Patent Publication Number: US-7898714-B2

Title: Methods and apparatus for actuating displays

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application is a divisional of U.S. patent application Ser. No. 11/656,307, entitled “Methods and Apparatus for Actuating Displays” and filed Jan. 19, 2007. U.S. patent application Ser. No. 11/656,307 is a continuation-in-part of U.S. patent application Ser. No. 11/251,035, entitled “Methods and Apparatus for Actuating Displays” and filed Oct. 14, 2005, which is a continuation-in-part of U.S. patent application Ser. No. 11/218,690, entitled “Methods and Apparatus for Spatial Light Modulation” and filed Sep. 2, 2005, both of which claim priority to and benefit of U.S. Provisional Patent Application No. 60/676,053, entitled “MEMS Based Optical Display” and filed on Apr. 29, 2005, and U.S. Provisional Patent Application No. 60/655,827, entitled MEMS Based Optical Display Modules” and filed on Feb. 23, 2005. The entirety of each of these provisional and non-provisional applications is incorporated by reference herein. 
    
    
     FIELD OF THE INVENTION 
     In general, the invention relates to the field of video displays, in particular, the invention relates to mechanically actuated display apparatus. 
     BACKGROUND OF THE INVENTION 
     Displays built from mechanical light modulators are an attractive alternative to displays based on liquid crystal technology. Mechanical light modulators are fast enough to display video content with good viewing angles and with a wide range of color and grey scale. Mechanical light modulators have been successful in projection display applications. Backlit displays using mechanical light modulators have not yet demonstrated sufficiently attractive combinations of brightness and low power. There is a need in the art for fast, bright, low-powered mechanically actuated displays. Specifically there is a need for mechanically actuated displays that include bi-stable mechanisms and that can be driven at low voltages for reduced power consumption. 
     SUMMARY OF THE INVENTION 
     In one aspect, the invention relates to a display apparatus that includes an array of electrowetting-based light modulators for transmissively displaying an image, a reflective aperture layer formed on a first substrate and defining a plurality of apertures, and a light guide, separated from the first substrate by a gap, for guiding light towards the reflective aperture layer. The image is formed by modulating light passing through apertures in the reflective aperture layer. The reflective aperture layer is positioned adjacent the array and reflects light not passing through the apertures away from the array. A second reflective layer may be positioned on a side of the reflective aperture layer opposite the array to reflect light towards the reflective aperture layer. In some implementations, the reflective aperture layer is a metal and/or a dielectric mirror. 
     In some embodiments, the gap is filled with a fluid. In some implementations, the fluid has a first refractive index and the light guide has a second refractive index, where the first refractive index is less than the second refractive index. In some implementations, the fluid is air. 
     In some embodiments, the array of electrowetting-based light modulators includes a first electrowetting-controlled color modulation layer and a electrowetting-controlled black layer. In some implementations, the first electrowetting-controlled color modulation layer includes a plurality of electrowetting-controlled color modulation cells each having at least one edge and the electrowetting-controlled black layer includes a plurality of electrowetting-controlled light-absorbing cells corresponding to one or more of the color modulation cells, each having at least one edge. The at least one edge of each light-absorbing cell overlaps the at least one edge of its corresponding one or more color modulation cells. The first electrowetting-controlled color modulation layer may include groups of electrowetting-controlled color modulation cells, where each cell in a given group modulates a different color of light. In some implementations, first, second, and third electrowetting-controlled color modulation layers each modulate a different color of light. 
     In some embodiments, a light source emits light into the light guide. In some implementations, the light source emits white light. In some implementations, the light source sequentially emits light of at least three different colors, where the light source may include at least three light sources, each corresponding to one of the three different colors. 
     In some embodiments, the array of electrowetting-based light modulators comprises a plurality of electrowetting-based light modulation cells, where each of the cells corresponds to an aperture in the reflective aperture layer. In some implementations, each of the apertures has at least one edge, each cell includes a layer of light modulating fluid, and when in a non-actuated state, the light modulating fluid of a cell overlaps the at least one edge of its corresponding aperture. 
     In some embodiments, the electrowetting-based light modulators include an electrode layer. In some implementations, a light modulating fluid is disposed between the reflective aperture layer and the electrode layer. In some implementations, the electrode layer is disposed between the reflective aperture layer and a light modulating fluid. In some implementations, the reflective aperture layer is disposed on a first substrate while the electrode layer is disposed on a second substrate that is separate from the first substrate. 
     In one aspect, the invention relates to a display apparatus that includes an array of electrowetting-based light modulators for transmissively displaying an image, a reflective aperture layer formed on a first substrate and defining a plurality of apertures, and a layer of low-refractive index material positioned on a side of the reflective aperture layer opposite the array. The image is formed by modulating light passing through apertures in the reflective aperture layer. The reflective aperture layer is positioned adjacent the array and reflects light not passing through the apertures away from the array. 
     In some embodiments, the layer of low-refractive index material is coupled to a light guide which guides light towards the reflective aperture layer, where the layer of low-refractive index material has a refractive index less than that of the light guide. In some implementations, the layer of low-refractive index material is interposed between the reflective aperture layer and the light guide. In some implementations, the light guide includes the first substrate. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The system and methods may be better understood from the following illustrative description with reference to the following drawings in which: 
         FIG. 1  is conceptual isometric view of a display apparatus, according to an illustrative embodiment of the invention; 
         FIGS. 2A-2B  are top views of dual compliant beam electrode actuator-based shutter assemblies for use in a display apparatus, according to an illustrative embodiment of the invention; 
         FIG. 3A  is a diagram illustrating various compliant electrode shapes suitable for inclusion in dual compliant electrode actuator-based shutter assemblies; 
         FIG. 3B  is a diagram illustrating the incremental energy needed to move dual compliant electrode actuator-based shutter assemblies having the shapes illustrated in  FIG. 3A ; 
         FIGS. 3C-3F  are top views of the compliant beam electrode actuator-based shutter assembly of  FIG. 2A  in various stages of actuation; 
         FIGS. 4A and 4B  are cross section views of a dual compliant electrode actuator-based mirror-based light modulator in an active and an inactive state, according to an illustrative embodiment of the invention; 
         FIG. 5  is a top view of a dual compliant beam electrode actuator-based shutter assembly having a beam with thickness which varies along its length, according to an illustrative embodiment of the invention; 
         FIG. 6  is an isometric view of a dual compliant beam electrode actuator-based shutter assembly, according to an illustrative embodiment of the invention; 
         FIG. 7  is a top view of a dual compliant beam electrode actuator-based shutter assembly including a return spring, according to an illustrative embodiment of the invention; 
         FIG. 8  is a top view of a dual compliant beam electrode actuator-based shutter assembly having separate open and close actuators, according to an illustrative embodiment of the invention; 
         FIG. 9  is a conceptual diagram of an active matrix array for controlling dual compliant electrode actuator based-light modulators, according to an illustrative embodiment of the invention; 
         FIG. 10  is a conceptual diagram of a second active matrix array for controlling dual compliant electrode actuator based-light modulators, according to an illustrative embodiment of the invention; 
         FIG. 11  is a cross sectional view of the dual compliant beam electrode actuator-based shutter assembly of  FIG. 8 ; 
         FIG. 12  is an energy diagram illustrating the energy characteristics of various dual compliant electrode based shutter assemblies, according to an illustrative embodiment of the invention; 
         FIG. 13A  is a top view of a bi-stable dual compliant beam electrode actuator based-shutter assembly, according to an illustrative embodiment of the invention; 
         FIG. 13B  shows the evolution of force versus displacement for a bi-stable shutter assembly; 
         FIG. 14  is a top view of a second bi-stable dual compliant beam electrode actuator based-shutter assembly, according to an illustrative embodiment of the invention; 
         FIG. 15  is a top view of a tri-stable shutter assembly incorporating dual compliant electrode actuators, according to an illustrative embodiment of the invention; 
         FIGS. 16A-C  are conceptual diagrams of another embodiment of a bi-stable shutter assembly, illustrating the state of the shutter assembly during a change in shutter position, according to an illustrative embodiment of the invention; 
         FIG. 17A  is a conceptual diagram of a bi-stable shutter assembly including substantially rigid beams, according to an illustrative embodiment of the invention; 
         FIG. 17B  is a top view of a rotational bi-stable shutter assembly; 
         FIG. 18  is a conceptual diagram of a bi-stable shutter assembly incorporating thermoelectric actuators, according to an illustrative embodiment of the invention; 
         FIG. 19  is a conceptual diagram of a passive matrix array for controlling bi-stable shutter assemblies, according to an illustrative embodiment of the invention; 
         FIGS. 20A and 20B  are conceptual tiling diagrams for arranging shutter assemblies in a display apparatus; 
         FIG. 21  is cross-sectional view of a display apparatus, according to an illustrative embodiment of the invention; 
         FIGS. 22A and 22B  are top views of the shutter assembly of  FIG. 8  in open and closed states, respectively, according to an illustrative embodiment of the invention; 
         FIGS. 23A-23D  are cross sectional views of shutter assemblies having shutters, which, when in a closed position, overlap apertures formed in an adjacent reflective surface, according to an illustrative embodiment of the invention; 
         FIG. 24  is a cross sectional view of a first electrowetting-based light modulation array, according to an illustrative embodiment of the invention; 
         FIG. 25  is a cross sectional view of a second electrowetting-based light modulation array, according to an illustrative embodiment of the invention; and 
         FIG. 26  is a cross sectional view of a third electrowetting-based light modulation array, according to an illustrative embodiment of the invention. 
     
    
    
     DESCRIPTION OF CERTAIN ILLUSTRATIVE EMBODIMENTS 
       FIG. 1  is an isometric view of a display apparatus  100 , according to an illustrative embodiment of the invention. The display apparatus  100  includes a plurality of light modulators, in particular, a plurality of shutter assemblies  102   a - 102   d  (generally “shutter assemblies  102 ”) arranged in rows and columns. In general, a shutter assembly  102  has two states, open and closed (although partial openings can be employed to impart grey scale). Shutter assemblies  102   a  and  102   d  are in the open state, allowing light to pass. Shutter assemblies  102   b  and  102   c  are in the closed state, obstructing the passage of light. By selectively setting the states of the shutter assemblies  102   a - 102   d , the display apparatus  100  can be utilized to form an image  104  for a projection or backlit display, if illuminated by lamp  105 . In another implementation the apparatus  100  may form an image by reflection of ambient light originating from the front of the apparatus. In the display apparatus  100 , each shutter assembly  102  corresponds to a pixel  106  in the image  104 . 
     Each shutter assembly  102  includes a shutter  112  and an aperture  114 . To illuminate a pixel  106  in the image  104 , the shutter  112  is positioned such that it allows light to pass, without any significant obstruction, through the aperture  114  towards a viewer. To keep a pixel  106  unlit, the shutter  112  is positioned such that it obstructs the passage of light through the aperture  114 . The aperture  114  is defined by an opening patterned through a reflective or light-absorbing material in each shutter assembly  102 . 
     In alternative implementations, a display apparatus  100  includes multiple shutter assemblies  102  for each pixel  106 . For example, the display apparatus  100  may include three color-specific shutter assemblies  102 . By selectively opening one or more of the color-specific shutter assemblies  102  corresponding to a particular pixel  106 , the display apparatus  100  can generate a color pixel  106  in the image  104 . In another example, the display apparatus  100  includes two or more shutter assemblies  102  per pixel  106  to provide grayscale in an image  104 . In still other implementations, the display apparatus  100  may include other forms of light modulators, such as micromirrors, filters, polarizers, interferometric devices, and other suitable devices, instead of shutter assemblies  102  to modulate light to form an image. 
     The shutter assemblies  102  of the display apparatus  100  are formed using standard micromachining techniques known in the art, including lithography; etching techniques, such as wet chemical, dry, and photoresist removal; thermal oxidation of silicon; electroplating and electroless plating; diffusion processes, such as boron, phosphorus, arsenic, and antimony diffusion; ion implantation; film deposition, such as evaporation (filament, electron beam, flash, and shadowing and step coverage), sputtering, chemical vapor deposition (CVD), plasma enhanced CVD, epitaxy (vapor phase, liquid phase, and molecular beam), electroplating, screen printing, and lamination. See generally Jaeger, Introduction to Microelectronic Fabrication (Addison-Wesley Publishing Co., Reading Mass. 1988); Runyan, et al., Semiconductor Integrated Circuit Processing Technology (Addison-Wesley Publishing Co., Reading Mass. 1990); Proceedings of the IEEE Micro Electro Mechanical Systems Conference 1987-1998; Rai-Choudhury, ed., Handbook of Microlithography, Micromachining &amp; Microfabrication (SPIE Optical Engineering Press, Bellingham, Wash. 1997). 
     More specifically, multiple layers of material (typically alternating between metals and dielectrics) are deposited on top of a substrate forming a stack. After one or more layers of material are added to the stack, patterns are applied to a top most layer of the stack marking material either to be removed from, or to remain on, the stack. Various etching techniques, including wet or dry etches or reactive ion etching, are then applied to the patterned stack to remove unwanted material. The etch process may remove material from one or more layers of the stack based on the chemistry of the etch, the layers in the stack, and the amount of time the etch is applied. The manufacturing process may include multiple iterations of layering, patterning, and etching. 
     In one implementation the shutter assemblies  102  are fabricated upon a transparent glass or plastic substrate. This substrate may be made an integral part of a backlight which acts to evenly distribute the illumination from lamp  105  before the light exits through apertures  114 . Alternatively, and optionally, the transparent substrate may be placed on top of a planar light guide, wherein the array of shutter assemblies  102  act as light modulation elements in the formation of an image. In one implementation the shutter assemblies  102  are fabricated in conjunction with or subsequent to the fabrication of a thin film transistor (TFT) array on the same glass or plastic substrate. The TFT array provides a switching matrix for distribution of electrical signals to the shutter assemblies. 
     The process also includes a release step. To provide freedom for parts to move in the resulting device, sacrificial material is interdisposed in the stack proximate to material that will form moving parts in the completed device. An etch removes much of the sacrificial material, thereby freeing the parts to move. 
