Patent Publication Number: US-7218429-B2

Title: Digital focus lens system

Description:
This application is a continuation of U.S. patent application Ser. No. 10/617,572, filed Jul. 11, 2003, now U.S. Pat. No. 7,072,086, and published as U.S. Patent Application Publication 2004/0114203 A1, the entire disclosures of which are incorporated herein by reference. Application Ser. No. 10/617,572 claims the benefit of U.S. Provisional Patent Application No. 60/395,849 filed Jul. 11, 2002, the entire disclosures of which are incorporated herein by reference. This application is also a continuation-in-part of co-pending U.S. patent application Ser. No. 10/029,399 filed Oct. 19, 2001 and published as U.S. Patent Application Publication 2002/0158866 A1, the entire contents of which are incorporated herein by reference. Application Ser. No. 10/029,399 claims the benefit of US Provisional Application 60/242,395 filed Oct. 20, 2000, the entire disclosures of which are incorporated herein by reference. This application claims the benefit of priority of application Ser. Nos. 10/617,572, 10/029,399, 60/395,849, and 60/242,395. 

   FIELD OF THE INVENTION 
   This invention relates to optical components such as optical lens complexes and more specifically, to variable-focus lenses such as liquid crystal lenses. 
   BACKGROUND OF THE INVENTION 
   Solid-state variable-focus lens systems are needed in a variety of applications such as in cameras deployed on aircraft and subjected to strong acceleration forces. It is often desirable to have a variable-focus lens system that is compact and capable of solid-state operation; and further, one in which the number of possible states of focusing is an exponential function of the number of optical elements in the system. Conventional variable-focus lens systems are bulky, require moving parts, or require numerous elements resulting in optical losses and aberration of the images. 
   Thus, there is a need in the art, for a lens system that overcomes the above disadvantages. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The teachings of the present invention can be readily understood by considering the following detailed description in conjunction with the accompanying drawings, in which: 
       FIG. 1  shows a digital focus lens system according to an embodiment of the present invention. 
       FIG. 2  shows another view of a digital focus lens system. 
       FIG. 3  shows yet another view of a digital focus lens system. 
       FIG. 4  shows yet another view of a digital focus lens system. 
       FIG. 5  shows another embodiment of a digital focus lens system. 
       FIG. 6  shows yet another embodiment of a digital focus lens system. 
       FIGS. 7A–7D  are a sequence of cross-sectional schematic diagrams that illustrate a method for fabricating the switchable elements of a digital focus lens system according to an embodiment of the invention. 
       FIG. 8  shows schematic diagram illustrating a method of controlling a digital focus lens system according to an embodiment of the invention. 
       FIG. 9  shows a digital focus lens system incorporated in a digital telescope according to an embodiment of the invention. 
   

   DESCRIPTION OF THE SPECIFIC EMBODIMENTS 
   Although the following detailed description contains many specific details for the purposes of illustration, anyone of ordinary skill in the art will appreciate that many variations and alterations to the following details are within the scope of the invention. Accordingly, the embodiments of the invention described below are set forth without any loss of generality to, and without imposing limitations upon, the claimed invention. 
   According to an embodiment of the present invention digital focus lens system may include a stack of switchable lens elements. The stack may include a plurality of optically transparent substrates symmetrically spaced apart, optically transparent electrodes deposited on the surfaces of each substrate, polymer layers deposited on the electrodes, and liquid crystal (LC) layers filling the gaps between adjacent pairs of polymer layers. Each polymer layer may be spatially patterned to provide a selected lens function having a selected focal length, and each is treated with alignment features to facilitate orientation of the LC monomers. When a selected voltage is applied across adjacent pairs of electrodes, the refractive index of the LC layer positioned between those electrodes is switched to a selected value and the focal lengths of the polymer layers adjacent to the LC layer are modulated. Thus, each group of electrode-polymer-LC-polymer-electrode layers may constitute a different switchable lens element where each can be switched between a first state, having a first focal length, and a second state, having a second focal length. 
   A control signal may be provided for selecting the states of the switchable lens elements. For a stack of N switchable lens elements, the control signal will include a digital word comprised of at least N bits. The control signal is demultiplexed and each bit used to modulate the voltage applied to a corresponding switchable lens element. The state of each switchable lens element is thus controlled by a corresponding bit of the digital word. 
   The switchable lens elements can be stacked coaxially and can be switched independently to either state. Thus, the system has a focal length that is determined by the instantaneous combination of states of the switchable lens elements. In the first state, the switchable lens elements may have identical focal lengths. In the second state, the focal lengths of the switchable lens elements increase, following a progression, similar to binary weighting, in which the focal length of each sequential switchable lens element in the stack increases by a factor of 2 from that of the previous element. For a system comprising a stack of N switchable lens elements, the focal length can be selected from a set of at least 2 N  values. The system thus has a focal power that is a function of the digital word contained in the control signal. 
   Referring now to  FIG. 1 , a digital focus lens system according to an embodiment of the present invention is shown and indicated generally at  110 . System  110  employs a first optical module, M 1  (first module)  120 . First module  120  incorporates a number of optical elements (elements) indicated schematically at  130 . The individual elements  130  in the module  140  are in optical communication with each other such that each element  130  in the module  140  may contribute to a cumulative optical affect. For example, the elements  130  may be oriented such that the optical axes  140  of elements  130  are generally collinear with the optical axis  150  of the system  110 . Some or all of elements  130  may be positioned in close proximity to adjacent elements. In this fashion, the elements in first module  120  may form a first stack of elements (first stack)  160 . One or more of Elements  130  may be similar to thin lenses whereby the standard thin lens approximation formulas may be applicable to portions of first stack  160  and/or first module  120 . First module  120  and/or first stack  160  may also be considered similar to a “lens group”, or to a “lens complex,” terms commonly used in the field of lens design. 
   At least one of elements  130  includes a first switchable element  170 . First switchable element  170  may be activated between a number of unique states, where for each state, first switchable element  170  is capable of performing a unique optical transform (or filter function). Preferably, first switchable element  170  may be activated between two states, however, in general, any number of states may be utilized by first switchable element  170 . Preferably, the transform performed by first switchable element  170  is similar to that of a thin lens. For example, first switchable element  170  may be activated into a first-switchable-element first-state (FSE 0-state), having the property of an FSE 0-state focal length  180 . Similarly, first switchable element  170  may be activated into a first-switchable-element second-state (FSE 1-state) having an FSE 1-state focal length  190 . First Module  120  may also incorporate a second switchable element  200 . Second switchable element  200  may be activated into a second-switchable-element first-state (SSE 0-state), having the property of an SSE 0-state focal length  210 . Similarly, second switchable element  200  may be activated into a second-switchable-element second-state (SSE 1-state) having an SSE 1-state focal length  220 . In this fashion, First Module  120  may also incorporate additional switchable elements  230 . In this fashion, first module  120  may incorporate a number of switchable elements, N, indicated generally at  236  whereby each switchable element may be activated between a first state (0-state) and a second state (1-state) corresponding to a first focal length and a second focal length, respectively. Examples of switchable elements  170 ,  200  include without limitation liquid crystals (LCs), holographic optical elements, polymer-dispersed liquid crystals, nonlinear optical lenses, electro-optic elements, electro-optic lenses, LC lenses, LC prisms, LC gratings, LC shutters, LC aperture stops, LC irises, polymer dispersed liquid crystals, switchable holographic optical elements (HOEs), polarization rotators, isotropic, uniaxial, biaxial and/or other anisotropic optical materials, deformable mirrors and deformable gratings, and micro-electro-mechanical systems (MEMS) and MEMS mirrors. Similarly, a second module  240  and third module  250 , and in general, any number of additional modules (not shown), may be incorporated in system  110 . 
   Turning now to  FIG. 2 , a further description of the optical properties of a digital focus lens system according to an embodiment of the present invention is shown and indicated generally at  254 . The same components as in  FIG. 1  have the same assigned number as in  FIG. 1 . System  110  includes a first module  120 . First module  120  includes a first stack  160 . First stack  160  includes a number of elements  130 . One or more of elements  130  comprise a number of switchable elements, N,  236 . For the purpose of the indexing the N switchable elements  236 , each switchable element may be assigned a unique subscript number n, where n may be chosen from the set:
 
nε{0,1 . . . N−1}.  Eq. 1
 
   Each switchable element n may be switched between two specified states, however, in general, any number of states may be specified. 
   An impulse (or “state”) variable, δ m,n (state), can be defined as corresponding to the state of switchable element n in module m where 
   
