FIGS. 7A ,  7 B, the GRIN lens  50  is disposed on a shuttle  56 , which is fabricated integrally with the in-plane contoured chevron TMAs  34  (and chevron flexures  35 ), e.g., by microfabrication as discussed hereinbelow. The optical bench  52  provides micrometer level precision alignment for small optical elements. Moreover, as shown, it is symmetrical, including a two-stage chevron amplification mechanism driven by symmetrical chevron TMA trains  34  actuatable in the direction perpendicular to the Z direction. The chevron flexures  35  that connect the TMA trains  34  and the GRIN lens shuttle  56  further amplify the motion provided from the TMA trains  34 . The symmetric design provides precision motion guidance and reduced parasitic motion for the GRIN lens  50 . 
     The transmission ratio, which is defined as the output displacement over input displacement, of the two-stage chevron mechanism was optimized by mechanically matching the axial and lateral stiffness of the individual chevron flexures, by disposing them at predetermined angles θ 1  and θ 2 , as shown in  FIG. 8A .  FIG. 8B  presents the surface plot of transmission ratio as functions of θ 1  and θ 2  for different numbers of contoured TMAs. Based on this plot, angles θ 1  and θ 2  and the number of TMAs  34  for a particular application may be selected. 
     Turning now to  FIG. 9 , an exemplary method of fabricating embodiments of the present invention is shown and described. This exemplary methodology may be used to fabricate any of the embodiments described herein, with particular reference to the 2D active silicon optical bench  52 . 
     Step 1: 150 mm SOI wafer includes three layers: Silicon handling wafer (500 μm)-Silicon Dioxide (2 μm)-Silicon device layer (200 μm). 
     Step 2: Aluminum electrical contact pads are patterned on the wafer. 
     Step 3: The mechanism-actuator geometry is etched through the device layer via deep reactive ion etching (DRIE). 
     Step 4: The wafer is mounted to a quartz handling and support wafer. 
     Step 5: A back-side DRIE process creates access to the silicon dioxide layer. 
     Step 6: The quartz wafer is removed from the device wafer. A concentrated vapor Hydrofluoric acid is used to release the mechanism  52  including shuttle  56  and actuators  34 . 
     It should be understood that any of the features described with respect to one of the embodiments described herein may be similarly applied to any of the other embodiments described herein without departing from the scope of the present invention. 
     In the preceding specification, the invention has been described with reference to specific exemplary embodiments for the purposes of illustration and description. It is not intended to be exhaustive or to limit the invention to the precise form disclosed. Many modifications and variations are possible in light of this disclosure. It is intended that the scope of the invention be limited not by this detailed description, but rather by the claims appended hereto.