     After release, one or more of the surfaces of the shutter assembly may be insulated so that charge does not transfer between moving parts upon contact. This can be accomplished by thermal oxidation and/or by conformal chemical vapor deposition of an insulator such as Al2O3, Cr2O3, TiO2, TiSiO4, HfO2, HfSiO4, V2O5, Nb2O5, Ta2O5, SiO2, or Si3N4 or by depositing similar materials using techniques such as atomic layer deposition and others. The insulated surfaces are chemically passivated to prevent problems such as stiction between surfaces in contact by chemical conversion processes such as fluoridation, silanization, or hydrogenation of the insulated surfaces. 
     Dual compliant electrode actuators make up one suitable class of actuators for driving the shutters  112  in the shutter assemblies  102 . A dual compliant beam electrode actuator, in general, is formed from two or more at least partially compliant beams. At least two of the beams serve as electrodes (also referred to herein as “beam electrodes”). In response to applying a voltage across the beam electrodes, the beam electrodes are attracted to one another from the resultant electrostatic forces. Both beams in a dual compliant beam electrode are, at least in part, compliant. That is, at least some portion of each of the beams can flex and/or bend to aid in the beams being brought together. In some implementations the compliance is achieved by the inclusion of flexures or pin joints. Some portion of the beams may be substantially rigid or fixed in place. Preferably, at least the majority of the length of the beams are compliant. 
     Dual compliant electrode actuators have advantages over other actuators known in the art. Electrostatic comb drives are well suited for actuating over relatively long distances, but can generate only relatively weak forces. Parallel plate or parallel beam actuators can generate relatively large forces but require small gaps between the parallel plates or beams and therefore only actuate over relatively small distances. R. Legtenberg et. al. ( Journal of Microelectromechanical Systems v.  6, p. 257, 1997) demonstrated how the use of curved electrode actuators can generate relatively large forces and result in relatively large displacements. The voltages required to initiate actuation in Legtenberg, however, are still substantial. As shown herein such voltages can be reduced by allowing for the movement or flexure of both electrodes. 
     In a dual compliant beam electrode actuator-based shutter assembly, a shutter is coupled to at least one beam of a dual compliant beam electrode actuator. As one of the beams in the actuator is pulled towards the other, the pulled beam moves the shutter, too. In doing so, the shutter is moved from a first position to a second position. In one of the positions, the shutter interacts with light in an optical path by, for example, and without limitation, blocking, reflecting, absorbing, filtering, polarizing, diffracting, or otherwise altering a property or path of the light. The shutter may be coated with a reflective or light absorbing film to improve its interferential properties. In the second position, the shutter allows the light to pass by, relatively unobstructed. 
       FIGS. 2A and 2B  are diagrams of two embodiments of cantilever dual compliant beam electrode actuator based-shutter assemblies for use in a display apparatus, such as display apparatus  100 . More particularly,  FIG. 2A  depicts a cantilever dual compliant beam electrode actuator-based shutter assembly  200   a . The shutter assembly  200   a  modulates light to form an image by controllably moving a shutter  202   a  in and out of an optical path of light. In one embodiment, the optical path begins behind a surface  204   a , to which the shutter  202   a  is attached. The surface  204   a  is illustrated as a boundary line. However, the surface  204   a  extends beyond the space delimited by the boundary line. Similar boundary lines are used in other figures and may also indicate surfaces which extend beyond the space delimited by the boundary line. The light passes through an aperture  206   a  in the surface  204   a  towards a viewer or towards a display screen. In another embodiment, the optical path begins in front of the surface  204   a  and is reflected back to the viewer from the surface of the aperture  206   a.    
     The shutter  202   a  of the shutter assembly  200   a  is formed from a solid, substantially planar, body. The shutter  202   a  can take virtually any shape, either regular or irregular, such that in a closed position the shutter  202   a  sufficiently obstructs the optical path through the aperture  206   a  in the surface  204   a . In addition, the shutter  202   a  must have a width consistent with the width of the aperture, that, in the open position (as depicted), sufficient light can pass through the aperture  206   a  in the surface  204   a  to illuminate a pixel, or contribute to the illumination of a pixel, in the display apparatus. 
     The shutter  202   a  couples to one end of a load beam  208   a . A load anchor  210   a , at the opposite end of the load beam  208   a  physically connects the load beam  208   a  to the surface  204   a  and electrically connects the load beam  208   a  to driver circuitry in the surface  204   a . Together, the load  208   a  beam and load anchor  210   a  serve as a mechanical support for supporting the shutter  202   a  over the surface  204   a.    
     The shutter assembly  200   a  includes a pair of drive beams  212   a  and  214   a , one located along either side of the load beam  210   a . Together, the drive beams  212   a  and  214   a  and the load beam  210   a  form an actuator. One drive beam  212   a  serves as a shutter open electrode and the other drive beam  214   a  serves as a shutter close electrode. Drive anchors  216   a  and  218   a  located at the ends of the drive beams  212   a  and  214   a  closest to the shutter  202   a  physically and electrically connects each drive beam  212   a  and  214   a  to the surface  204   a . In this embodiment, the other ends and most of the lengths of the drive beams  212   a  and  214   a  remain unanchored or free. The free ends of the drive beams  212   a  and  214   a  are closer to the anchored end of the load beam  208   a  than the anchored ends of the drive beams  212   a  and  214   a  are to the shutter end of the load beam  208   a.    
     The load beam  208   a  and the drive beams  212   a  and  214   a  are compliant. That is, they have sufficient flexibility and resiliency that they can be bent out of their unstressed (“rest”) position or shape to at least some useful degree, without fatigue or fracture. As the load beam  208   a  and the drive beams  212   a  and  214   a  are anchored only at one end, the majority of the lengths of the beams  208   a ,  212   a , and  214   a  is free to move, bend, flex, or deform in response to an applied force. The operation of the cantilever dual compliant beam electrode actuator based-shutter assembly  200   a  is discussed further below in relation to  FIG. 3 . 
       FIG. 2B  is a second illustrative embodiment of a cantilever dual compliant beam electrode actuator-based shutter assembly  200   b . Like the shutter assembly  200   a , the shutter assembly  200   b  includes a shutter  202   b , coupled to a load beam  208   b , and two drive beams  212   b  and  214   b . The shutter  202   b  is positioned in between its fully open position and its fully closed position. The load beam  208   b  and the drive beams  212   b  and  214   b , together, form an actuator. Drive anchors  210   b ,  216   b  and  218   b , coupled to each end of the beams connect the beams to a surface  204   b . In contrast to the shutter assembly  200   a , the shutter of shutter assembly  200   b  includes several shutter apertures  220 , in the form of slots. The surface  204   b , instead of only having one aperture, includes one surface aperture  206   b  corresponding to each shutter aperture  220 . In the open position, the shutter apertures  220  substantially align with the apertures  206   b  in the surface  204   b , allowing light to pass through the shutter  202   b . In the closed position, the surface apertures  206   b  are obstructed by the remainder of the shutter  202   b , thereby preventing the passage of light. 
     Changing the state of a shutter assembly that includes multiple shutter apertures with a corresponding number of surface apertures requires less shutter movement than changing the state of a shutter assembly incorporating a solid shutter and single surface aperture, while still providing for the same aperture area. Reduced required motion corresponds to lower required actuation voltage. More particularly, a decrease in required motion by ⅓ reduces the necessary actuation voltage of the actuator by a factor of about ⅓. Reduced actuation voltage further corresponds to reduced power consumption. Since the total aperture area for either shutter assembly is about the same, each shutter assembly provides a substantially similar brightness. 
     In other implementations, the shutter apertures and corresponding surface apertures have shapes other than slots. The apertures may be circular, polygonal or irregular. In alternative implementations, the shutter may include more shutter apertures than there are surface apertures in the shutter assembly. In such implementations, one or more of the shutter apertures may serve as a filter, such as color filter. For example, the shutter assembly may have three shutter apertures for every surface aperture, each shutter aperture including a red, blue, or green colored filter. 
       FIGS. 3A and 3B  are diagrams illustrating the relationship between the displacement at the end of the load beam and the relative voltage needed to move the load beam closer to the drive beam. The displacement that can be achieved at any given voltage depends, at least in part, on the curvature or shape of the drive beam, or more precisely, on how the separation, d, and the bending stress along the drive beam and the load beam varies as a function of position x along the load beam A separation function d(x), shown in  FIG. 3A  can be generalized to the form of d=ax n , where y is the distance between the beams. For example, if n=1, the distance between drive electrode and load electrode increase linearly along the length of the load electrode. If n=2, the distance increases parabolically. In general, assuming a constant voltage, as the distance between the compliant electrodes decreases, the electrostatic force at any point on the beams increases proportional to 1/d. At the same time, however, any deformation of the load beam which might decrease the separation distance may also result in a higher stress state in the beam. Below a minimum threshold voltage a limit of deformation will be reached at which any electrical energy released by a closer approach of the electrodes is exactly balanced by the energy which becomes stored in the deformation energy of the beams. 
     As indicated in the diagram  3 B, for actuators having separation functions in which n is less than or equal to 2, the application of a minimum actuation voltage (V 2 ) results in a cascading attraction of the load beam to the drive beam without requiring the application of a higher voltage. For such actuators, the incremental increase in electrostatic force on the beams resulting from the load beam getting closer to the drive beam is greater than the incremental increase in stress on the beams needed for further displacement of the beams. 
     For actuators having separation functions in which x is greater than 2, the application of a particular voltage results in a distinct partial displacement of the load electrode. That is, the incremental increase in electrostatic force on the beams resulting from a particular decrease in separation between the beams, at some point, fails to exceed the incremental deformation force needed to be imparted on the load beam to continue reducing the separation. Thus, for actuators having separation functions having n greater than 2, the application of a first voltage level results in a first corresponding displacement of the load electrode. A higher voltage results in a greater corresponding displacement of the load electrode. How the shapes and relative compliance of thin beam electrodes effects actuation voltage is discussed in more detail in the following references: (R. Legtenberg et. al.,  Journal of Microelectromechanical Systems v.  6, p. 257 (1997) and J. Li et. al.  Transducers &#39; 03 , The  12 11    International Conference on Solid State Sensors, Actuators and Microsystems , p. 480 (2003), each of which is incorporated herein by reference. 
     Referring back to  FIGS. 2A and 2B , a display apparatus incorporating the shutter assemblies  202   a  and  202   b  actuates, i.e., changes the position of the shutter assemblies  202   a  and  202   b , by applying an electric potential, from a controllable voltage source, to one of the drive beams  212   a ,  212   b ,  214   a , or  214   b  via its corresponding drive anchor  216   a ,  216   b ,  218   a , or  218   b , with the load beam  208   a  or  208   b  being electrically coupled to ground, resulting in a voltage across the beams  208   a ,  208   b ,  212   a ,  212   b ,  214   a ,  214   b . The controllable voltage source, such as an active matrix array driver, is electrically coupled to load beam  208   a  or  208   b  via an active matrix array (see  FIGS. 9 and 10  below). The display apparatus may instead apply an electric potential to the load beam  208   a  or  208   b  via the load anchor  210   a  or  210   b  of the shutter assembly  202   a  or  202   b  to increase the voltage. An electrical potential difference between the drive beams and the load beams, regardless of sign or ground potential, will generate an electrostatic force between the beams. 
     With reference back to  FIG. 3 , the shutter assembly  200   a  of  FIG. 2A  has a second order separation function (i.e., n=2). Thus, if the voltage or potential difference between the beams  208   a  and  212   a  or  214   a  of the shutter assembly  202   a  at their point of least separation exceeds the minimum actuation voltage (V 2 ) the deformation of the beams  208   a  and  212   a  or  214   a  cascades down the entire lengths of the beams  208   a  and  212   a  or  214   a , pulling the shutter end of the load beam  208   a  towards the anchored end of the drive beam  212   a  or  214   a . The motion of the load beam  208   a  displaces the shutter  202   a  such that it changes its position from either open to closed, or visa versa, depending on to which drive beam  212   a  or  214   a  the display apparatus applied the potential. To reverse the position change, the display apparatus ceases application of the potential to the energized drive beam  212   a  or  214   a . Upon the display apparatus ceasing to apply the potential, energy stored in the form of stress on the deformed load beam  208   a  restores the load beam  208   a  to its original or rest position. To increase the speed of the restoration and to reduce any oscillation about the rest position of the load beam  208   a , the display apparatus may return the shutter  202   a  to its prior position by applying an electric potential to the opposing drive beam  212   a  or  214   a.    
     The shutter assemblies  200   a  and  200   b , as well as shutter assemblies  500  (see  FIG. 5  below),  600  (see  FIG. 6  below),  700  (see  FIG. 7  below) and  800  (see  FIG. 8  below) have the property of being electrically bi-stable. Generally, this is understood to encompass, although not be limited to, devices wherein the electrical potential V 2  that initiates movement between open and closed states is generally greater than the electrical potential (V 1 ) required to keep the shutter assembly in a stable state. Once the load beam  208   a  and one of the drive beams are in contact, a substantially greater electrical force is to be applied from the opposing drive beam to move or separate the load beam, such electrical force being greater than would be necessary if the load beam  208   a  were to begin in a neutral or non-contact position. The bistable devices described herein may employ a passive matrix driving scheme for the operation of an array of shutter assemblies such as  200   a . In a passive matrix driving sequence it is possible to preserve an image by maintaining a stabilization voltage V 1  across all shutter assemblies (except those that are being actively driven to a state change). With no or substantially no electrical power required, maintenance of a potential V 1  between the load beam  208   a  and drive beam  212   a  or  214   a  is sufficient to maintain the shutter assembly in either its open or closed states. In order to effect a switching event the voltage between load beam  208   a  and the previously affected drive beam (for instance  212   a ) is allowed to return from V 1  to zero while the voltage between the load beam  208   a  and the opposing beam (for instance  212   b ) is brought up to the switching voltage V 2 . 