     
       
         
           
             
               
                 
                   
                     δ 
                     
                       m 
                       , 
                       n 
                     
                   
                   ⁡ 
                   
                     ( 
                     state 
                     ) 
                   
                 
                 = 
                 
                   { 
                   
                     
                       
                         
                           0 
                         
                         
                           
                             0 
                             - 
                             state 
                           
                         
                       
                       
                         
                           1 
                         
                         
                           
                             1 
                             - 
                             state 
                           
                         
                       
                     
                     . 
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 2 
               
             
           
         
       
     
   
   For the remainder of this discussion, the parenthesis (state) will be dropped from the symbol δ m,n  for simplification. It follows that a δ m,n  will be specified for each switchable element n in module m. Also, it will be seen that δ m,n  is similar to delta functions commonly used in the field of Fourier analysis. 
   By way of example, δ 1,0 , corresponding to the state of switchable element 0 in module 1, may have the value δ m,n =0 while the 0 th  element is activated in the 0-state, and the value 1 while activated in 1-state. Now, a switchable element focal length variable, f m,n   δ     m,n    is given to specify the focal length of switchable element n in module m. The parameters of f m,n   δ     m,n    are: a first subscript m specifying the module number; a second subscript n specifying the switchable element number and a superscript δ m,n  specifying the state of the n th  element in module m. It follows that a variable f m,n   δ     m,n    will be specified for each switchable element n in each module m. As illustrated in  FIG. 2 , the 0-state and 1-state switchable element focal lengths of switchable element n=2 in module m=1 are identified by f 1,2   0  and f 1,2   1 , respectively. As a further example of this nomenclature, the 0-state focal length of the first switchable element  170  in the first module  120  will be referred to by the symbol f 1,0   0    300 . The 1-state focal length of the first switchable element  170  in the first module  120  will be referred to by the symbol f 1,0   1    310 . Similarly, the 0-state focal length of the second switchable element  200  in the first module  120  will be referred to by the symbol f 1,1   0    320 . As a final example of the nomenclature, the 1-state focal length of the second switchable element  200  in the first module  120  will be referred to by the symbol f 1,1   1    330 . Next, a module focal length, F m ,  256  is given for the focal length of a module where the module number is indicated by the subscript m. For example, F 1    260  is the symbol for the module focal length  256  of the first module  120 , m=1. From the above discussion, it follows that, for the case of the switchable elements being approximated as a stack of thin lenses (such when each element can be approximated as a thin lens and each is in approximate contact with any adjacent elements) and the paraxial approximation applies to the stack, the module focal length, F m , can be expressed as 
   
     
       
         
           
             
               
                 
                   F 
                   m 
                 
                 = 
                 
                   
                     
                       ( 
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 δ 
                                 
                                   m 
                                   , 
                                   n 
                                 
                               
                               
                                 f 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 1 
                               
                             
                             + 
                             
                               
                                 1 
                                 - 
                                 
                                   δ 
                                   
                                     m 
                                     , 
                                     n 
                                   
                                 
                               
                               
                                 f 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 0 
                               
                             
                           
                           ) 
                         
                       
                       ) 
                     
                     
                       - 
                       1 
                     
                   
                   . 
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 3 
               
             
           
         
       
     
   
   Eq. 3 can be rearranged as 
   
     
       
         
           
             
               
                 
                   F 
                   m 
                 
                 = 
                 
                   
                     
                       [ 
                       
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               1 
                               
                                 f 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 0 
                               
                             
                           
                           ) 
                         
                         + 
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             
                               δ 
                               
                                 m 
                                 , 
                                 n 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   1 
                                   
                                     f 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     1 
                                   
                                 
                                 - 
                                 
                                   1 
                                   
                                     f 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     0 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                     
                       - 
                       1 
                     
                   
                   . 
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 4 
               
             
           
         
       
     
   
   In one embodiment of the invention, the N switchable elements in module m may be constructed such that their 1-state focal lengths, f m,n   1 , follow the mathematical form 
                   f     m   ,   n     1     =       (         2   n       Δ   m       +     1     f     m   ,   n     0         )       -   1               Eq   .           ⁢   5               
where Δ m  is a constant for module m, is independent of n, and has the dimension of length. Generally, however, in other embodiments of the invention, f m,n   1  may be expressed by other mathematical forms.
 
   Substituting Eq. 5 into Eq. 4 gives 
   
     
       
         
           
             
               
                 
                   F 
                   m 
                 
                 = 
                 
                   
                     
                       [ 
                       
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               1 
                               
                                 f 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 0 
                               
                             
                           
                           ) 
                         
                         + 
                         
                           
                             1 
                             
                               Δ 
                               m 
                             
                           
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               
                                 δ 
                                 
                                   m 
                                   , 
                                   n 
                                 
                               
                               ⁢ 
                               
                                 2 
                                 n 
                               
                             
                           
                         
                       
                       ] 
                     
                     
                       - 
                       1 
                     
                   
                   . 
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 6 
               
             
           
         
       
     
   
   Now it can be seen that the second summand term in Eq. 6 will have a unique value for each possible combination of states δ m,n  for the N switchable elements  236  in module m. 
   Neglecting the 1/Δ m  factor in front of the summand term of Eq. 6, the combination of possible values for the summand define the set of integers 
   
     
       
         
           
             
               
                 
                   
                     ∑ 
                     
                       n 
                       = 
                       0 
                     
                     
                       N 
                       - 
                       1 
                     
                   
                   ⁢ 
                   
                     
                       ξ 
                       n 
                     
                     ⁢ 
                     
                       2 
                       n 
                     
                   
                 
                 ∈ 
                 
                   
                     { 
                     
                       0 
                       , 
                       
                         
                           1 
                           ⁢ 
                           … 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             2 
                             N 
                           
                         
                         - 
                         1 
                       
                     
                     } 
                   
                   . 
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 7 
               
             
           
         
       
     
   
   The summand of Eq. 7 can be replaced with a dimensionless indexing variable, k, 
   where k represents any value of the set of integers corresponding to the 2 N  possible combinations of states for the N switchable elements in module m. Substituting Eq. 8 into Eq. 6 gives 
   
     
       
         
           
             
               
                 
                   
                     
                       F 
                       m 
                     
                     ⁡ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                   ⁢ 
                   
                     | 
                     
                       
                         k 
                         = 
                         0 
                       
                       , 
                       
                         
                           1 
                           ⁢ 
                           … 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             2 
                             N 
                           
                         
                         - 
                         1 
                       
                     
                   
                 
                 = 
                 
                   
                     
                       ( 
                       
                         
                           k 
                           
                             Δ 
                             m 
                           
                         
                         + 
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             1 
                             
                               f 
                               
                                 m 
                                 , 
                                 n 
                               
                               0 
                             
                           
                         
                       
                       ) 
                     
                     
                       - 
                       1 
                     
                   
                   . 
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 9 
               
             
           
         
       
     
   
   Further, substituting Eq. 8 into Eq. 9 gives an expanded form for the module focal length 
   