     In  FIG. 2B , the actuator has a third order separation function (i.e., n=3). Thus applying a particular potential to one of the drive beams  212   b  or  214   b  results in an incremental displacement of the shutter  202   b . The display apparatus takes advantage of the ability to incrementally displace the shutter  202   b  to generate a grayscale image. For example, the application of a first potential to a drive beam  212  or  214   b  displaces the shutter  202   b  to its illustrated position, partially obstructing light passing through the surface apertures  206   b , but still allowing some light to pass through the shutter  202   b . The application of other potentials results in other shutter  202   b  positions, including fully open, fully closed, and other intermediate positions between fully open and fully closed. In such fashion electrically analog drive circuitry may be employed in order to achieve an analog grayscale image. 
       FIGS. 3C through 3F  demonstrate the stages of motion of the load beam  208   a , the shutter close electrode  214   a , and the shutter  202   a  of the shutter assembly  200   a  of  FIG. 2A . The initial separation between the compliant beams  208   a  and  214   a  fits a second order separation function.  FIG. 3C  shows the load beam  208   a  in a neutral position with no voltage applied. The aperture  206   a  is half-covered by the shutter  202   a.    
       FIG. 3D  demonstrates the initial steps of actuation. A small voltage is applied between the load beam  208   a  and the shutter close electrode  214   a . The free end of the shutter close electrode  214   a  has moved to make contact with the load beam  208   a.    
       FIG. 3E  shows the shutter assembly  200   a  at a point of actuation after the shutter  202   a  begins to move towards the shutter close electrode  214   a.    
       FIG. 3F  shows the end state of actuation of the shutter assembly  200   a . The voltage has exceeded the threshold for actuation. The shutter assembly  200   a  is in the closed position. Contact is made between the load beam  208   a  and the shutter close electrode  214   a  all along its length. 
       FIG. 4A  is a first cross sectional diagram of dual compliant electrode mirror-based light modulator  400  for inclusion in a display apparatus, such as display apparatus  100 , instead of, or in addition to, the shutter assemblies  102 . The mirror-based-based light modulator  400  includes a mechanically compliant reflection platform  402 . At least a portion of the reflection platform  402  is itself reflective or is coated with or is connected to a reflective material. 
     The reflection platform  402  may or may not be conductive. In implementations in which the reflection platform  402  is conductive, the reflection platform serves as a load electrode for the mirror-based light modulator  400 . The reflection platform  402  is physically supported over, and is electrically coupled to, a substrate  404  via a compliant support member  406 . If the reflection platform  402  is formed from a non-conductive material, the reflection platform  402  is coupled to a compliant conductive load beam or other form of compliant load electrode. A compliant support member  406  physically supports the combined reflection platform  402  and electrode over the substrate  404 . The support member  406  also provides an electrical connection from the electrode to the substrate  404 . 
     The mirror-based light modulator  400  includes a second compliant electrode  408 , which serves a drive electrode  408 . The drive electrode  408  is supported between the substrate  404  and the reflection platform  402  by a substantially rigid second support member  410 . The second support member  410  also electrically connects the second compliant electrode  408  to a voltage source for driving the mirror-based light modulator  400 . 
     The mirror-based light modulator  400  depicted in  FIG. 4A  is in rest position in which neither of the electrodes  402  or  408  carry a potential.  FIG. 4B  depicts the mirror-based light modulator  400  in an activated state. When a potential difference is generated between the drive electrode  408  and the load electrode  402  (be it the reflective platform  402  or an attached load beam), the load electrode  402  is drawn towards the drive electrode  408 , thereby bending the compliant support beam  406  and angling the reflective portion of the reflection platform  402  to be at least partially transverse to the substrate  404 . 
     To form an image, light  412  is directed at an array of mirror-based light modulators  400  at a particular angle. Mirror-based light modulators  400  in their rest states reflect the light  412  away from the viewer or the display screen, and mirror-based light modulators in the active state reflect the light  412  towards a viewer or a display screen, or visa versa. 
       FIG. 5  is a diagram of another cantilever dual compliant beam electrode actuator-based shutter assembly  500 . As with the shutter assemblies  200   a  and  200   b , the shutter assembly  500  includes a shutter  502  coupled to a compliant load beam  504 . The compliant load beam  504  is then physically anchored to a surface  506 , and electrically coupled to ground, at its opposite end via a load anchor  508 . The shutter assembly  500  includes only one compliant drive beam  510 , located substantially alongside the load beam  504 . The drive beam  510 , in response to being energized with an electric potential from a controllable voltage source draws the shutter  502  from a first position (in which the load beam  504  is substantially unstressed) in a plane substantially parallel to the surface, to a second position in which the load beam  504  is stressed. When the potential is removed, the stored stress in the load beam  504  restores the load beam  504  to its original position. 
     In addition, in comparison to the shutter assemblies  202   a  and  202   b , the load beam  504  has a width which varies along its length. The load beam  504  is wider near its anchor  508  than it is nearer to the shutter  502 . In comparison to the shutter assemblies  202   a  and  202   b  and because of its tailored width, the load beam  504  typically has an overall greater stiffness. Shutter assemblies incorporating stiffer beams typically require higher voltages for actuation, but in return, allow for higher switching rates. For example, the shutter assemblies  202   a  and  202   b  may be switched up to about 10 kHz, while the stiffer shutter assembly  500  may be switched up to about 100 kHz. 
       FIG. 6  is a diagram of a shutter assembly  600  incorporating two dual compliant electrode beam actuators  602  (“actuators  602 ”), according to an illustrative embodiment of the invention. The shutter assembly  600  includes a shutter  604 . The shutter  604  may be solid, or it may include one or more shutter apertures as described in relation to  FIG. 2B . The shutter  604  couples on one side to the beam actuators  602 . Together, the actuators  602  move the shutter transversely over a surface in plane of motion which is substantially parallel to the surface. 
     Each actuator  602  includes a compliant load member  606  connecting the shutter  604  to a load anchor  608 . The compliant load members  606  each include a load beam  610  and an L bracket  612 . The load anchors  608  along with the compliant load members  606  serve as mechanical supports, keeping the shutter  604  suspended proximate to the surface. The load anchors  608  physically connect the compliant load members  606  and the shutter  604  to the surface and electrically connect the load beams  610  of the load members  606  to ground. The coupling of the shutter  604  from two positions on one side of the shutter  604  to load anchors  608  in positions on either side of the shutter assembly  600  help reduce twisting motion of the shutter  604  about its central axis  614  during motion. 
     The L brackets  612  reduce the in-plane stiffness of the load beam  610 . That is, the L brackets  612  reduce the resistance of actuators  602  to movement in a plane parallel to the surface (referred to as “in-plane movement”  615 ), by relieving axial stresses in the load beam. 
     Each actuator  602  also includes a compliant drive beam  616  positioned adjacent to each load beam  610 . The drive beams  616  couple at one end to a drive beam anchor  618  shared between the drive beams  616 . The other end of each drive beam  616  is free to move. Each drive beam  616  is curved such that it is closest to the load beam  610  near the free end of the drive beam  616  and the anchored end of the load beam  610 . 
     In operation, a display apparatus incorporating the shutter assembly  600  applies an electric potential to the drive beams  616  via the drive beam anchor  618 . As a result of a potential difference between the drive beams  616  and the load beam  610 , the free ends of the drive beams  616  are pulled towards the anchored ends of the load beams  610  and the shutter ends of the load beams  610  are pulled toward the anchored ends of the drive beams  616 . The electrostatic force draws the shutter  604  towards the drive anchor  618 . The compliant members  606  act as springs, such that when the electrical potentials are removed from the drive beams  616 , the load beams compliant members  606  push the shutter  604  back into its initial position, releasing the stress stored in the load beams  610 . The L brackets  612  also serve as springs, applying further restoration force to the shutter  604 . 
     In fabrication of shutter assemblies  200  through  800 , as well as for shutter assemblies  1300  through  1800 , it is preferable to provide a rectangular shape for the cross section of the load beams (such as load beams  610 ) and the drive beams (such as drive beams  616 ). By providing a beam thickness (in the direction perpendicular to surface) which is 1.4 times or more larger in dimension than the beam width (in a direction parallel to the surface) the stiffness of the load beam  610  will be increased for out-of-plane motion  617  versus in-plane motion  615 . Such a dimensional and, by consequence, stiffness differential helps to ensure that the motion of the shutter  604 , initiated by the actuators  602 , is restricted to motion along the surface and across the surface apertures as opposed to out-of-plane motion  617  which would a wasteful application of energy. It is preferable for certain applications that the cross section of the load beams (such as  610 ) be rectangular as opposed to curved or elliptical in shape. The strongest actuation force is achieved if the opposing beam electrodes have flat faces so that upon actuation they can approach and touch each other with the smallest possible separation distance. 
       FIG. 7  is a diagram of a second shutter assembly  700  incorporating two dual compliant electrode beam actuators  702 , according to an illustrative embodiment of the invention. The shutter assembly  700  takes the same general form of the shutter assembly  600 , other than it includes a return spring  704 . As with the shutter assembly  600 , in the shutter assembly  700 , two actuators  702  couple to a first side of a shutter  706  to translate the shutter  706  in a plane parallel to a surface over which the shutter is physically supported. The return spring  704  couples to the opposite side of the shutter  706 . The return spring  704  also couples to the surface at a spring anchor  708 , acting as an additional mechanical support. By physically supporting the shutter  706  over the surface at opposite sides of the shutter  706 , the actuators  702  and the return spring  704  reduce motion of the shutter  706  out of the plane of intended motion during operation. In addition, the return spring  704  incorporates several bends which reduce the in-plane stiffness of the return spring  704 , thereby further promoting in-plane motion over out-of-plane motion. The return spring  704  provides an additional restoration force to the shutter  706 , such that once an actuation potential is removed, the shutter  706  returns to its initial position quicker. The addition of the return spring  704  increases only slightly the potential needed to initiate actuation of the actuators  702 . 
       FIG. 8  is a diagram of a shutter assembly  800  including a pair of shutter open actuators  802  and  804  and a pair of shutter close actuators  806  and  808 , according to an illustrative embodiment of the invention. Each of the four actuators  802 ,  804 ,  806 , and  808  take the form of a dual compliant beam electrode actuator. Each actuator  802 ,  804 ,  806 , and  808  includes a compliant load member  810  coupling a shutter  812 , at one end, to a load anchor  814 , at the other end. Each compliant load member  810  includes a load beam  816  and an L bracket  818 . Each actuator  802 ,  804 ,  806 , and  808  also includes a drive beam  820  with one end coupled to a drive anchor  822 . Each pair of actuators  802 / 804  and  806 / 808  share a common drive anchor  822 . The unanchored end of each drive beam  820  is positioned proximate to the anchored end of a corresponding compliant load member  810 . The anchored end of each drive beam  820  is located proximate to the L bracket end of a corresponding load beam  816 . In a deactivated state, the distance between a load beam  816  and its corresponding drive beam  820  increases progressively from the anchored end of the load beam  816  to the L bracket  818 . 
     In operation, to open the shutter  812 , a display apparatus incorporating the shutter assembly  800  applies an electric potential to the drive anchor  822  of the shutter open actuators  802  and  804 , drawing the shutter  812  towards the open position. To close the shutter  812 , the display apparatus applies an electric potential to the drive anchor  822  of the shutter close actuators  806  and  808  drawing the shutter  812  towards the closed position. If neither pair of actuators  802 / 804  or  806 / 808  are activated, the shutter  812  remains in an intermediate position, somewhere between fully open and fully closed. 
     The shutter open actuators  802 / 804  and shutter closed actuators  806 / 808  couple to the shutter  812  at opposite ends of the shutter. The shutter open and closed actuators have their own load members  810 , thus reducing the actuation voltage of each actuator  802 ,  804 ,  806  and  808 . Because of the electrical bi-stability described in reference to  FIG. 3 , it is advantageous to find an actuation method or structure with more leverage for separating the compliant load member  810  from a drive beam  820  with which it might be in contact. By positioning the open and closed actuators  802 / 804  and  806 / 808  on opposite sides of the shutter  812 , the actuation force of the actuator-to-be-actuated is transferred to the actuator-to-be-separated through the shutter. The actuation force is therefore applied to the task of separation at a point close to the shutter (for instance near the L-bracket end of the load beam  816 ) where its leverage will be higher. 
     For shutter assemblies such as in  FIG. 8  typical shutter widths (along the direction of the slots) will be in the range of 20 to 800 microns. The “throw distance” or distance over which the shutter will move between open and closed positions will be in the range of 4 to 100 microns. The width of the drive beams and load beams will be in the range of 0.2 to 40 microns. The length of the drive beams and load beams will be in the range of 10 to 600 microns. Such shutter assemblies may be employed for displays with resolutions in the range of 30 to 1000 dots per inch. 
     Each of the shutter assemblies  200   a ,  200   b ,  500 ,  600 ,  700  and  800 , and the mirror-based light modulator  400 , described above fall into a class of light modulators referred to herein as “elastic light modulators.” Elastic light modulators have one mechanically stable rest state. In the rest state, the light modulator may be on (open or reflecting), off (closed or not reflecting), or somewhere in between (partially open or partially reflecting). If the generation of a voltage across beams in an actuator forces the light modulator out of its rest state into a mechanically unstable state, some level of voltage across the beams must be maintained for the light modulator to remain in that unstable state. 
       FIG. 9  is a diagram of an active matrix array  900  for controlling elastic light modulators  902  in a display apparatus. In particular, the active matrix array  900  is suitable for controlling elastic light modulators  902 , such as the mirror-based light modulator  400  or shutter-based light modulators  500 ,  600 , and  700 , that include only a passive restoration force. That is, these light modulators  902  require electrical activation of actuators to enter a mechanically unstable state, but then utilize mechanical mechanisms, such as springs, to return to the rest state. 
     The active matrix array is fabricated as a diffused or thin-film-deposited electrical circuit on the surface of a substrate on which the elastic light modulators  902  are formed. The active matrix array  900  includes a series of row electrodes  904  and column electrodes  906  forming a grid like pattern on the substrate, dividing the substrate into a plurality of grid segments  908 . The active matrix array  900  includes a set of drivers  910  and an array of non-linear electrical components, comprised of either diodes or transistors that selectively apply potentials to grid segments  908  to control one or more elastic light modulators  902  contained within the grid segments  908 . The art of thin film transistor arrays is described in  Active Matrix Liquid Crystal Displays: Fundamentals and Applications  by Willem den Boer (Elsevier, Amsterdam, 2005). 
     Each grid segment  908  contributes to the illumination of a pixel, and includes one or more elastic light modulators  902 . In grid segments  908 , including only a single elastic light modulator  902 , the grid segment  908  includes, in addition to the elastic light modulator  902 , at least one diode or transistor  912  and optionally a capacitor  914 . The capacitor  914  shown in  FIG. 9  can be explicitly added as a design element of the circuit, or it can be understood that the capacitor  914  represents the equivalent parallel or parasitic capacitance of the elastic light modulator. The emitter  916  of the transistor  912  is electrically coupled, to either the drive electrode or the load electrode of the elastic light modulator  902 . The other electrode of the actuator is coupled to a ground or common potential. The base  918  of the transistor  912  electrically couples to a row electrode  904  controlling a row of grid segments. When the base  918  of the transistor receives a potential via the row electrode  904 , current can run through the transistor  912  from a corresponding column electrode  906  to generate a potential in the capacitor  914  and to apply a potential to the drive electrode of the elastic light modulator  902  activating the actuator. 
     The active matrix array  900  generates an image, in one implementation by, one at a time, applying a potential from one of the drivers  910  to a selected row electrode  904 , activating a corresponding row of grid segments  908 . While a particular row is activated, the display apparatus applies a potential to the column electrodes corresponding to grid segments in the active row containing light modulators which need to be switched out of a rest state. 
     When a row is subsequently deactivated, a stored charge will remain on the electrodes of the actuator  902  (as determined by the equivalent capacitance of the actuator) as well as, optionally, on the parallel capacitor  914  that can be designed into the circuit, keeping the elastic shutter mechanisms  902  in their mechanically unstable states. The elastic shutter mechanism  902  remains in the mechanically unstable state until the voltage stored in the capacitor  914  dissipates or until the voltage is intentionally reset to ground potential during a subsequent row selection or activation step. 
       FIG. 10  is diagram of another implementation of an active matrix array  1000  for controlling elastic light modulators  1002  in a display apparatus. In particular, the active matrix array  1000  is suitable for controlling elastic light modulators, such as shutter-based light modulators  200   a ,  200   b , and  800 , which include one set of actuators for forcing the light modulators from a rest state to a mechanically unstable state and a second set of actuators for driving the light modulators back to the rest state and possibly to a second mechanically unstable state. Active matrix array  1000  can also be used for driving non-elastic light modulators described further in relation to  FIGS. 12-20 . 
     The active matrix array  1000  includes one row electrode  1004  for each row in the active matrix array  1000  and two column electrodes  1006   a  and  1006   b  for each column in the active matrix array  1000 . For example, for display apparatus including shutter-based light modulators, one column electrode  1006   a  for each column corresponds to the shutter open actuators of light modulators  1002  in the column. The other column electrode  1006   b  corresponds to the shutter close actuators of the light modulators  1002  in the column. The active matrix array  1000  divides the substrate upon which it is deposited into grid sections  1008 . Each grid section  1008  includes one or more light modulators  1002  and at least two diodes or transistors  1010   a  and  1010   b  and optionally two capacitors  1012   a  and  1012   b . The bases  1014   a  and  1014   b  of each transistor  1101   a  and  1010   b  are electrically coupled to a column electrode  1006   a  or  1006   b . The emitters  1016   a  and  1016   b  of the transistors  1010   a  and  1010   b  are coupled to a corresponding capacitor  1012   a  or  1012   b  and a drive electrode of the light modulator(s)  1002  in the grid section  1008 . 
     In operation, a driver applies a potential to a selected row electrode  1004 , activating the row. The active matrix array  1000  selectively applies potentials to one of the two column electrodes  1006   a  or  1006   b  of each column in which the state of the light modulator(s)  1002  in the grid section  1008  needs to be changed. Alternatively, the active matrix array  1000  may also apply a potential to column electrodes  1006   a  or  1006   b  for grid sections  1008  previously in an active state which are to remain in an active state. 
     For both active matrix arrays  900  and  1000 , the drivers powering the column electrodes, in some implementations, select from multiple possible potentials to apply to individual column electrodes  1006   a  and  1006   b . The light modulator(s)  1002  in those columns can then be opened or closed different amounts to create grayscale images. 
       FIG. 11  is a cross sectional view of the shutter-assembly  800  of  FIG. 8  along the line labeled A-A′. Referring to  FIGS. 8 ,  10 , and  11 , the shutter assembly  800  is built on substrate  1102  which is shared with other shutter assemblies of a display apparatus, such as display apparatus  100 , incorporating the shutter assembly  800 . The voltage signals to actuate the shutter assembly, are transmitted along conductors in underlying layers of the shutter assembly. is The voltage signals are controlled by an active matrix array, such as active matrix array  1000 . The substrate  1102  may support as many as 4,000,000 shutter assemblies, arranged in up to about 2000 rows and up to about 2000 columns. 
     In addition to the shutter  812 , the shutter open actuators  802  and  804 , the shutter close actuators  806  and  808 , the load anchors  814  and the drive anchors  822 , the shutter assembly  800  includes a row electrode  1104 , a shutter open electrode  1106 , a shutter close electrode  1108 , and three surface apertures  1110 . The depicted shutter assembly has at least three functional layers, which may be referred to as the row conductor layer, the column conductor layer, and the shutter layer. The shutter assembly is preferably made on a transparent substrate such as glass or plastic. Alternatively the substrate can be made from an opaque material such as silicon, as long as through holes are provided at the positions of each of the surface apertures  1110  for the transmission of light. The first metal layer on top of the substrate is the row conductor layer which is patterned into row conductor electrodes  1104  as well as reflective surface sections  1105 . The reflective surface sections  1105  reflect light passing through the substrate  1102  back through the substrate  1102  except at the surface apertures  1110 . In some implementations the surface apertures may include or be covered by red, green, or blue color filtering materials. 
     The shutter open electrode  1106  and the shutter close electrode  1108  are formed in a column conductor layer  1112  deposited on the substrate  1102 , on top of the row conductor layer  1104 . The column conductor layer  1112  is separated from the row conductor layer  1104  by one or more intervening layers of dielectric material or metal. The shutter open electrode  1104  and the shutter close electrode  1106  of the shutter assembly  800  are shared with other shutter assemblies in the same column of the display apparatus. The column conductor layer  1112  also serves to reflect light passing through gaps in the ground electrode  1104  other than through the surface apertures  1110 . The row conductor layer  1104  and the column conductor layer  1112  are between about 0.1 and about 2 microns thick. In alternative implementations, the column conductor  1112  layer can be located below the row conductor layer  1104 . In another alternative implementation both the column conductor layer and the row conductor layer may be located above the shutter layer. 
     The shutter  812 , the shutter open actuators  802  and  804 , the shutter close actuators  806  and  808 , the load anchors  814  and the drive anchors  822  are formed from a third functional layer of the shutter assembly  800 , referred to as the shutter layer  1114 . The actuators  802 ,  804 ,  806 , and  808  are formed from a deposited metal, such as, without limitation, Au, Cr or Ni, or a deposited semiconductor, such as, without limitation, polycrystalline silicon, or amorphous silicon, or from single crystal silicon if formed on top of a buried oxide (also known as silicon on insulator). The beams of the actuators  802 ,  804 ,  806 , and  808  are patterned to dimensions of about 0.2 to about 20 microns in width. The shutter thickness is typically in the range of 0.5 microns to 10 microns. To promote the in-plane movement of the shutters (i.e. reduce the transverse beam stiffness as opposed to the out-of-plane stiffness), it is preferable to maintain a beam dimensional ratio of about at least 1.4:1, with the beams being thicker than they are wide. 
     Metal or semiconductor vias electrically connect the row electrode  1104  and the shutter open electrode  1106  and the shutter close electrode  1108  of the column conductor layer  1112  to features on the shutter layer  1114 . Specifically, vias  116  electrically couple the row electrode  1104  to the load anchors  814  of the shutter assembly  800 , keeping the compliant load member  810  of the shutter open actuators  802  and  804  and the shutter close actuators  806  and  808 , as well as the shutter  812 , at the row conductor potential. Additional vias electrically couple the shutter open electrode  1106  to the drive beams  820  of the shutter open actuators  802  and  804  via the drive anchor  822  shared by the shutter open actuators  802  and  804 . Still other vias electrically couple the shutter close electrode  1108  to the drive beams  820  of the of the shutter close actuators  806  and  808  via the drive anchor  822  shared by the shutter close actuators  806  and  808 . 
     The shutter layer  1114  is separated from the column conductor layer  1112  by a lubricant, vacuum or air, providing the shutter  812  freedom of movement. The moving pieces in the shutter layer  1114  are mechanically separated from neighboring components (except their anchor points  814 ) in a release step, which can be a chemical etch or ashing process, which removes a sacrificial material from between all moving parts. 
     The diodes, transistors, and/or capacitors (not shown for purpose of clarity) employed in the active matrix array may be patterned into the existing structure of the three functional layers, or they can be built into separate layers that are disposed either between the shutter assembly and the substrate or on top of the shutter layer. The reflective surface sections  1105  may be patterned as extensions of the row and column conductor electrodes or they can be patterned as free-standing or electrically floating sections of reflective material. Alternatively the reflective surface sections  1105  along with their associated surface apertures  1110  can be patterned into a fourth functional layer, disposed between the shutter assembly and the substrate, and formed from either a deposited metal layer or a dielectric mirror. Grounding conductors may be added separately from the row conductor electrodes in layer  1104 . These separate grounding conductors may be required when the rows are activated through transistors, such as is the case with an active matrix array. The grounding conductors can be either laid out in parallel with the row electrodes (and bussed together in the drive circuits), or the grounding electrodes can be placed into separate layers between the shutter assembly and the substrate. 
     In addition to elastic light modulators, display apparatus can include bi-stable light modulators, for example bi-stable shutter assemblies. As described above, a shutter in an elastic shutter assembly has one mechanically stable position (the “rest position”), with all other shutter positions being mechanically unstable. The shutter of a bi-stable shutter assembly, on the other hand, has two mechanically stable positions, for example, open and closed. Mechanically bi-stable shutter assemblies have the advantage that no voltage is required to maintain the shutters in either the open or the closed positions. Bi-stable shutter assemblies can be further subdivided into two classes: shutter assemblies in which each stable position is substantially energetically equal, and shutter assemblies in which one stable position is energetically preferential to the other mechanically stable position. 
       FIG. 12  is a diagram  1200  of potential energy stored in three types of shutter assemblies in relation to shutter position. The solid line  1202  corresponds to an elastic shutter assembly. The first dashed line  1204  corresponds to a bi-stable shutter assembly with equal energy stable states. The second dashed line  1206  corresponds to a bi-stable shutter assembly with non-equal energy stable states. As indicated in the energy diagram  1200 , the energy curves  1204  and  1206  for the two types of bi-stable shutter assemblies each include two local minima  1208 , corresponding to stable shutter positions, such as fully open  1210  and fully closed  1211 . As illustrated, energy must be added to the assembly in order to move its shutters out of the positions corresponding to one of the local minima. For the bi-stable shutter assemblies with non-equal-energy mechanically stable shutter positions, however, the work needed to open a shutter  1212  is greater than the work required to close the shutter  1214 . For the elastic shutter assembly, on the other hand, opening the shutter requires work  1218 , but the shutter closes spontaneously after removal of the control voltage. 
       FIG. 13A  is a top view of a shutter layer  1300  of a bi-stable shutter assembly. The shutter layer  1300  includes a shutter  1302  driven by two dual compliant electrode actuators  1304  and  1306 . The shutter  1302  includes three slotted shutter apertures  1308 . One dual compliant electrode actuator  1304  serves as a shutter open actuator. The other dual compliant electrode actuator  1306  serves as a shutter close actuator. 
     Each dual compliant electrode actuator  1304  and  1306  includes a compliant member  1310  connecting the shutter  1302 , at about its linear axis  1312 , to two load anchors  1314 , located in the corners of the shutter layer  1300 . The compliant members  1310  each include a conductive load beam  1316 , which may have an insulator disposed on part of, or the entirety of its surface. The load beams  1316  serve as mechanical supports, physically supporting the shutter  1302  over a substrate on which the shutter assembly is built. The actuators  1304  and  1306  also each include two compliant drive beams  1318  extending from a shared drive anchor  1320 . Each drive anchor  1320  physically and electrically connects the drive beams  1318  to the substrate. The drive beams  1318  of the actuators  1304  and  1306  curve away from their corresponding drive anchors  1320  towards the points on the load anchors  1314  at which load beams  1316  couple to the load anchors  1314 . These curves in the drive beams  1318  act to reduce the stiffness of the drive beams, thereby helping to decrease the actuation voltage. 
     Each load beam  1316  is generally curved, for example in a bowed (or sinusoidal) shape. The extent of the bow is determined by the relative distance between the load anchors  1314  and the length of the load beam  1316 . The curvatures of the load beams  1316  provide the bi-stability for the shutter assembly  1300 . As the load beam  1316  is compliant, the load beam  1316  can either bow towards or away from the drive anchor  1320 . The direction of the bow changes depending on what position the shutter  1302  is in. As depicted, the shutter  1302  is in the closed position. The load beam  1316  of the shutter open actuator  1304  bows away from the drive anchor  1320  of the shutter open actuator  1304 . The load beam  1316  of the shutter closed actuator  1306  bows towards the drive anchor  1320  of the shutter close actuator  1306 . 