     
       
         
           
             
               
                 
                   F 
                   m 
                 
                 ∈ 
                 
                   
                     { 
                     
                       
                         
                           ( 
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               1 
                               
                                 f 
                                 
                                   m 
                                   , 
                                   n 
                                 
                                 0 
                               
                             
                           
                           ) 
                         
                         
                           - 
                           1 
                         
                       
                       , 
                       
                         
                           ( 
                           
                             
                               1 
                               
                                 Δ 
                                 m 
                               
                             
                             + 
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 1 
                                 
                                   f 
                                   
                                     m 
                                     , 
                                     n 
                                   
                                   0 
                                 
                               
                             
                           
                           ) 
                         
                         
                           - 
                           1 
                         
                       
                       , 
                       
                         
                           
                             ( 
                             
                               
                                 2 
                                 
                                   Δ 
                                   m 
                                 
                               
                               + 
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                   1 
                                   
                                     f 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     0 
                                   
                                 
                               
                             
                             ) 
                           
                           
                             - 
                             1 
                           
                         
                         ⁢ 
                         ⋯ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                       N 
                                     
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                 
                                   Δ 
                                   m 
                                 
                               
                               + 
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     N 
                                     - 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                   1 
                                   
                                     f 
                                     
                                       m 
                                       , 
                                       n 
                                     
                                     0 
                                   
                                 
                               
                             
                             ) 
                           
                           
                             - 
                             1 
                           
                         
                       
                     
                     } 
                   
                   . 
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 10 
               
             
           
         
       
     
   
   From Eq. 10 it can be seen that F m , the effective module focal length for module m, may consist of a set of 2 N  unique focal lengths. 
   In one embodiment, the 0-state focal lengths of all N switchable elements  236  within the same module may be identical. For this case f m,n   0  will be constant for all values of n. Therefore, in the current embodiment, the 0-state focal lengths of all N switchable elements  236  may be expressed in the shortened form
 
f m,n   0 =f m   0   Eq. 11
 
where the second subscript, n, has been dropped for simplicity due to the fact that the 0-state focal length is now independent of the value of n. Generally, however, as described above, the 0-state focal length of the N switchable elements  236  within a module may have any value.
 
   Substituting Eq. 11 into Eqs. 9 and 10 gives simplified forms for F m   
                         F   m     ⁡     (   k   )       ⁢     |       k   =   0     ,       1   ⁢   …   ⁢           ⁢     2   N       -   1           =       (       N     f   m   0       +     k     Δ   m         )       -   1         ,           Eq   .           ⁢   12               
and in expanded form
 
   
     
       
         
           
             
               
                 
                   F 
                   m 
                 
                 ∈ 
                 
                   
                     { 
                     
                       
                         
                           ( 
                           
                             N 
                             
                               f 
                               m 
                               0 
                             
                           
                           ) 
                         
                         
                           - 
                           1 
                         
                       
                       , 
                       
                         
                           ( 
                           
                             
                               1 
                               
                                 Δ 
                                 m 
                               
                             
                             + 
                             
                               N 
                               
                                 f 
                                 m 
                                 0 
                               
                             
                           
                           ) 
                         
                         
                           - 
                           1 
                         
                       
                       , 
                       
                         
                           
                             ( 
                             
                               
                                 2 
                                 
                                   Δ 
                                   m 
                                 
                               
                               + 
                               
                                 N 
                                 
                                   f 
                                   m 
                                   0 
                                 
                               
                             
                             ) 
                           
                           
                             - 
                             1 
                           
                         
                         ⁢ 
                         ⋯ 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           
                             ( 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                       N 
                                     
                                     - 
                                     1 
                                   
                                   ) 
                                 
                                 
                                   Δ 
                                   m 
                                 
                               
                               + 
                               
                                 N 
                                 
                                   f 
                                   m 
                                   0 
                                 
                               
                             
                             ) 
                           
                           
                             - 
                             1 
                           
                         
                       
                     
                     } 
                   
                   . 
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 13 
               
             
           
         
       
     
   
   A module focal power for module m may be introduced as
 
 P   m   =F   m   −1 .  Eq. 14
 
   Substituting Eq. 14 into Eqs. 12 and 13 gives 
                       P   m     ⁡     (   k   )       ⁢     |       k   =   0     ,       1   ⁢   …   ⁢           ⁢     2   N       -   1           =       N     f   m   0       +     k     Δ   m                 Eq   .           ⁢   15               
and in expanded form
 
   
     
       
         
           
             
               
                 
                   P 
                   m 
                 
                 ∈ 
                 
                   
                     { 
                     
                       
                         ( 
                         
                           N 
                           
                             f 
                             m 
                             0 
                           
                         
                         ) 
                       
                       , 
                       
                         ( 
                         
                           
                             1 
                             
                               Δ 
                               m 
                             
                           
                           + 
                           
                             N 
                             
                               f 
                               m 
                               0 
                             
                           
                         
                         ) 
                       
                       , 
                       
                         
                           ( 
                           
                             
                               2 
                               
                                 Δ 
                                 m 
                               
                             
                             + 
                             
                               N 
                               
                                 f 
                                 m 
                                 0 
                               
                             
                           
                           ) 
                         
                         ⁢ 
                         … 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ( 
                           
                             
                               
                                 ( 
                                 
                                   
                                     2 
                                     N 
                                   
                                   - 
                                   1 
                                 
                                 ) 
                               
                               
                                 Δ 
                                 m 
                               
                             
                             + 
                             
                               N 
                               
                                 f 
                                 m 
                                 0 
                               
                             
                           
                           ) 
                         
                       
                     
                     } 
                   
                   . 
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 16 
               
             
           
         
       
     
   
   From Eqs. 12 and 15 it can be seen that F m  and P m  (where k has been dropped from both functions for simplicity) are functions of the indexing variable k. It thus follows from these equations, and from Eqs. 13 and 16, that the module focal length F m , and module focal power P m , of module m are selectable from a set of 2 N  focal lengths for the switchable elements. It also follows that each focal length (or focal power) corresponds to a unique combination of states for the N switchable elements in module m. In this sense, module m is capable of performing a “quantized zoom” function in that its module focal length (or focal power) may be varied between a number of quantized focal lengths (or focal powers). The addition of other optical elements, such as conventional lenses, in the module will affect the F m  and P m  in a fashion that will be understood by those skilled in the art. 
   In another embodiment of the invention the 0-state focal length of each of the N switchable elements in module m may be fixed at a distance of infinity,
 
f m   0 =∞mm.  Eq. 17
 
   Substituting Eq. 17 into Eqs. 12, 13 and 16 gives the following expressions for module focal length and module focal power. For the focal length, 
                         F   m     ⁡     (   k   )       ⁢     |       k   =   0     ,       1   ⁢   …   ⁢           ⁢     2   N       -   1           =       Δ   m     k       ,           Eq   .           ⁢   18               
and in expanded form
 
                     F   m     ∈     {         Δ   m     0     ,       Δ   m     1     ,         Δ   m     2     ⁢   …   ⁢           ⁢       Δ   m         2   N     -   1           }       ;           Eq   .           ⁢   19               
and, in terms of focal power
 
                         P   m     ⁡     (   k   )       ⁢     |       k   =   0     ,       1   ⁢   …   ⁢           ⁢     2   N       -   1           =     k     Δ   m         ,           Eq   .           ⁢   20               
and in expanded form
 
   
     
       
         
           
             
               
                 