     In operation, to change states, for example from closed to open, a display apparatus applies a potential to the drive beams  1318  of the shutter open actuator  1304 . The display apparatus may also apply a potential to the load beams  1316  of the shutter open actuator. Any electrical potential difference between the drive beams and the load beams, regardless of sign with respect to a ground potential, will generate an electrostatic force between the beams. The resultant voltage between the drive beams  1318  and the load beams  1316  of the shutter open actuator  1304  results in an electrostatic force, drawing the beams  1316  and  1318  together. If the voltage is sufficiently strong, the load beam  136  deforms until its curvature is substantially reversed, as depicted in the shutter close actuator in  FIG. 13A . 
       FIG. 13B  shows the evolution of force versus displacement for the general case of bi-stable actuation, including that for  FIG. 13A . Referring to  FIGS. 13A and 13B , generally the force required to deform a compliant load beam will increase with the amount of displacement. However, in the case of a bi-stable mechanism, such as illustrated in  FIG. 13A , a point is reached (point B in  FIG. 13B ) where further travel leads to a decrease in force. With sufficient voltage applied between the load beam  1316  and the drive beam  1318  of the shutter open actuator  1304 , a deformation corresponding to point B of  FIG. 13B  is reached, where further application of force leads to a large and spontaneous deformation (a “snap through”) and the deformation comes to rest at point C in  FIG. 13B . Upon removal of a voltage, the mechanism will relax to a point of stability, or zero force. Point D is such a relaxation or stable point representing the open position. To move the shutter  1302  in the opposite direction it is first necessary to apply a voltage between the load beam  1316  and the drive beam  1318  of the shutter close actuator  1306 . Again a point is reached where further forcing results in a large and spontaneous deformation (point E). Further forcing in the closed direction results in a deformation represented by point F. Upon removal of the voltage, the mechanism relaxes to its initial and stable closed position, point A. 
     In  FIG. 13A , the length of the compliant member is longer than the straight-line distance between the anchor and the attachment point at the shutter. Constrained by the anchor points, the load beam finds a stable shape by adapting a curved shape, two of which shapes constitute configurations of local minima in the potential energy. Other configurations of the load beam involve deformations with additional strain energy. 
     For load beams fabricated in silicon, typical as-designed widths are about 0.2 μm to about 10 μm. Typical as-designed lengths are about 20 μm to about 1000 μm. Typical as-designed beam thicknesses are about 0.2 μm to about 10 μm. The amount by which the load beam is pre-bent is typically greater than three times the as-designed width 
     The load beams of  FIG. 13A  can be designed such that one of the two curved positions is close to a global minimum, i.e. possesses the lowest energy or relaxed state, typically a state close to zero energy stored as a deformation or stress in the beam. Such a design configuration may be referred to as “pre-bent,” meaning, among other things, that the shape of the compliant member is patterned into the mask such that little or no deformation is required after release of the shutter assembly from the substrate. The as-designed and curved shape of the compliant member is close to its stable or relaxed state. Such a relaxed state holds for one of the two shutter positions, either the open or the closed position. When switching the shutter assembly into the other stable state (which can be referred to as a metastable state) some strain energy will have to be stored in the deformation of the beam; the two states will therefore have unequal potential energies; and less electrical energy will be required to move the beam from metastable to stable states as compared to the motion from the stable state to the metastable state. 
     Another design configuration for  FIG. 13A , however, can be described as a pre-stressed design. The pre-stressed design provides for two stable states with equivalent potential energies. This can be achieved for instance by patterning the compliant member such that upon release of the shutter assembly, the compliant member will substantially and spontaneously deform into its stable shape (i.e. the initial state is designed to be unstable). Preferably the two stable shapes are similar such that the deformation or strain energy stored in the compliant member of each of those stable states will be similar. The work required to move between open and closed shutter positions for a pre-stressed design will be similar. 
     The pre-stress condition of the shutter assembly can be provided by a number of means. The condition can be imposed post-manufacture by, for instance, mechanically packaging the substrate to induce a substrate curvature and thus a surface strain in the system. A pre-stressed condition can also be imposed as a thin film stress imposed by surface layers on or around the load beams. These thin film stresses result from the particulars of a deposition process. Deposition parameters that can impart a thin film stress include thin film material composition, deposition rate, and ion bombardment rate during the deposition process. 
     In  FIG. 13A , the load beam is curved in each of its locally stable states and the load beam is also curved at all points of deformation in between the stable states. The compliant member may be comprised, however, of any number of straight or rigid sections of load beam as will be described in the following figures. In  FIG. 18 , furthermore, will be shown the design of a bi-stable shutter assembly in which neither of the two equivalent stable positions possesses, requires, or accumulates any significant deformation or strain energy. Stress is stored in the system temporarily as it is moved between the stable states. 
       FIG. 14  is a top view of the shutter layer  1400  of a second bi-stable shutter assembly. As described above in relation to  FIG. 6 , reducing resistance to in-plane motion tends to reduce out-of-plane movement of the shutter. The shutter layer  1400  is similar to that of the shutter layer  1300 , other than the shutter layer  1400  includes an in-plane stiffness-reducing feature, which promotes in-plane movement, and a deformation promoter which promotes proper transition between states. As with the shutter layer  1300  of  FIG. 13A , the shutter layer  1400  of  FIG. 14  includes load beams  1402  coupling load anchors  1404  to a shutter  1406 . To reduce the in-plane stiffness of the shutter assembly and to provide some axial compliance to the load beams  1402 , the load anchors  1404  couple to the load beams  1402  via springs  1408 . The springs  1408  can be formed from flexures, L brackets, or curved portions of the load beams  1402 . 
     In addition, the widths of the load beams  1402  vary along their lengths. In particular, the beams are narrower along sections where they meet the load anchors  1404  and the shutter  1406 . The points along the load beams  1402  at which the load beams  1402  become wider serve as pivot points  1410  to confine deformation of the load beams  1402  to the narrower sections  1410 . 
       FIG. 15  is a top view of a shutter layer  1500  of a tri-stable shutter assembly incorporating dual compliant electrode actuators, according to an illustrative embodiment of the invention. The shutter layer  1500  includes a shutter open actuator  1502  and a shutter close actuator  1504 . Each actuator  1502  and  1504  includes two compliant drive beams  1506  physically and electrically coupled to a substrate of a display apparatus by a drive anchor  1508 . 
     The shutter open actuator  1502 , by itself, is an elastic actuator, having one mechanically stable state. Unless otherwise constrained, the shutter open actuator  1502 , after actuation would return to its rest state. The shutter open actuator  1502  includes two load beams  1510  coupled to load anchors  1512  by L brackets  1514  at one end and to the shutter  1516  via L brackets  1518  at the other end. In the rest state of the shutter open actuator  1502 , the load beams  1510  are straight. The L brackets  1514  and  1518  allow the load beams  1510  to deform towards the drive beams  1506  of the shutter open actuator  1502  upon actuation of the shutter open actuator  1502  and away from the drive beams  1506  upon actuation of the shutter close actuator  1504 . 
     The shutter close actuator  1504  is similarly inherently elastic. The shutter close actuator  1504  includes a single load beam  1520  coupled to a load anchor  1522  at one end. When not under stress, i.e., in its rest state, the load beam  1520  is straight. At the opposite end of the load beam  1520  of the shutter close actuator  1504 , the load beam  1520  is coupled to a stabilizer  1524  formed from two curved compliant beams  1526  connected at their ends and at the center of their lengths. The beams  1526  of the stabilizer  1524  have two mechanically stable positions: bowed away from the shutter close actuator  1504  (as depicted) and bowed towards the shutter close actuator  1504 . 
     In operation, if either the shutter open actuator  1502  or the shutter close actuator are activated  1504 , the load beam  1520  of the shutter close actuator  1504  is deformed to bow towards the shutter open actuator  1504  or towards the drive beams  1528  of the shutter close actuator  1504 , respectively, as the shutter  1516  is moved into an actuated position. In either case, the length of the shutter close actuator  1504  load beam  1520  with respect to the width of the shutter layer  1500  as a whole, is reduced, pulling the beams  1526  of the stabilizer  1524  to bow towards the shutter close actuator  1504 . After the activated actuator is deactivated, the energy needed to deform the beams  1526  of the stabilizer  1524  back to its original position is greater than the energy stored in the load beams  1510  and  1520  and of the actuators  1502  and  1504 . Additional energy must be added to the system to return the shutter  1516  to its rest position. Thus, the shutter  1516  in the shutter assembly has three mechanically stable positions, open, half open, and closed. 
       FIGS. 16A-C  are diagrams of another embodiment of a bi-stable shutter assembly  1600 , illustrating the state of the shutter assembly  1600  during a change in shutter  1602  position. The shutter assembly  1600  includes a shutter  1602  physically supported by a pair of compliant support beams  1604 . The support beams couple to anchors  1603  as well as to the shutter  1602  by means of rotary joints  1605 . These joints may be understood to consist of pin joints, flexures or thin connector beams. In the absence of stress being applied to the support beams  1604 , the support beams  1604  are substantially straight. 
       FIG. 16A  depicts the shutter  1602  in an open position,  FIG. 16B  depicts the shutter  1602  in the midst of a transition to the closed position, and  FIG. 16C  shows the shutter  1602  in a closed position. The shutter assembly  1600  relies upon an electrostatic comb drive for actuation. The comb drive is comprised of a rigid open electrode  1608  and a rigid closed electrode  1610 . The shutter  1602  also adopts a comb shape which is complementary to the shape of the open and closed electrodes. Comb drives such as are shown in  FIG. 16  are capable of actuating over reasonably long translational distances, but at a cost of a reduced actuation force. The primary electrical fields between electrodes in a comb drive are aligned generally perpendicular to the direction of travel, therefore the force of actuation is generally not along the lines of the greatest electrical pressure experienced by the interior surfaces of the comb drive. 
     Unlike the bi-stable shutter assemblies described above, instead of relying upon a particular curvature of one or more beams to provide mechanical stability, the bi-stable actuator  1600  relies on the straight relaxed state of its support beams  1604  to provide mechanical stability. For example, in its two mechanically stable positions, depicted in  FIGS. 16A and 16C , the compliant support beams  1604  are substantially straight at an angle to the linear axis  1606  of the shutter assembly  1600 . As depicted in  FIG. 16B , in which the shutter  1602  is in transition from one mechanically stable position to the other, the support beams  1604  physically deform or buckle to accommodate the movement. The force needed to change the position of the shutter  1602  must therefore be sufficient to overcome the resultant stress on the compliant support beams  1604 . Any energy difference between the open and closed states of shutter assembly  1600  is represented by a small amount of elastic energy in the rotary joints  1605 . 
     The shutter  1602  is coupled to two positions on either side of the shutter  1602  through support beams  1604  to anchors  1603  in positions on either side of the shutter assembly  1600 , thereby reducing any twisting or rotational motion of the shutter  1602  about its central axis. The use of compliant support beams  1604  connected to separate anchors on opposite sides of the shutter  1602  also constrains the movement of the shutter along a linear translational axis. In another implementation, a pair of substantially parallel compliant support beams  1604  can be coupled to each side of shutter  1602 . Each of the four support beams couples at independent and opposing points on the shutter  1602 . This parallelogram approach to support of the shutter  1602  helps to guarantee that linear translational motion of the shutter is possible. 
       FIG. 17A  depicts a bi-stable shutter assembly  1700   a , in which the beams  1702   a  incorporated into the shutter assembly  1700   a  are substantially rigid as opposed to compliant, in both of the shutter assembly&#39;s stable positions  17 A- 1  and  17 A- 3  as well as in a transitional position  17 A- 2 . The shutter assembly  1700   a  includes a shutter  1704   a  driven by a pair of dual compliant beam electrode actuators  1706   a . Two compliant members  1710   a  support the shutter  1704   a  over a surface  1712   a . The compliant members  1710   a  couple to opposite sides of the shutter  1704   a . The other ends of the compliant members  1710   a  couple to anchors  1714   a , connecting the compliant members  1710   a  to the surface  1712   a . Each compliant member  1710   a  includes two substantially rigid beams  1716   a  coupled to a flexure or other compliant element  1718   a , such as a spring or cantilever arm. Even though the beams  1716   a  in the compliant members are rigid, the incorporation of the compliant element  1718   a  allows the compliant member  1710   a  as a whole to change its shape in a compliant fashion to take on two mechanically stable shapes. The compliant element is allowed to relax to its rest state in either of the closed or open positions of the shutter assembly (see  17 A- 1  and  17 A- 3 ), so that both of the end states possess substantially identical potential energies. No permanent beam bending or beam stressing is required to establish the stability of the two end states, although strain energy is stored in the compliant element  1718   a  during the transition between states (see  17 A- 2 ). 
     The shape of the compliant element  1718   a  is such that a relatively easy in-plane translation of the shutter  1704   a  is allowed while out-of-plane motion of the shutter is restricted. 
     The actuation of the bi-stable shutter assembly  1700   a  is accomplished by a pair of elastic dual compliant beam electrode actuators  1706   a , similar to the actuators employed in  FIG. 15 . In shutter assembly  1700   a  the actuators  1706   a  are physically separated and distinct from the compliant members  1710   a . The compliant members  1710   a  provide a relatively rigid support for the shutter  1704   a  while providing the bi-stability required to sustain the open and closed states. The actuators  1706   a  provide the driving force necessary to switch the shutter between the open and closed states. 
     Each actuator  1706   a  comprises a compliant load member  1720   a . One end of the compliant load member  1720   a  is coupled to the shutter  1704   a , while the other end is free. In shutter assembly  1700   a  the compliant load members in actuators  1706   a  are not coupled to anchors or otherwise connected to the surface  1712   a . The drive beams  1722   a  of the actuators  1706   a  are coupled to anchors  1724   a  and thereby connected to the surface  1712   a . In this fashion the voltage of actuation is reduced. 
       FIG. 17B  is a diagram of a bi-stable shutter assembly  1700   b  in which the shutter  1702   b  is designed to rotate upon actuation. The shutter  1702   b  is supported at four points along its periphery by 4 compliant support beams  1704   b  which are coupled to four anchors  1706   b . As in  FIG. 16 , the compliant support beams  1704   b  are substantially straight in their rest state. Upon rotation of the shutter  1702   b  the compliant members will deform as the distance between the anchors and the shutter periphery decreases. There are two low energy stable states in which the compliant support beams  1704   b  are substantially straight. The shutter mechanism in  1700   b  has the advantage that there is no center of mass motion in the shutter  1702   b.    
     The shutter  1702   b  in shutter assembly  1700   b  has a plurality of shutter apertures  1708   b , each of which possesses a segmented shape designed to make maximum use of the rotational motion of the shutter. 
       FIG. 18  is a diagram of a bi-stable shutter assembly  1800  incorporating thermoelectric actuators  1802  and  1804 . The shutter assembly  1800  includes a shutter  1806  with a set of slotted shutter apertures  1808 . Thermoelectric actuators  1802  and  1804  couple to either side of the shutter  1806  for moving the shutter  1806  transversely in a plane substantially parallel to a surface  1808  over which the shutter  1806  is supported. The coupling of the shutter  1806  from two positions on either side of the shutter  1806  to load anchors  1807  in positions on either side of the shutter assembly  1800  help reduce any twisting or rotational motion of the shutter  1806  about its central axis. 
     Each thermoelectric actuator  1802  and  1804  includes three compliant beams  1810 ,  1812 , and  1814 . Compliant beams  1810  and  1812  are each thinner than compliant beam  1814 . Each of the beams  1810 ,  1812 , and  1814  is curved in an s-like shape, holding the shutter  1806  stably in position. 
     In operation, to change the position of the shutter from open (as depicted) to closed, current is passed through a circuit including beams  1810  and  1814 . The thinner beams  1810  in each actuator  1802  and  1804  heat, and therefore also expands, faster than the thicker beam  1814 . The expansion forces the beams  1810 ,  1812 , and  1814  from their mechanically stable curvature, resulting in transverse motion of the shutter  1806  to the closed position. To open the shutter  1806 , current is run through a circuit including beams  1812  and  1814 , resulting in a similar disproportionate heating and expansion of beams  1812 , resulting in the shutter  1806  being forced back to the open position. 
     Bi-stable shutter assemblies can be driven using a passive matrix array or an active matrix array.  FIG. 19  is a diagram of a passive matrix array  1900  for controlling bi-stable shutter assemblies  1902  to generate an image. As with active matrix arrays, such as active matrix arrays  900  and  1000 , the passive matrix array  1900  is fabricated as a diffused or thin-film-deposited electrical circuit on a substrate  1904  of a display apparatus. In general, passive matrix arrays  1900  require less circuitry to implement than active matrix arrays  900  and  1000 , and are easier to fabricate. The passive matrix array  1900  divides the shutter assemblies  1902  on the substrate  1904  of the display apparatus into rows and columns of grid segments  1906  of a grid. Each grid segment  1906  may include one or more bi-stable shutter assemblies  1902 . In the display apparatus, all grid segments  1906  in a given row of the grid share a single row electrode  1908 . Each row electrode  1908  electrically couples a controllable voltage source, such as driver  1910  to the load anchors of the shutter assemblies  1902 . All shutter assemblies  1902  in a column share two common column electrodes, a shutter open electrode  1912  and a shutter close electrode  1914 . The shutter open electrode  1912  for a given column electrically couples a driver  1910  to the drive electrode of the shutter open actuator of the shutter assemblies  1902  in the column. The shutter close electrode  1914  for a given column electrically couples a driver  1910  to the drive electrode of the shutter close actuator of the shutter assemblies  1902  in the column. 
     The shutter assemblies  1300 ,  1400 ,  1500 ,  1600 ,  1700   a , and  1800  are amenable to the use of a passive matrix array because their property of mechanical bi-stability makes it possible to switch between open and closed states if the voltage across the actuator exceeds a minimum threshold voltage. If the drivers  1910  are programmed such that none of them will output a voltage that by itself is sufficient to switch the shutter assemblies between open and closed states, then a given shutter assembly will be switched if its actuator receives voltages from two opposing drivers  1910 . The shutter assembly at the intersection of a particular row and column can be switched if it receives voltages from its particular row and column drivers whose difference exceeds the minimum threshold voltage. 
     To change the state of a shutter assembly  1902  from a closed state to an open state, i.e., to open the shutter assembly  1902 , a driver  1910  applies a potential to the row electrode  1908  corresponding to the row of the grid in which the shutter assembly  1902  is located. A second driver  1910  applies a second potential, in some cases having an opposite polarity, to the shutter open electrode  1912  corresponding to the column in the grid in which the shutter assembly  1902  is located. To change the state of a shutter assembly  1902  from an open state to a closed state, i.e., to close the shutter assembly  1902 , a driver  1910  applies a potential to the row electrode  1908  corresponding to the row of the display apparatus in which the shutter assembly  1902  is located. A second driver  1910  applies a second potential, in some cases having an opposite polarity, to the shutter close electrode  1914  corresponding to the column in the display apparatus in which the shutter assembly  1902  is located. In one implementation, a shutter assembly  1902  changes state in response to the difference in potential applied to the row electrode  1908  and one of the column electrodes  1912  or  1914  exceeding a predetermined switching threshold. 
     To form an image, in one implementation, a display apparatus sets the state of the shutter assemblies  1902  in the grid, one row at a time in sequential order. For a given row, the display apparatus first closes each shutter assembly  1902  in the row by applying a potential to the corresponding row electrodes  1908  and a pulse of potential to all of the shutter close electrodes  1914 . Then, the display apparatus opens the shutter assemblies  1902  through which light is to pass by applying a potential to the shutter open electrode  1912  and applying a potential to the row electrodes  1908  for the rows which include shutter assemblies  1902  in the row which are to be opened. In one alternative mode of operation, instead of closing each row of shutter assemblies  1902  sequentially, after all rows in the display apparatus are set to the proper position to form an image, the display apparatus globally resets all shutter assemblies  1902  at the same time by applying potentials to all shutter close electrodes  1914  and all row electrodes  1908  concurrently. In another alternative mode of operation, the display apparatus forgoes resetting the shutter assemblies  1902  and only alters the states of shutter assemblies  1902  that need to change state to display a subsequent image. A number of alternate driver control schemes for images have been proposed for use with ferroelectric liquid crystal displays, many of which can be incorporated for use with the mechanically bi-stable displays herein. These technologies are described in  Liquid Crystal Displays: Driving Schemes and Electro - Optical Effects , Ernst Lieder (Wiley, New York, 2001). 
     The physical layout of the display is often a compromise between the characteristics of resolution, aperture area, and driving voltage. Small pixel sizes are generally sought to increase the resolution of the display. As pixels become smaller, however, proportionally the room available for shutter apertures decreases. Designers seek to maximize aperture ratio as this increases the brightness and power efficiency of the display. Additionally, the combination of a small pixels and large aperture ratios implies large angular deformations in the compliant members that support the shutters, which tends to increase the drive voltages required and the energy dissipated by the switching circuitry. 
       FIGS. 20A and 20B  demonstrate two methods of tiling shutter assemblies into an array of pixels to maximize the aperture ratios in dense arrays and minimize the drive voltages. 
       FIG. 20A , for example, depicts a tiling  2000  of two cantilever dual beam electrode actuator-based shutter assemblies  2002  and  2004  tiled to form a rhombehedral pixel  2006  from two generally triangular shutter assemblies  2002  and  2004 . The shutter assemblies  2002  and  2004  may be independently or collectively controlled. The rhombehedral tiling of  FIG. 20A  is quite close to a rectangular tiling arrangement, and in fact adapted to a rectangular pixel with aspect ratio of 2:1. Since two shutter assemblies can be established within each rectangle, such a 2:1 rectangular tiling arrangement can further be attached or built on top of an active matrix array which possesses a square repeating distance between rows and columns. A 1 to 1 correlation between pixels in the two arrays can therefore be established. Square pixel arrays are most commonly employed for the display of text and graphic images. The advantage of the layout in  FIG. 20B  is that it is understood to maximize the length of the load beams in each triangular pixel to reduce the voltage required for switching shutters between open and closed states. 
       FIG. 20B  is an illustrative tiling of a plurality of bi-stable dual compliant beam electrode-actuator-based shutter assemblies  1300  of  FIG. 13A . In comparison, for example, to the bi-stable dual compliant beam electrode-actuator-based shutter assembly  1400  of  FIG. 14 , the width of the shutter  1302  of the shutter assembly  1300  is substantially less than the distance between the load anchors  1314  of the shutter assembly  1300 . While the narrower shutter  1302  allows for less light to pass through each shutter assembly  1300 , the extra space can be utilized for tighter packing of shutter assemblies  1300 , as depicted in  FIG. 20B , without loss of length in the load beams. The longer load beams makes it possible to switch the shutters in the array at reduced voltages. In particular, the narrower shutter  1302  enables portions of the actuators  1304  and  1306  of the shutter assemblies  1300  to interleave with the gaps between actuators  1302  and  1304  of neighboring shutter assemblies  1300 . The interleaved arrangement of  FIG. 20B  can nevertheless still be mapped onto a square arrangement of rows and columns, which is the common pixel configuration for textual displays. 
     The tiling or pixel arrangements for shutter assemblies need not be limited to the constraints of a square array. Dense tiling can also be achieved using rectangular, rhombehedral, or hexagonal arrays of pixels, all of which find applications, for example in video and color imaging displays. 
       FIG. 21  is a cross sectional view of a display apparatus  2100  incorporating dual compliant electrode actuator-based shutter assemblies  2102 , according to an illustrative embodiment of the invention. The shutter assemblies  2102  are disposed on a glass substrate  2104 . A rear-facing reflective layer, reflective film  2106 , disposed on the substrate  2104  defines a plurality of surface apertures  2108  located beneath the closed positions of the shutters  2110  of the shutter assemblies  2102 . The reflective film  2106  reflects light not passing through the surface apertures  2108  back towards the rear of the display apparatus  2100 . The reflective aperture layer  2106  can be a fine-grained metal film without inclusions formed in thin film fashion by a number of vapor deposition techniques including sputtering, evaporation, ion plating, laser ablation, or chemical vapor deposition. In another implementation, the rear-facing reflective layer  2106  can be formed from a mirror, such as a dielectric mirror. A dielectric mirror is fabricated as a stack of dielectric thin films which alternate between materials of high and low refractive index. 
     The display apparatus  2100  includes an optional diffuser  2112  and/or an optional brightness enhancing film  2114  which separate the substrate  2104  from a backlight  2116 . The backlight  2116  is illuminated by one or more light sources  2118 . The light sources  2118  can be, for example, and without limitation, incandescent lamps, fluorescent lamps, lasers, or light emitting diodes. A front-facing reflective film  2120  is disposed behind the backlight  2116 , reflecting light towards the shutter assemblies  2102 . Light rays from the backlight that do not pass through one of the shutter assemblies  2102  will be returned to the backlight and reflected again from the film  2120 . In this fashion light that fails to leave the display to form an image on the first pass can be recycled and made available for transmission through other open apertures in the array of shutter assemblies  2102 . Such light recycling has been shown to increase the illumination efficiency of the display. 
     In one implementation the light sources  2118  can include lamps of different colors, for instance, the colors red, green, and blue. A color image can be formed by sequentially illuminating images with lamps of different colors at a rate sufficient for the human brain to average the different colored images into a single multi-color image. The various color-specific images are formed using the array of shutter assemblies  2102 . In another implementation, the light source  2118  includes lamps having more than three different colors. For example, the light source  2118  may have red, green, blue and white lamps or red, green, blue, and yellow lamps. 
     A cover plate  2122  forms the front of the display apparatus  2100 . The rear side of the cover plate  2122  can be covered with a black matrix  2124  to increase contrast. The cover plate  2122  is supported a predetermined distance away from the shutter assemblies  2102  forming a gap  2126 . The gap  2126  is maintained by mechanical supports or spacers and/or by an epoxy seal  2128  attaching the cover plate  2122  to the substrate  2104 . The epoxy  2128  should have a curing temperature preferably below about 200° C., it should have a coefficient of thermal expansion preferably below about 50 ppm per degree C. and should be moisture resistant. An exemplary epoxy  2128  is EPO-TEK B9021-1, sold by Epoxy Technology, Inc. 
     The epoxy seal  2128  seals in a working fluid  2130 . The working fluid  2130  is engineered with viscosities preferably below about 10 centipoise and with relative dielectric constant preferably above about 2.0, and dielectric breakdown strengths above about 10 4  V/cm. The working fluid  2130  can also serve as a lubricant. Its mechanical and electrical properties are also effective at reducing the voltage necessary for moving the shutter between open and closed positions. In one implementation, the working fluid  2130  preferably has a low refractive index, preferably less than about 1.5. In another implementation the working fluid  2130  has a refractive index that matches that of the substrate  2104 . In another implementation the working fluid  2130  has a refractive index greater than that of the substrate. In another implementation the working fluid has a refractive index greater than 2.0. Suitable working fluids  2130  include, without limitation, de-ionized water, methanol, ethanol, silicone oils, fluorinated silicone oils, dimethylsiloxane, polydimethylsiloxane, hexamethyldisiloxane, and diethylbenzene. 
     Illustrative methods and materials for forming the reflective apertures  2106  on the same substrate as the shutter assemblies  2102  are disclosed in co-owned U.S. patent application Ser. No. 11/361,785, filed Feb. 23, 2006, incorporated herein by reference. 
     A sheet metal or molded plastic assembly bracket  2132  holds the cover plate  2122 , shutter assemblies  2102 , the substrate  2104 , the backlight  2116  and the other component parts together around the edges. The assembly bracket  2132  is fastened with screws or indent tabs to add rigidity to the combined display apparatus  2100 . In some implementations, the light source  2118  is molded in place by an epoxy potting compound. 