                   
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   Referring now to  FIG. 3 , a digital focus lens system according to another embodiment of the present invention is shown and indicated generally at  700 . Lens system  700  includes an optical module  710 . Although only a single module  710  is shown in the figure, any number of modules may be incorporated in the system  700 . Module  710  has an input face  712  for receiving input light generally indicated by  714  directed into module  710 . Light  714  may coherent or incoherent, and may originate from light sources including without limitation, light emitting diodes (LEDs), spatial light modulators, scanners, lasers, light bulbs, natural lighting (for example, sunlight), images (such as those generated by such LED or liquid crystal arrays or other optical systems such as telescopes, displays or microscopes). Similarly, module  710  has an output face  716  for emitting output light generally indicated at  718  which has been transmitted through module  710 . Module  710  comprises an optical element stack  719 . Stack  719  includes a number of optical elements, preferably in generally close proximity and orientation to one another such that the standard analytic approximations well known in the field of optics may apply. Such approximations include without limitation, thin lens and paraxial approximations. 
   Stack  719  may comprise a first sub-stack  720 . First sub-stack  720  may comprise a number of non-switchable elements  740 ,  742 . While only two non-switchable elements  740 ,  742  are shown in  FIG. 3 , any number of non-switchable elements may be incorporated. Preferably, each of the non-switchable elements  740 ,  742  comprises a number of optical elements which are capable of performing the functions of a thin lens, however, any refractive, diffractive, reflective or other conventional optical elements for the modulation of phase, frequency and/or amplitude of electromagnetic radiation may be employed. For example, non-switchable elements  740 ,  742  may include without limitation, mirrors, lenses, diffraction gratings, prisms, polarizers, faraday rotators, biaxial crystals, optical films and coatings, optical gain media, nonlinear optical materials, spatial filters, wavelength-selective filters, holographic optical elements, or other conventional on-axis or off-axis optical elements. Preferably, each of the non-switchable elements  740 ,  742  are capable of performing phase modulation functions similar to that of a lens, and will have a specific F#, optical axis  748 ,  749 , and focal length. Some or all of the focal lengths of non-switchable elements  740 ,  742  may be identical or unique from the others. Preferably, the optical axes  748 ,  749  are collinear. Alternatively, non-switchable elements  740 ,  742  may comprise a single non-switchable element  760 . 
   Lens stack  719  may further comprise a second sub-stack  730 . Second sub-stack  730  may comprise a stack of switchable elements, indicated at  750 ,  752 ,  754 . While only three switchable elements  750 ,  752 ,  754  are shown, any number of switchable elements may be incorporated. Preferably, each switchable element  750 ,  752 ,  754  comprises a variable focal length- or switchable-lens, however, any switchable refractive, diffractive, reflective or other optical elements for the modulation of phase, frequency and/or amplitude of electromagnetic radiation may be employed. As discussed previously, examples of switchable elements include without limitation liquid crystals (LCs), holographic optical elements, polymer-dispersed liquid crystals, nonlinear optical lenses, electro-optic elements, electro-optic lenses, LC lenses, LC prisms, LC gratings, LC shutters, LC aperture stops, LC irises, polymer dispersed liquid crystals, switchable holographic optical elements (HOEs), polarization rotators, isotropic, uniaxial, biaxial and/or other anisotropic optical materials, deformable mirrors and deformable gratings, and micro-electro-mechanical systems (MEMS) and MEMS mirrors. The number of switchable elements  750 ,  752 ,  754  and the number of non-switchable elements  740 ,  742  may be identical or different. Preferably, each of the switchable elements  750 ,  752 ,  754  are capable of performing phase modulation functions similar to that of a number of lenses, and will have specific F#&#39;s, optical axes, and focal lengths. Each of the switchable elements  750 ,  752 ,  754  may be switched between at least a specific first state (“0-state”) and a specific second state (“1-state”). 
   While only two states (0-state and 1-state) are discussed here for each of switchable elements  750 ,  752 ,  754 , any number of states may be employed. Preferably, for each of the switchable elements  750 ,  752 ,  754  the 0-state corresponds to a specific first focal length (0-state focal length) having a 0-state optical axis  758 ,  759 ,  760  and 0-state F#. Likewise, for each of the switchable elements  750 ,  752 ,  754  the 1-state corresponds to and a specific second focal length (1-state focal length) having a 1-state optical axis  762 ,  763 ,  764  and a 1-state F#. Preferably, the 0-state focal lengths for the switchable elements  750 ,  752 ,  754  are identical and at a distance of infinity. However, any 0-state focal lengths may be employed by switchable elements  750 ,  752 ,  754 . Each of the switchable elements  750 ,  752 ,  754  will preferably have a specific F#, optical axis  758 , and focal length for each of the 0-states and 1-states. Preferably, 1-state focal length for each of switchable elements  750 ,  752 ,  754  will be unique and will follow the relation similar to that described in Eq. 5 above. Preferably, the optical axes  748 ,  749 ,  758 ,  759 ,  760 ,  762 ,  763 ,  764  are collinear. Preferably, first sub-stack  720  and second sub-stack  730  are preferably of nominal thickness and in close contact with one another such that approximations, well known in the field of optics, including without limitation the thin-lens-close-contact approximations may apply to elements in both stacks  720 ,  730 . Preferably, switchable elements  750 ,  752 ,  754  may be activated in any combination of 1-states and 0-states simultaneously. In this fashion, module  710  may have a module focal length generally corresponding to the inverse of the sum of inverse focal lengths of switchable elements  750 ,  752 ,  754 . It also follows that the module focal length will be selectable from a prescribed set of possible values. Preferably, the module focal length will follow relations similar to those described in Eqs. 3, 4, 6, 9, 10, 12, 13, 18 and 19 above. 
   Switchable elements  750 ,  752 ,  754  are connected to control cable  770 . Control cable  770  connects to controller  772  which provides energy (such as voltage, current, or charge) and control signals for activating and selecting the states of switchable elements  750 ,  752 ,  754 . Preferably, each of switchable elements  750 ,  752 ,  754  are fabricated and arranged such that each of the 1-state focal lengths serves to focus light  718  at a corresponding focal point which is at a unique distance from output face  716 . When input light  714  is generally collimated, or originates from a light source an approximately infinite distance from input face  712 , output light  718  will be focused at a point located at a distance from output face  716  approximately equal to the module focal length. 
   The following examples will use the previously discussed notation F m (k) to describe the module focal length; the subscript m indicates the module number, and the number “k” in parentheses specifies the index number for the module focal length that has been selected from the set of possible values (see Eqs. 8, 9, 10, 12, 13, 18 and 19, above). For example, in the preferred embodiment, when all elements  750 ,  752 ,  754  are activated in the 0-states, module  710  will have a module focal length F m (0) and the transmitted light may be focused at a point A located a generally infinite distance from output face  716  and indicated by ray  780 . 
   Alternatively, however, system  700  may be configured such that point A is located at a finite distance from output face  716 . When element  750  is in the 1-state and elements  752 ,  754  are in the 0-states, the module focal length will correspond to F m (1) and light  718  may be focused at a point B. When element  752  is in the 1-state and elements  750 ,  754  are in the 0-states, the module focal length will correspond to F m (2) and light  718  may be focused at a point C. When elements  750 ,  752  are in the 1-states and element  754  is in the 0-state, the module focal length will correspond to F m (3) and light  718  may be focused at a point D. When element  754  is in the 1-state, and elements  752 ,  754  are in the 0-states, the module focal length will correspond to F m (4) and light  718  may be focused at a point E. When elements  750 ,  754  are in the 1-states and element  752  is in the 0-state, the module focal length will correspond to F m (5) and light  718  may be focused at a point F. When elements  752 ,  754  are in the 1-states and element  750  is in the 0-state, the module focal length will correspond to F m (6) and light  718  may be focused at a point G. 