     Display apparatus  2100  is referred to as the MEMS-up configuration, wherein the MEMS based light modulators are formed on a front surface of substrate  2104 , i.e. the surface that faces toward the viewer. The shutter assemblies  2102  are built directly on top of the reflective aperture layer  2106 . In an alternate embodiment of the invention, referred to as the MEMS-down configuration, the shutter assemblies are disposed on a substrate separate from the substrate on which the reflective aperture layer is formed. The substrate on which the reflective aperture layer is formed is referred to herein as the aperture plate. For the MEMS-down configuration, the substrate on which the MEMS-based light modulators are formed takes the place of cover plate  2122  in display apparatus  2100 . In the MEMS-down configuration, the substrate that carries the MEMS-based light modulators is oriented such that the MEMS-based light modulators are positioned on the rear surface of the top substrate, i.e. the surface that faces away from the viewer and toward the back light  2116 . The MEMS-based light modulators are thereby disposed directly opposite to and across a gap from the reflective aperture layer. Display apparatus corresponding to the MEMS-down configuration are described further in U.S. patent application Ser. No. 11/361,785, filed Feb. 23, 2006 and U.S. patent application Ser. No. 11/528,191, filed Sep. 26, 2006, both of which are incorporated herein by reference. 
     In various embodiments, it is advantageous for the shutters used in shutter assemblies to overlap the apertures to which they correspond, when the shutters are in the closed position.  FIGS. 22A and 22B  are top views of a shutter assembly  2200 , similar to the shutter assembly  800  of  FIG. 8 , in opened and closed positions, respectively, illustrating such an overlap. The shutter assembly  2200  includes a shutter  2202  supported over a reflective aperture layer  2204  by anchors  2206  via portions of opposing actuators  2208  and  2210 . The shutter assembly  2200  is suitable for inclusion in an array of light modulators included in a display apparatus. 
     The shutter  2202  includes three shutter apertures  2212 , through which light can pass. The remainder of the shutter  2202  obstructs the passage of light. In various embodiments, the side of the shutter  2202  facing the reflective aperture layer  2204  is coated with a light absorbing material or a reflective material to absorb or reflect, respectively, obstructed light. 
     The reflective aperture layer  2204  is deposited on a transparent substrate, preferably formed from plastic or glass. The reflective aperture layer  2204  can be formed from a film of metal deposited on the substrate, a dielectric mirror, or other highly reflective material or combination of materials. The reflective aperture layer  2204  has a set of apertures  2214  formed in it to allow light to pass through the apertures, from the transparent substrate, towards the shutter  2202 . The reflective aperture layer  2204  has one aperture corresponding to each shutter aperture  2212 . For example, for an array of light modulators including shutter assemblies  2200 , the reflective aperture layer includes three apertures  2214  for each shutter assembly  2200 . Each aperture has at least one edge around its periphery. For example, the rectangular apertures  2214  have four edges. In alternative implementations in which circular, elliptical, oval, or other curved apertures are formed in the reflective aperture layer  2204 , each aperture may have only a single edge. 
     In  FIG. 22A , the shutter assembly  2200  is in an open state. Actuator  2208  is in an open position, and actuator  2210  is in a collapsed position. Apertures  2214  are visible through the shutter apertures  2212 . As visible, the shutter apertures  2212  are larger in area than the apertures  2214  formed in the reflective aperture layer  2204 . The size differential increases the range of angles at which light can pass through the shutter apertures  2212  towards an intended viewer. 
     In  FIG. 22B , the shutter assembly is a closed state. Actuator  2208  is in a collapsed position and actuator  2210  is in an open position. Light blocking portions of the shutter  2202  cover the apertures  2214  in the reflective aperture layer  2204 . The light blocking portions of the shutter  2202  overlap the edges of the apertures  2214  in the reflective aperture layer  2204  by a predefined overlap  2216 . In some implementations, even when a shutter is in a closed state, some light, at angles far from an axis normal to the shutter  2202 , may leak through the apertures  2214 . The overlap included in shutter assembly  2200  reduces or eliminates this light leakage. While, as depicted in  FIG. 22B , the light blocking portions of shutter  2202  overlap all four edges of the aperture, having the light blocking portions of shutter  2202  overlap even one of the edges reduces light leakage. 
       FIG. 23A-23C  are cross-sectional views of various configurations of the shutter assembly  2200  in relation to the transparent substrate on which the reflective aperture layer  2204  is formed. The cross sectional views correspond to the line labeled B-B′ on  FIGS. 22A and 22B . For purposes of illustration, the shutter  2202  is illustrated in  FIGS. 23A-23C  as having only a single shutter aperture  2323  and two light blocking portions  2324 . 
       FIG. 23A  is a cross section of a first configuration of a display apparatus  2300  including a shutter assembly  2301  similar to that depicted in  FIG. 22  in the closed state taken across line B-B′, according to an illustrative embodiment of the invention. In the first configuration, the shutter assembly  2301  is formed on a reflective aperture layer  2302 . The reflective aperture layer  2302  is formed from a thin metal film deposited on a transparent substrate  2304 . Alternately, the reflective aperture layer  2302  can be formed from a dielectric mirror, or other highly reflective material or combination of materials. The reflective aperture layer  2302  is patterned to form apertures  2306 . The transparent substrate  2304  is positioned proximate a light guide  2308 . The transparent substrate  2304  and the light guide  2308  are separated by a gap  2309  filled with a fluid, such as air. The refractive index of the fluid is preferably less than that of the light guide  2308 . Suitable light guides  2308  for display apparatus  2300  are described further in U.S. patent application Ser. No. 11/528,191, the entirety of which is herein incorporated by reference. The display apparatus  2300  also includes a front-facing rear reflective layer  2310  positioned adjacent the rear side of the light guide  2308 . 
     The shutter assembly  2301  includes the shutter  2314  supported proximate to the reflective aperture layer  2302  by anchors  2316  via portions of opposing actuators  2318  and  2320 . The anchors  2316  and actuators  2318  and  2320  suspend the shutter  2314  at about a constant distance H 1  (measured from the bottom of the shutter  2314 ) over the reflective aperture layer  2302 . In addition, the display apparatus  2300  includes a cover plate  2311  supported over the transparent substrate  2304  by spacer posts  2312 . The spacer posts  2312  keep the cover plate at about a second constant distance H 2  away from the top of the shutter  2314 . The gap between the cover plate  2311  and the transparent substrate  2304  is filled with a working fluid  2322 , such as working fluid  2130 , described above. The working fluid  2322  preferably has a refractive index greater than that of the transparent substrate  2304 . In another implementation the working fluid has a refractive index greater than 2.0. In another implementation the working fluid  2322  has a refractive index that is equal to or less than the index of refraction of the transparent substrate  2304 . 
     As indicated above, the shutter assembly  2301  is in the closed state. Light blocking portions  2324  of the shutter  2314  overlap the edges of the apertures  2306  formed in the reflective aperture layer  2302 . The light blocking properties of shutter  2314  are improved when the gap between the shutter and the aperture, i.e. the distance H 1 , is made as small as possible. In one implementation, H 1  is less than about 100 μm. In another implementation, H 1  is less than about 10 μm. In still another implementation, H 1  is about 1 μm. In an alternative embodiment the distance H 1  is greater than 0.5 mm, but remains smaller than the display pitch. The display pitch is defined as the distance between pixels (measured center to center), and in many cases is established as the distance between apertures, such as apertures  2306 , measured center to center, in the rear-facing reflective layer  2302 . 
     The size of the overlap W 1  is preferably proportional to the distance H 1 . While the overlap W 1  may be smaller, preferably the overlap W 1  is greater than or equal to the distance H 1 . In one implementation the overlap W 1  is greater than or equal to 1 micron. In another implementation the overlap W 1  is between about micron and 10 microns. In another implementation the overlap W 1  is greater than 10 microns. In one particular implementation, the shutter  2314  is about 4 μm thick. H 1  is about 2 μm, H 2  is about 2 μm, and W 1 &gt;=2 μm. By having the overlap W 1  being greater than or equal to H 1 , if the shutter assembly  2301  is in the closed state as depicted in  FIG. 23A , most light having a sufficient angle to escape the light guide  2308  through the apertures  2306  impacts the light blocking portions  2324  of the shutter  2314 , thereby improving the contrast ratio of the display apparatus  2300 . 
     H 2  is preferably about the same distance as H 1 . The spacer posts  2312  are preferably formed from a polymer material that is lithographically patterned, developed, and/or or etched into cylindrical shapes. The height of the spacer is determined by the cured thickness of the polymer material. Methods and materials for formation of spacers  2312  are disclosed in co-owned U.S. patent application Ser. No. 11/361,785, filed Feb. 23, 2006, incorporated herein by reference. In an alternative embodiment the spacer  2312  can be formed from a metal which is electrochemically deposited into a mold made from a sacrificial material. 
       FIG. 23B  is a cross section of a second configuration of a display apparatus  2340  including a shutter assembly  2341  similar to that depicted in  FIG. 22  in the closed state, according to an illustrative embodiment of the invention. This second configuration is referred to as the MEMS-down configuration, in which the reflective aperture layer  2344  is formed on a transparent substrate called the aperture plate  2346 , which is distinct from the light modulator substrate  2342  to which shutter assembly  2341  is anchored. The shutter assembly includes a shutter  2354  having light blocking portions  2362  and shutter apertures  2363  formed therein. Like the aperture plate  2346 , the light modulator substrate  2342  is also transparent. The two substrates  2342  and  2346  are separated by a gap. The two substrates  2342  and  2346  are aligned during assembly such that a one to one correspondence exists, as indicated in  FIG. 22 , between each of the apertures  2347  and the light blocking portions  2362  of shutter  2354  when that shutter is in the closed position, and/or between the apertures  2347  and the shutter apertures  2363  when that shutter is in the open position. In alternative embodiments, the correspondence between apertures and either light blocking portions  2362  or shutter apertures  2363  of a shutter  2354  is a one to many or many to one correspondence. 
     In the MEMS-down display apparatus  2340 , the shutter assembly  2341  is formed on the rear-facing surface of the light modulator substrate  2342 , i.e. on the side which faces the light guide  2348 . In display apparatus  2340 , the aperture plate  2346  is positioned between the light modulator substrate  2342  and the light guide  2348  The reflective aperture layer  2344  is formed from a thin metal film deposited on the front-facing surface of transparent aperture plate  2346 . The reflective aperture layer  2344  is patterned to form apertures  2347 . In another implementation, the reflective layer  2344  can be formed from a mirror, such as a dielectric mirror. A dielectric mirror is fabricated from a stack of dielectric thin films with different refractive indices, or from combinations of metal layers and dielectric layers. 
     The aperture plate  2346  is positioned proximate to a backlight or light guide  2348 . The aperture plate  2346  is separated from the light guide  2348  by a gap  2349  filled with a fluid, such as air. The refractive index of the fluid is preferably less than that of the light guide  2348 . Suitable backlights  2348  for display apparatus  2340  are described further in U.S. patent application Ser. No. 11/528,191, the entirety of which is herein incorporated by reference. The display apparatus  2340  also includes a front-facing rear reflective layer  2350  positioned adjacent the rear side of the backlight  2348 . The front-facing reflective layer  2350  combined with the rear-facing reflective layer  2344  forms an optical cavity which promotes recycling of light rays which do not initially pass through apertures  2347 . The shutter assembly  2341  includes the shutter  2354  supported proximate to the transparent substrate  2342  by anchors  2356  via portions of opposing actuators  2358  and  2360 . The anchors  2356  and actuators  2358  and  2360  suspend the shutter  2354  at about a constant distance H 4  (measured from the top of the shutter  2354 ) below the light modulator substrate  2342 . In addition, display apparatus includes spacer posts  2357 , which support the light modulator substrate  2342  over the aperture plate  2346 . The spacer posts  2357  keep the light modulator substrate  2342  at about a second constant distance H 5  away from the aperture plate  2346 , thereby keeping the bottom surface of shutter  2354  at a third about constant distance H 6  above the reflective aperture layer  2344 . The spacer posts  2357  are formed in a fashion similar to those of spacers  2312 . 
     The gap between the light modulator substrate  2342  and the aperture plate  2346  is filled with a working fluid  2352 , such as working fluid  2130 , described above. The working fluid  2352  preferably has a refractive index greater than that of the transparent aperture plate  2346 . In another implementation the working fluid has a refractive index greater than 2.0. In another implementation the working fluid  2352  preferably has a refractive index that is equal to or less than the index of refraction of the aperture plate  2346 . 
     As indicated above, the shutter assembly  2341  is in the closed state. Light blocking portions  2362  of the shutter  2354  overlap the edges of the apertures  2347  formed in the reflective aperture layer  2344 . The size of the overlap W 2  is preferably proportional to the distance H 6 . While the overlap W 2  may be smaller, preferably the overlap W 2  is greater than or equal to the distance H 6 . In one implementation, H 6  is less than about 100 μm. In another implementation, H 6  is less than about 10 μm. In still another implementation, H 6  is about 1 μm. In an alternative embodiment the distance H 6  is greater than 0.5 mm, but remains smaller than the display pitch. The display pitch is defined as the distance between pixels (measured center to center), and in many cases is established as the distance between the centers of apertures in the rear-facing reflective layer, such as apertures  2347 . H 4  is preferably about the same distance as H 6 . In one particular implementation, the shutter  2354  is about 4 μm thick, H 6  is about 2 μm, H 4  is about 2 μm, H 5  is about 8 μm and W 2  &gt;=2 μm. By having the overlap W 2  being greater than or equal to H 6 , if the shutter assembly  2341  is in the closed state as depicted in  FIG. 23B , most light having a sufficient angle to escape the backlight  2348  through the apertures  2347  impacts the light blocking portions  2362  of the shutter  2354 , thereby improving the contrast ratio of the display apparatus  2340 . 