   Finally, when all elements  750 ,  752 ,  754  are activated in the 1-states and no elements are in the 0-states, the module focal length will correspond to F m (7) and light  718  may be focused at a point H. Controller  772  may also provide signals such that any portion of switchable elements  750 ,  752 ,  754  are activated simultaneously in any combination of states. In this fashion, elements  750 ,  752 ,  754  may perform any combination of 0-state and 1-state optical functions simultaneously. Further, the relative portions of light  714 ,  718  that is modified by 0-states and 1-states of elements  750 ,  752 ,  754  may be determined by controller  772 . In this fashion, the module  710  may simultaneously have a plurality of module focal lengths and light  718  may be focused simultaneously at combinations of points A, B, C, D, E, F, G, and H. The module focal length can also be expressed in terms of a module focal power, P m (k), similar to the relationships described in Eqs. 14, 15, 16, 20, 21 above. 
   The module focal power, P m (k), may be selected from a set of values that are determined by the combination of states of elements  750 ,  752 ,  754 . The possible values for P m (k) comprise a sequence of values, similar to the relation described in Eq. 20; in this fashion, the value (or, “state”) of the module focal power is a linear function of the value of k. It follows from this that, since k is a value (or “state”) indicating the combination of states of the switchable elements, the module focal power is thus a function of the combination of states of the switchable elements. 
   Referring now to  FIG. 4 , a stack of switchable elements according to an embodiment of the present invention is shown and indicated generally at  800 . Stack  800  includes switchable elements, generally indicated at  750 ,  752 ,  754 . While three elements  750 ,  752 ,  754  are shown, any number may be employed in stack  800 . Elements  750 ,  752 ,  754  may each include a liquid crystal lens interposed between substrates  832 ,  834 ,  836 ,  838 . Substrates  832 ,  834 ,  836 ,  838  are at least partially transparent to light  820 ,  830  transmitted through stack  800 . Substrates  832 ,  834 ,  836 ,  838  may comprise glass, plastic, acrylic resin, polymer, crystal, thin films or other materials known to provide a structure for layered electro-optic devices. Substrates  832 ,  834 ,  836 ,  838  each have a first substrate surface and a second substrate surface  839  and  840 ,  842  and  844 ,  846  and  848 ,  850  and  852 , respectively. At least a portion of substrate surfaces  839 ,  840 ,  842 ,  844 ,  846 ,  848 ,  850 ,  852  can include an antireflection coating as may be desirable for minimizing the loss of light  820 ,  830  transmitted through stack  800 . At least a portion of substrate surfaces  840 ,  842 ,  844 ,  846 ,  848 ,  850  are deposited with a generally transparent electrical conductors such as indium tin oxide or conducting polymer. Deposited on the conductive substrate surfaces  840 ,  842 ,  844 ,  846 ,  848 ,  850  are lens function layers  860 ,  862 ,  864 ,  866 ,  868 ,  870 . Lens function layers  860 ,  862 ,  864 ,  866 ,  868 ,  870  may consist of materials that may patterned and include without limitation polymer, epoxy, polymer-dispersed liquid crystal, poly (methyl methacrylate) (PMMA) or photoresist. Lens function layers  860 ,  862 ,  864 ,  866 ,  868 ,  870  are at least partially transparent to light  820 ,  830  transmitted through stack  800 . A portion of the lens function layers  860 ,  862 ,  864 ,  866 ,  868 ,  870  also are patterned such that the thickness, index of refraction, transmittance, scattering, absorption or other optical property of each layer spatially varies, and, in turn, may perform a phase, amplitude and/or frequency modifying function on light transmitted through the layers. Lens function layers  860 ,  862 ,  864 ,  866 ,  868 ,  870  may be patterned using techniques that include without limitation as optical lithography, electron-beam lithography, UV light exposure, holographic, laser or other interferometry, or contact pattern transfer from a patterned substrate to a portion of the lens function layers. Preferably, lens function layers  860 ,  862 ,  864 ,  866 ,  868 ,  870  are patterned with a lens function including without limitation, the optical properties of lenses such as thin, thick, Fresnel, concave, convex, binary, diffracting, aspheric, on-axis, off-axis, cylindrical, holographic and other lenses. Lens function layers  860 ,  862 ,  864 ,  866 ,  868 ,  870  may also include alignment grooves or additional alignment layers to provide a desired orientation or alignment of liquid crystal monomers. Lens function layers  860 ,  862 ,  864 ,  866 ,  868 ,  870  are preferably separated by spacers  880 ,  881 ,  882 ,  883 ,  884 ,  885 . Spacers  880 ,  881 ,  882 ,  883 ,  884 ,  885  serve to provide cells  890 ,  892 ,  894  between adjacent pairs of layers  860 ,  862 ,  864 ,  866 ,  868 ,  870 , and may comprise such materials as Mylar, photoresist, glass fiber, glass or plastic spheres or other films or materials of generally uniform or controlled thickness. At least a portion of cells  890 ,  892 ,  894  are filled with liquid crystal fluid  900 ,  902 ,  904 . Liquid crystal  900 ,  902 ,  904  may include without limitation one or more of a liquid crystal material, liquid crystal, doped liquid crystal, doped liquid crystal material, a nematic liquid crystal, a nematic liquid crystal material, a smectic liquid crystal, a smectic liquid crystal material, a ferroelectric liquid crystal, a ferroelectric liquid crystal material or a polymer dispersed liquid crystal material. Conductor surfaces  840 ,  842 ,  844 ,  846 ,  848 ,  850  are connected to control cables  910 ,  912 ,  914 . Control cables  910 ,  912 ,  914  are connected to controller  772 . Controller  772  provides voltage to control cables  910 ,  912 ,  914  and provides electric fields across pairs of conducting surfaces  840 ,  842 ,  844 ,  846 ,  848 ,  850  which control the molecular orientation of liquid crystal  900 ,  902 ,  904 . 
   By way of example, switchable elements  750 ,  752 ,  754  may be configured similar to conventional nematic liquid crystal cells having parallel homogeneous alignment. Considering element  750  when no electric field is applied across conducting surfaces  840 ,  842 , molecules of liquid crystal  900  are aligned such there exists a first refractive-index mismatch between liquid crystal  900  and layers  860 ,  862 . This first refractive-index mismatch results in element  750  having a first focal length (0-state focal length) for light  820  of a specific polarization. In the presence of an electric field across conducting surfaces  840 ,  842 , molecules of liquid crystal  900  are aligned such there exists a second refractive-index mismatch between liquid crystal  900  and layers  860 ,  862 . This second refractive-index mismatch results in element  750  having a second focal length (1-state focal length) for light  820  of a specific polarization. Similarly, element  752  will have a 0-state focal length with no electric field applied across conducting surfaces  844 ,  846 , and will have a 1-state focal length in the presence of an electric field. Likewise, element  754  will have a 0-state focal length with no electric field applied across conducting surfaces  848 ,  850 , and will have a 1-state focal length in the presence of an electric field. Additional switchable elements  920  may be included in stack  800 . Alternately, the 0-state and 1-state focal lengths Nay correspond to the presence and absence of electric fields, respectively. The relationship between the state of focal length and the absence, or presence, of the applied electric field may depend on factors including without limitation, orientation of alignment grooves, types of liquid crystal, refractive indexes of lens function layers, the amplitude, frequency and direction of displacement fields in the liquid crystal, amplitude and frequency of applied electric fields and voltage potentials across the cells and the polarization of light  820 . Additional elements  920  may include without limitation liquid crystal lenses similar to those described above, polarizers, liquid crystal—or other—variable apertures or field stops, tunable color filters, variable polarization rotators and retarders, deformable mirrors and MEMS devices. Further, additional non-switchable elements  760  may be included in stack  800 . Preferably, 0-state focal lengths of each of elements  750 ,  752 ,  754  will be approximately infinity; this may be accomplished, for example, when the lens function layers  860 ,  862 ,  864 ,  866 ,  868 ,  870  are generally index-matched to the extra-ordinary index of the corresponding liquid crystal  900 ,  902 ,  904 . Alternatively, switchable elements  750 ,  752 ,  754  may be configured similar to other liquid crystal configurations, including without limitation twisted or super-twisted nematic liquid crystal cells whereby the focusing properties of the switchable lenses are generally-independent of the polarization of the light  820 . 
   Preferably, 1-state focal lengths of each of elements  750 ,  752 ,  754  will follow relationships similar to the 2 n  relationships described in Eq. 5 above. For example, the 0-state focal lengths of elements  750 ,  752 ,  754  may all be infinite. However, elements  750 ,  752 ,  754  may have 1-state focal lengths with values of Δ m /2 0 , Δ m /2 1  and Δ m /2 2 , respectively, where Δ m  is a constant having the dimension of length. It follows that, in the present example, Δ m  may be equal to the 1-state focal length of element  750  (f m,0   1 ). In this fashion, elements  750 ,  752 ,  754  may have 1-state focal lengths of f m,0   1 , f m,0   1 /2, f m,0   1 /4, respectively. 
   Turning now to  FIG. 5 , an alternative embodiment of the stack of switchable elements is shown and generally indicated at  1000 . The same components as in  FIG. 4  have the same assigned number as in  FIG. 4 . Stack  1000  includes a plurality of switchable elements, indicated generally at  750 ,  752 ,  754 . A first transparent substrate  832  has a first conductive surface  840  that is at least partially coated with an optically transparent, electrically conductive layer such as indium tin oxide. Conductive surface  840  is at least partially deposited with a first lens function layer  860 . First lens function layer  860  may have a number of optical phase- and/or amplitude-modifying functions embedded in it. Additionally, first lens function layer  860  may have alignment grooves or features for providing liquid crystal monomer alignment. First spacers  880 ,  881  are deposited on first lens function layer  860  and have a controlled thickness. Second lens function layer  862  is deposited on spacers  880 ,  881  thereby forming a first cell  890 . Second lens function layer  862  may also have a number of optical phase- and/or amplitude-modifying functions imbedded in it. Second lens function layer  862  is deposited on a second conductive surface  1180 . Second conductive surface  1180  is deposited on a first transparent film  1100 . First transparent film  1100  may be comprised of optically transparent materials including without limitation glass, vinyl-acetate, thin coat sputtered- or evaporated-films, plastic or polymer. First transparent film  1100  may include an optical phase- and/or amplitude-modifying function, such as a lens function, imbedded in it. A third lens function layer  864  is deposited on first transparent film  1100 . Third lens function layer  864  may include a number of optical phase- and/or amplitude-modifying functions imbedded in it. Second spacers  882 ,  883  are deposited on third polymer layer  864 . 
   A fourth lens function layer  866  is deposited on second spacers  882 ,  883  thereby forming a second cell  892 . Fourth lens function layer  866  may include a number of optical phase- and/or amplitude-modifying functions. Fourth lens function layer  866  is deposited on a third conductive surface  1182 . Third conductive surface  1182  is deposited on a second transparent film  1102 . Second transparent film  1102  may be comprised of optically transparent materials including without limitation glass, vinyl-acetate, plastic or polymer. Second transparent film  1102  may include an optical phase modifying function imbedded in it. A fifth lens function layer  868  is deposited on second transparent film  1102 . Fifth lens function layer  868  may include a number of optical phase- and/or amplitude-modifying functions. Third spacers  884 ,  885  are deposited on fifth polymer layer  868 . A sixth lens function layer  870  is deposited on third spacers  884 ,  885  thereby forming a third cell  894 . Sixth lens function layer  870  may include a number of optical phase- and/or amplitude-modifying functions imbedded in it. Sixth polymer layer  870  is deposited on a fourth conductive surface  1184 . Fourth conductive surface  1184  is deposited on a second transparent substrate  1200 . Liquid crystal material  1206 ,  1207 ,  1208  is deposited in cells  890 ,  892 ,  894 , respectively. Liquid crystal may include one or more of a liquid crystal material, liquid crystal, doped liquid crystal, doped liquid crystal material, a nematic liquid crystal, a nematic liquid crystal material, a smectic liquid crystal, a smectic liquid crystal material, a ferroelectric liquid crystal, or a ferroelectric liquid crystal material. 
   Conductor surfaces  840 ,  1180 ,  1182 ,  1184  are connected to control cables indicated generally at  1210 . Control cables  1210  are connected to controller  1220 . Second conducting surface  1180  functions as a common electrode to switchable elements  750  and  752 . Likewise, third conducting surface  1182  functions as a common electrode to switchable elements  752  and  754 . Controller  1220  provides voltages to control cables  1210  such that electric fields formed across elements  750 ,  752 ,  754  are of appropriate modulation, amplitude and sign such that the liquid crystal monomers in cells  890 ,  892 ,  894 , become aligned to desired orientations. In this fashion, switchable elements  750 ,  752 ,  754  function as independently switchable lenses. Also, in this fashion, any number of similar switchable elements may be incorporated in stack  1000 . 
   Preferably the thickness of the optical components in between first substrate  832  and second substrate  1200  is of appropriate thickness, relative to parameters such as the numerical apertures of the lens functions of switchable elements, and the wavelengths of light transmitted through stack  1000 , such that the thin-lens-close-contact approximations, known in the field of geometric optics, can be applied. For example, with no electric field applied across conducting surfaces  1180 ,  1182  and no electric field applied across surfaces  1182 ,  1184 , switchable elements  752 ,  754  are in the 0-states, and hence may function as lenses having infinite focal lengths. With the proper electric field applied across conducting surfaces  840 ,  1180 , switchable element  750  is switched to the 1-state, and hence may function as a lens having a finite focal length, of, for example, f m,0   1 . In this fashion, light  1210  emitted from light source  1238 , and is transmitted through stack  1000 , will be focused at a point B. Under these same conditions, but with an electric field now also applied across conducting surfaces  1182 ,  1184 , liquid crystal monomers  1230  become aligned such that switchable element  754  is switched to the 1-state, and hence may function as a lens having a finite focal length, of, for example, f m,0   1 /4. In this fashion, for example, light input light indicated at  1239  is emitted from light source  1238 . Input light  1239  is transmitted through stack  1000  and is transmitted as light generally indicated as  1250 . In this fashion transmitted light  1250  may therefore be redirected by switchable elements  750 ,  754 , and may be focused at a point F. Generally, in this fashion, for the various combinations of states for the three switchable elements  750 ,  752 ,  754 , given in this example, transmitted light  1250  may be focused at focal points indicated at A, B, C, D, E, F, G, and H. 
   Alternatively, some or all of switchable elements  750 ,  752 ,  754  may be configured such that, with appropriate applied voltages, the focal lengths may be continuously tunable instead of being selectable for a discrete set of focal lengths. For example, such continuously tunable configurations may include without limitation nematic liquid crystal cells in parallel homogeneous alignment and electro-optic lenses. 
   Alternatively, focal points A, B, C, D, E, F, G, and H may comprise focal planes whereby the transmitted light  1250  forms a virtual or real image at focal planes A, B, C, D, E, F, G, and H. While only three switchable elements  750 ,  752 ,  754  are described here, any number of N switchable elements may be incorporated in embodiments of the present invention. In this fashion, the number of selectable focal points may increase proportionally with the function 2 N . 
   Turning now to  FIG. 6 , an alternative embodiment of the stack of switchable elements is shown and generally indicated at  1000 ′. The same components as in  FIG. 5  have the same assigned number as in  FIG. 5 . Stack  1000  includes a plurality of switchable elements, indicated generally at  750 ,  752 ,  754 . A first transparent substrate (first substrate)  832  has a first optically transparent, electrically conductive surface  840 . First conductive surface  840  is at least partially deposited with a first lens function layer  860 . First lens function layer  860  has an optical phase- and/or amplitude-modifying function, such as a lens, prism, grating, or other optical function, imbedded in it and may include without limitation materials such polymer, epoxy, PMMA and photoresist. 
   First spacers  880 ,  881  are deposited on lens function layer  860  and have a controlled thickness. A second lens function layer  862  is deposited on spacers  880 ,  881  thereby forming a first cell  890 . Second lens function layer  862  is deposited on a first electrically conductive substrate  1410 . First electrically conductive substrate (first conductive substrate)  1410  provides both electrical conductivity and structural support to element  750  and to stack  1400  in general. A third lens function layer  864  is deposited on first conductive substrate  1410 . Second spacers  882 ,  883  are deposited on third lens function layer  864 . 