       FIG. 23C  is a cross section of a third configuration of a display apparatus  2370  including a shutter assembly  2371  similar to that depicted in  FIG. 22  in the closed state, according to an illustrative embodiment of the invention. In comparison to the second configuration of the display apparatus  2340  described above, the display apparatus  2370  is designed to account for minor misalignments that may occur during the aligning and bonding of a light modulator substrate  2372  (similar to light modulator substrate  2342 ) on which a shutter assembly  2371  is formed to an aperture plate  2374  (similar to the aperture plate  2346 ) on which a reflective aperture layer  2376  is deposited. To address this potential issue, the display apparatus  2370  includes an additional layer of light absorbing material  2377 , deposited on the light modulator substrate  2372 . The light absorbing material  2377  may be part of a black mask, though at least some of the light absorbing material  2377  is preferably located in the interior of a pixel to which the shutter assembly  2371  corresponds. The light absorbing material  2377  absorbs light  2378  that would otherwise pass through the light modulator substrate  2372  while the shutter  2382  is in the closed state. Additional light absorbing material  2377  may be deposited on the front side of reflective aperture layer  2376  to absorb light, for example light  2380  deflected from a shutter  2382 . 
       FIG. 23D  is a cross section of a fourth configuration of a display apparatus  2390  including a shutter assembly  2385  similar to that depicted in  FIG. 22  in the closed state, according to an illustrative embodiment of the invention. In comparison to the second configuration of shutter assembly  2354  described above, the shutter assembly  2385  is fabricated according to a different process resulting in different cross sectional thicknesses for some of its members. The resulting shutter  2393  is referred to herein as a corrugated shutter. The design guidelines for gap distances, e.g. H 8  and H 10 , and for the overlap parameter W 4 , however, are preferably unchanged from the corresponding gap distances and the overlap parameters described above. The display apparatus  2390  includes a transparent light modulator substrate  2386 , oriented in the MEMS down configuration, and to which the shutter assembly  2385  is attached. The display apparatus  2390  also includes a transparent aperture plate  2387  on which a rear-facing reflective aperture layer  2388  is deposited. The display apparatus  2390  includes a fluid  2389  which fills the gap between substrates  2386  and  2387 . The fluid  2389  preferably has a refractive index higher than that of the aperture plate  2387 . The display apparatus also includes a backlight  2348  along with front-facing reflective layer  2350 . 
     The shutter assembly  2385  is in the closed state. Light blocking portions  2391  of the corrugated shutter  2393  overlap the edges of apertures  2394  formed in the reflective aperture layer  2388 . The corrugated shutter  2393  is comprised of two connected flat plate sections: section  2391  which is oriented horizontally and section  2392  which is oriented vertically. Each flat plate  2391  and  2392  is comprised of thin film materials with thicknesses in the range of 0.2 to 2.0 μm. In a particular embodiment the thickness of the horizontal section  2391  is 0.5 μm. The vertical section  2392  provides a stiffness to the corrugated shutter  2393  and a height which matches that of actuator  2358  without requiring the deposition of a bulk materials thicker than about 2 μm. Methods and materials for formation of shutters with a corrugated and/or three dimensional structures are disclosed in co-owned U.S. patent application Ser. No. 11/361,785, filed Feb. 23, 2006, incorporated herein by reference. 
     Similar to dimensions described for display apparatus  2340 , in a particular example the dimensions of H 8 , H 9 , and H 10  of display apparatus  2390  can be 2, 8, and 2 μm respectively. The overlap W 4  is preferably greater than or equal to the distance H 10 . In another example, the distance H 10  and the overlap W 4  can be &gt;=1 μm. Using the materials and methods for a corrugated shutter  2393 , however, the thickness of section  2391  can be as thin as 0.5 μm. By having the overlap W 4  greater than or equal to H 10 , if the shutter assembly  2385  is in the closed state as depicted in  FIG. 23A , most light having a sufficient angle to escape the backlight  2348  through the apertures  2394  impacts the light blocking portions  2391  of the shutter  2393 , thereby improving the contrast ratio of the display apparatus  2390 . 
       FIG. 24  is a cross sectional view of a first electrowetting-based light modulation array  2400 , according to an illustrative embodiment of the invention. The light modulation array  2400  includes a plurality of electrowetting-based light modulation cells  2402   a - 2402   d  (generally “cells  2402 ”) formed on an optical cavity  2404 . The light modulation array  2400  also includes a set of color filters  2406  corresponding to the cells  2402 . 
     Each cell  2402  includes a layer of water (or other transparent conductive or polar fluid)  2408 , a layer of light absorbing oil  2410 , a transparent electrode  2412  (made, for example, from indium-tin oxide) and an insulating layer  2414  positioned between the layer of light absorbing oil  2410  and the transparent electrode  2412 . Illustrative implementation of such cells are described further in U.S. Patent Application Publication No. 2005/0104804, published May 19, 2005 and entitled “Display Device,” incorporated herein by reference. In the embodiment described herein, the transparent electrode  2412  takes up only a portion of a rear surface of a cell  2402 . 
     The remainder of the rear surface of a cell  2402  is formed from a reflective aperture layer  2416  that forms the front surface of the optical cavity  2404 . The rear-facing reflective layer  2416  is patterned to form apertures, which in the embodiment of cell  2402  are coincident with the transparent electrode  2412 . Preferably, when in the closed position, the layer of light absorbing oil  2410  overlaps one or more edges of its corresponding aperture in the reflective aperture layer  2416 . The reflective aperture layer  2416  is formed from a reflective material, such as a reflective metal or a stack of thin films forming a dielectric mirror. For each cell  2402 , an aperture is formed in the reflective aperture layer  2416  to allow light to pass through. In an alternate embodiment, the electrode  2412  for the cell is deposited in the aperture and over the material forming the reflective aperture layer  2416 , separated by another dielectric layer. 
     The remainder of the optical cavity  2404  includes a light guide  2418  positioned proximate the reflective aperture layer  2416 , and a second reflective layer  2420  on a side of the light guide  2418  opposite the reflective aperture layer  2416 . A series of light redirectors  2421  are formed on the rear surface of the light guide, proximate the second reflective layer. The light redirectors  2421  may be either diffuse or specular reflectors. One of more light sources  2422  inject light  2424  into the light guide  2418 . 
     In an alternate implementation the light sources  2422  can include lamps of different colors, for instance, the colors red, green, and blue. A color image can be formed by sequentially illuminating images with lamps of different colors at a rate sufficient for the human brain to average the different colored images into a single multi-color image. The various color-specific images are formed using the array of electrowetting modulation cells  2402 . In another implementation, the light source  2422  includes lamps having more than three different colors. For example, the light source  2422  may have red, green, blue and white lamps or red, green, blue, and yellow lamps. 
     In an alternative implementation, the cells  2402  and the reflective aperture layer  2416  are formed on an additional light modulator substrate which is distinct from light guide  2418  and separated from it by a gap. (See for example the light modulator substrate  2513  of  FIG. 25 .) In yet another implementation, a layer of material with a refractive index less than that of the light guide  2418  is interposed between the reflective aperture layer  2416  and the light guide  2418 . The layer of material with lower refractive index may help to improve the uniformity of light emitted from the light guide  2418 . 
     In operation, application of a voltage to the electrode  2412  of a cell (for example, cell  2402   b  or  2402   c ) causes the light absorbing oil  2410  in the cell to collect in one portion of the cell  2402 . As a result, the light absorbing oil  2410  no longer obstructs the passage of light through the aperture formed in the reflective aperture layer  2416  (see, for example, cells  2402   b  and  2402   c ). Light escaping the backlight at the aperture is then able to escape through the cell and through a corresponding color (for example, red, green, or blue) filter in the set of color filters  2406  to form a color pixel in an image. When the electrode  2412  is grounded, the light absorbing oil  2410  covers the aperture in the reflective aperture layer  2416 , absorbing any light  2424  attempting to pass through it (see for example cell  2402   a ). 
     The area under which oil  2410  collects when a voltage is applied to the cell  2402  constitutes wasted space in relation to forming an image. This area cannot pass light through, whether a voltage is applied or not, and therefore, without the inclusion of the reflective portions of reflective apertures layer  2416 , would absorb light that otherwise could be used to contribute to the formation of an image. However, with the inclusion of the reflective aperture layer  2416 , this light, which otherwise would have been absorbed, is reflected back into the light guide  2420  for future escape through a different aperture. 
       FIG. 25  is a cross sectional view of a second electrowetting-based light modulation array  2500 , according to an illustrative embodiment of the invention. The second electrowetting-based light modulation array  2500  includes three sub-arrays  2501   a ,  2501   b , and  2501   c  of colored electrowetting-based light modulation cells  2502  (generally “cells  2502 ”), positioned on top of one another. Each cell  2502  includes a transparent electrode  2504 , and a colored oil  2506  separated by an insulator  2508 . In one implementation, the oil  2506  in the cells  2502  of sub-array  2501   a  is colored cyan, the oil  2506  in the cells  2502  of sub-array  2501   b  is colored yellow, and the oil  2506  in the cells  2502  of sub-array  2501   c  is colored magenta. The cells  2502  in sub-array  2501   a  and the cells  2502  of sub-array  2501   b  share a common layer of water  2520 . The cells  2502  of sub-array  2501   c  include their own layer of water  2520 . 
     The electrowetting-based light modulation array  2500  includes a light-recycling optical cavity  2510  coupled to the three sub-arrays  2501   a - 2501   c . The optical cavity  2510  includes a light guide  2512  and a light modulator substrate  2513 , separated from the light guide  2512  by a gap  2515 . The front surface of the light modulator substrate  2513  includes a rear-facing reflective aperture layer  2514 . The reflective aperture layer  2514  is formed from a layer of metal or a stack of thin films forming a dielectric mirror. Apertures  2516  are patterned into the reflective aperture layer beneath the cells  2502  of the sub-arrays  2501   a - 2501   c  to allow light to escape the light guide and pass through the sub-arrays  2501   a - 2501   c  to form an image. The transparent electrodes  2504  of cells  2502  are formed over the top of the reflective aperture layer  2514 . 
     The substrates, i.e., light guide  2512  and modulator substrate  2513 , are separated by a gap  2515  filled with a fluid, such as air. The refractive index of the fluid is less than that of the light guide  2512 . A front-facing reflective layer  2518  is formed on, or positioned proximate to, the opposite side of the light guide  2512 . The light modulation array  2500  includes at least one light source  2522  for injecting light into the light guide  2512 . Suitable light guides  2618  for display apparatus  2600  are described further in U.S. patent application Ser. No. 11/528,191, the entirety of which is herein incorporated by reference. 
       FIG. 26  is a cross sectional view of a third electrowetting-based light modulation array  2600 , according to an illustrative embodiment of the invention. The light modulation array  2600  includes a plurality of electrowetting-based light modulation cells  2602   a - 2602   c  (generally “cells  2602 ”) formed on an optical cavity  2604 . The light modulation array  2600  also includes a set of color filters  2606  corresponding to the cells  2602 . 
     While the array  2400  might be considered an example of an array in a MEMS-up configuration, the array  2600  is an example of an electrowetting-based array assembled in a MEMS-down configuration. Each cell  2602  includes a layer of water (or other transparent conductive or polar fluid)  2608 , a layer of light absorbing oil  2610 , a transparent electrode  2612  (made, for example, from indium-tin oxide) and an insulating layer  2614  positioned between the layer of light absorbing oil  2610  and the transparent electrode  2612 . In the MEMS-down configuration of light modulator array  2600 , however, both the insulating layer  2614  and the transparent electrode  2612  are disposed on a light modulator substrate  2630  distinct from an aperture plate  2632 . Like the light modulator substrate  2630 , the aperture plate  2632  is also a transparent substrate. The light modulator substrate  2630  is the topmost substrate and is oriented such that control electrodes such as transparent electrode  2612  are disposed on the rear surface of substrate  2630 , i.e. the surface that faces away from the viewer and toward the light guide. In addition to transparent electrode  2612 , the rear surface of light modulator substrate  2630  can carry other common components of a switching or control matrix for the modulator array, including without limitation, row electrodes, column electrodes, transistors for each pixel and capacitors for each pixel. The electrodes and switching components formed on light modulator substrate  2630 , which govern the actuation of light modulators in the array, are disposed opposite to and across a gap  2636  from a reflective aperture layer  2616 , disposed on the front surface of aperture plate  2632 . The gap  2636  is filled with the electrowetting fluid components water  2608  and oil  2610 . 
     The reflective aperture layer  2616  is deposited on transparent substrate  2632 , preferably formed from plastic or glass. The reflective aperture layer  2616  can be formed from a film of metal deposited on the substrate, a dielectric mirror, or other highly reflective material or combination of materials. The reflective aperture layer  2616  is a rear-facing reflective layer, forming the front surface of optical cavity  2604 . The reflective aperture layer  2616  has a set of apertures  2617  formed in it to allow light to pass through the apertures toward the electrowetting fluid components  2608  and  2610 . Optionally, the aperture plate  2632  includes a set of color filters  2606  deposited on the top surface of reflective aperture  2616  and filling the apertures  2617 . 
     The aperture plate  2632  is positioned between the light modulator substrate  2630  and the light guide  2618 . The substrates  2632  and  2618  are separated from each other by a gap  2634  filled with a fluid (such as air). The refractive index of the fluid is less than that of the light guide  2618 . Suitable light guides  2618  for display apparatus  2600  are described further in U.S. patent application Ser. No. 11/528,191, the entirety of which is herein incorporated by reference. The optical cavity  2604  also includes substrates  2632 ,  2618 , and the front-facing rear reflective layer  2620  positioned adjacent the rear side of the light guide  2618 . One or more light sources  2622  inject light into the light guide  2618 . 
     The reflective aperture layer  2616  has one aperture  2617  corresponding to each light modulator cell  2602  in the array  2600 . Similarly, the light modulator substrate  2630  has one transparent electrode  2612  or one set of pixel transistors and capacitors for each light modulator cell  2602 . The substrates  2630  and  2632  are aligned during assembly to ensure that corresponding apertures  2617  are positioned where light will not be obstructed by the oil  2610  when cells are actuated or held in the open state, e.g. cell  2602   b.    
     The invention may be embodied in other specific forms without departing from the spirit or essential characteristics thereof. The forgoing embodiments are therefore to be considered in all respects illustrative, rather than limiting of the invention.