   A fourth lens function layer  866  is deposited on second spacers  882 ,  883  thereby forming a second cell  892 . Fourth lens function layer  866  is deposited on a second electrically conductive substrate  1420 . Second electrically conductive substrate (second conductive substrate)  1420  provides both electrical conductivity and structural support to element  752  and to stack  1400  in general. A fifth lens function layer  868  is deposited on second conductive substrate  1420 . Third spacers  884 ,  885  are deposited on fifth layer  868 . A sixth lens function layer  870  is deposited on third spacers  884 ,  885  thereby forming a third cell  894 . Sixth lens function layer  870  is deposited on a second conductive surface  1184 . Fourth conductive surface  1184  may be deposited on a second transparent substrate (second substrate)  1200 . Generally, in a similar fashion, first substrate  832  may be at least partially electrically conductive, such that first substrate  832  and first conductive surface  840  may be combined into a single substrate (not shown). Likewise, second substrate  1200  may be at least partially electrically conductive, such that second substrate  1200  and second conductive surface  1184  may be combined into a single substrate (not shown). Liquid crystal material  1206 ,  1207 ,  1208  is deposited in cells  890 ,  892 ,  894 , respectively. Conductive surfaces (and conductive substrates)  840 ,  1410 ,  1420 ,  1184  are connected to control cables indicated generally at  1210 . Control cables  1210  are connected to controller  1220 . Alternatively, a portion of lens function layers  860 ,  862 ,  864 ,  866 ,  868 ,  870  may include a partially conductive surface (not shown) or may be coated with a conducting film (not shown) such that the conducting surface or film is in near contact with a portion of the liquid crystal material  1206 ,  1207 ,  1208 . Additionally, a portion of lens function layers  860 ,  862 ,  864 ,  866 ,  868 ,  870  may have alignment grooves, coatings or features for providing liquid crystal monomer alignment. 
   Turning now to  FIGS. 7   a – 7   d , a die-stamping replication method for fabricating the portions of the switchable elements, specifically, the layered structure that includes a substrate, a conductive layer and a lens function layer. The same components as in  FIG. 4  have the same assigned number as in  FIG. 4 . As shown in  FIG. 7   a , a transparent substrate  832  has a first substrate surface  1305 . First substrate surface  1305  has deposited on it an optically transparent, electrically conductive surface (or, conductive layer)  840  such as ITO. Conductive layer  840  may be deposited by sputtering or by other known techniques. Deposited on conductive layer  840  is a lens function layer  860 . Lens function layer  860  may include patternable materials including without limitation polymer, epoxy, photoresist or PMMA. Lens function layer  860  may be spin-coated on conductive layer  840 . Lens function layer  860  may deposited in such a fashion as to have a generally uniform thickness, yet will be soft or viscous for a period of time before it is hardened by baking, exposure to ultraviolet (UV) light or other hardening processes. A die substrate  1300 , is comprised of substrate material that is capable of being patterned or micromachined, including without limitation, glass, plastic, silicon or other substrate materials. Die substrate  1300  has a first die surface  1310 . First die surface  1310  is has a spatially-varying thickness pattern  1320 . While lens function layer  860  is in its soft or viscous state, die substrate  1300  is brought toward it, indicated schematically by an arrow  1330 . 
   Now referring to  FIG. 7   b , die substrate  1300  is brought into contact with lens function layer  860 . In this fashion, first die surface  1310  is stamped ( FIG. 7   b ) onto phase modifying layer  860  so as to transfer an inverse-copy of spatially-varying thickness pattern  1320  into lens function layer  860 . A release agent (not shown), such a silicone spray, may be deposited on one or more of the first die surface  1310  and the lens function layer  860 . The release agent may serve to assist in release the of the die substrate  1300  from the lens function layer  860  in later steps of the process. This arrangement is then subjected to a hardening force  1340 , such as heat that emanates from a heat source (not shown), or from UV light emanating from a UV source (not shown). Hardening force  1340  serves to harden the lens function layer  860 . Now referring to  FIG. 7   c , after lens function layer  860  has been sufficiently hardened, die substrate  1300  (not shown) is removed. Lens function layer  860  will now have stamped into it an inverse-copy of spatially-varying thickness pattern  1350 . With appropriate die, lens function layer  860  can perform phase-modifying functions such as lens functions and other functions including refraction, diffraction, reflection and scattering. Now turning to  FIG. 7   d , this method can be generally repeated using a second substrate surface  1306  of substrate  832 , or using a plurality of substrates (not shown). In this fashion, a second lens function layer  1360 , or a plurality of lens function layers (not shown) can be patterned, each its own specific phase- and/or amplitude-modifying properties. 
   Referring now to  FIG. 8 , a method for controlling the states of the switchable elements of embodiments of the present invention is shown and generally indicated at  1500 . The same components as in  FIG. 3  have the same assigned number as in  FIG. 3 . A signal, generally indicated at S, is generated and provided to controller  772 . Signal S contains information for controlling the states of switchable elements  750 ,  752 ,  754 . Signal S may be either generated either internally or externally to controller  772 . A portion of signal S includes a serial data stream comprising a control word, indicated generally at  1520 . Control word  1520  is digital word having a bit field length of N bits where N may be a number equal to the number of switchable elements  750 ,  752 ,  754  in stack  716 . 
   In the current example ( FIG. 8 ), control word  1520  may comprise a 3-bit field length where the bits are generally indicated at A, B, C, and has the base-two value “101”. However, in general, control word  1520  can have any bit field length and may be comprised of any number of groups of bits. A demultiplexer  1530  serves to demultiplex control word  1520  whereby each of bits A, B, C are routed to a separate port, indicated generally at A′, B′, C′. Each port A′, B′, C′ is connected to additional electronics (not shown) including a voltage source (not shown) that are, in turn, connected to a separate switchable element  750 ,  752 ,  754 . In this fashion, bit A provides a signal for controlling the state of switchable element  750 , bit B provides a signal for controlling the state of switchable element  752 , and bit C provides a signal for controlling the state of switchable element  754 . Thus, control word  1520 , serves to control the states (or the, “combination of states”) of the switchable elements  750 ,  752 ,  754 . As was described in  FIG. 3 , the module focal power (or the state thereof) is a function of the combination of states of the switchable elements. Therefore, the state of the module focal power is a function of the value of control word  1520 . 
   Digital Telescope Lens System 
   Referring now to  FIG. 9 , the digital focus lens system is applied to a two-lens telescope system. It will be seen that the present embodiment of the invention similar to a simple Galilean telescope having digitally variable focal length, or zoom, properties. A digital zoom lens system (system)  400  incorporates a first module  410  having a first focal length, F 1    420 . System  400  further incorporates a second module  430  having a second focal length, F 2    440 . Second module  430  is located a first distance, d 1    450 , from first module  410 . One or more of the first module  410  and second module  430  may incorporate a number of optical elements (elements)  460 ,  464 . One or more of elements  460  may include switchable elements  470 ,  474  and may be activated into a number of states. The optical axes, generally indicated at  480 , of first module  410 , second module  430 , and elements  460 ,  470 ,  474 ,  464  may be generally collinear. Alternatively, the optical axes of first module  410 , second module  430 , and elements  460 ,  470 ,  474 ,  464  may be arranged at any relative orientations. In the present embodiment, one or more of elements  470  are similar to thin lenses in close proximity or in contact with one another, and the thin lens and/or the paraxial approximations may apply to portions of first module  410  and/or second module  430 . 
   A first object  490  and a first image  500  are located at distances s o1    510  and s i1    520 , respectively, from first module  410 . Likewise, a second object  530  and second image  540 , are located at distances s o2    550  and s i2    560 , respectively, from second module  430 . Using the standard lens makers formula, s i1    520  and s i2    560  can be expressed as 
   
     
       
         
           
             
               
                 
                   
                     s 
                     
                       o 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       1 
                     
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             1 
                             
                               F 
                               1 
                             
                           
                           - 
                           
                             1 
                             
                               s 
                               
                                 i 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 1 
                               
                             
                           
                         
                         ) 
                       
                       
                         - 
                         1 
                       
                     
                     ⁢ 
                     and 
                   
                 
                 ⁢ 
                 
                     
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 22 
               
             
           
           
             
               
                 
                   s 
                   
                     o 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                   
                 
                 = 
                 
                   
                     
                       ( 
                       
                         
                           1 
                           
                             F 
                             2 
                           
                         
                         - 
                         
                           1 
                           
                             s 
                             
                               i 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                             
                           
                         
                       
                       ) 
                     
                     
                       - 
                       1 
                     
                   
                   . 
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 23 
               
             
           
         
       
     
   
   Using the definition of s o2    550  similar to that conventionally used in simple two-lens systems
 
 s   i1   ≡d   1   −s   o2 ,  Eq. 24
 
s o1    510  can be expressed in a form similar to that of common two-lens systems
 
   
     
       
         
           
             
               
                 
                   s 
                   
                     o 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                   
                 
                 = 
                 
                   
                     
                       
                         F 
                         1 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               s 
                               
                                 i 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 2 
                               
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   d 
                                   1 
                                 
                                 - 
                                 
                                   F 
                                   2 
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             
                               d 
                               1 
                             
                             ⁢ 
                             
                               F 
                               2 
                             
                           
                         
                         ] 
                       
                     
                     
                       
                         
                           s 
                           
                             i 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                           
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               d 
                               1 
                             
                             - 
                             
                               F 
                               1 
                             
                             - 
                             
                               F 
                               2 
                             
                           
                           ) 
                         
                       
                       + 
                       
                         
                           F 
                           2 
                         
                         ⁡ 
                         
                           ( 
                           
                             
                               F 
                               1 
                             
                             - 
                             
                               d 
                               1 
                             
                           
                           ) 
                         
                       
                     
                   
                   . 
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 25 
               
             
           
         
       
     
   
   A parameter of system  400 , the magnification, M, (or, transverse magnification, M T ), is similar to the magnification of a simple two-lens system, i.e., 
   
     
       
         
           
             
               
                 
                   M 
                   ≡ 
                   
                     M 
                     T 
                   
                 
                 = 
                 
                   
                     
                       - 
                       
                         s 
                         
                           i 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                         
                       
                     
                     
                       s 
                       
                         o 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                     
                   
                   · 
                   
                     
                       
                         - 
                         
                           s 
                           
                             i 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                           
                         
                       
                       
                         s 
                         
                           o 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                         
                       
                     
                     . 
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 26 
               
             
           
         
       
     
   
   Substituting for s i1    520  and s o2    550 , M can now be expressed as 
                   M   =         F   2     ⁢     F   1             s     o   ⁢           ⁢   1       ⁡     (       d   1     -     F   1     -     F   1       )       +       F   1     ⁡     (       F   2     -     d   1       )             ,           Eq   .           ⁢   27               
again, similar to the transverse magnification for standard two-lens systems.
 
   Now, for a zoom lens system such as a telescope or telephoto lens, it may be desirable for s i2    560  to be a generally fixed distance from second module  430  while s o1    510  is variable over a specified range of distances from first module  410 . In this fashion, a system focal length  570 , given as the distance between s o1    510  and S i2    560 , is a variable. However, it is often difficult to construct such a system in which M, s i2    560  and d 1    450  are all constant while s o1    510  is variable. In a present embodiment of the invention, a telescope based on combinatorial optics is enabled in which M, s i2    560  and d 1    450  are constant while s o1    510  is variable. 
   To accomplish this, for example, F 1    420  may be constant and F 2    440  may be variable and expressed as a function of k, i.e., F 2 (k), and given a form similar to that described in Eq. 12, 
                         F   2     ⁡     (   k   )       ⁢     |       k   =   0     ,       1   ⁢   …   ⁢           ⁢     2   N       -   1           =       (       N     f   2   0       +     k     Δ   m         )       -   1         ,           Eq   .           ⁢   28               
where, for this example, the 0-states of the N switchable elements  474  of the second module  430  are identical. Substituting Eq. 28 into Eq. 27 gives an expression for M as a function of the variable k, i.e., M(k),
 
   
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           M 
                           ⁡ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                         ⁢ 
                         
                           | 
                           
                             
                               k 
                               = 
                               0 
                             
                             , 
                             
                               
                                 1 
                                 ⁢ 
                                 … 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   2 
                                   N 
                                 
                               
                               - 
                               1 
                             
                           
                         
                       
                       = 
                         
                       ⁢ 
                       
                         ( 
                         
                           
                             
                               
                                 
                                   - 
                                   
                                     N 
                                     
                                       f 
                                       2 
                                       0 
                                     
                                   
                                 
                                 - 
                                 
                                   k 
                                   
                                     Δ 
                                     m 
                                   
                                 
                                 + 
                               
                             
                           
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         
                                           d 
                                           1 
                                         
                                         ⁢ 
                                         N 
                                       
                                       
                                         f 
                                         2 
                                         0 
                                       
                                     
                                     - 
                                     
                                       
                                         
                                           
                                             f 
                                             0 
                                             2 
                                           
                                           ⁢ 
                                           
                                             Δ 
                                             m 
                                           
                                         
                                         - 
                                         
                                           
                                             d 
                                             1 
                                           
                                           ⁢ 
                                           
                                             kf 
                                             2 
                                             0 
                                           
                                         
                                       
                                       
                                         
                                           f 
                                           2 
                                           0 
                                         
                                         ⁢ 
                                         
                                           Δ 
                                           m 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   1 
                                   
                                     F 
                                     1 
                                   
                                 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
                 
                   
                     
                         
                       ⁢ 
                       
                         
                           s 
                           
                             i 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                           
                         
                         + 
                         1 
                         - 
                         
                           
                             
                               d 
                               1 
                             
                             
                               F 
                               1 
                             
                           
                           . 
                         
                       
                     
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 29 
               
             
           
         
       
     
   
   Setting the first derivative of M(k) with respect to k equal to 0, i.e., 
                       ∂   M     ⁢           ⁢     (   k   )         ∂   k       =   0           Eq   .           ⁢   30               
a solution is found for F 1    420   F 1 =d 1 .  Eq. 31 
   Substituting Eqs. 31 and 28 into Eq. 25 gives 
   
     
       
         
           
             
               
                 
                   s 
                   
                     o 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                     ⁢ 
                     
                       ( 
                       k 
                       ) 
                     
                   
                 
                 = 
                 
                   
                     
                       ( 
                       
                         
                           
                             - 
                             k 
                           
                           
                             Δ 
                             m 
                           
                         
                         + 
                         
                           1 
                           
                             s 
                             i2 
                           
                         
                         - 
                         
                           N 
                           
                             f 
                             2 
                             0 
                           
                         
                       
                       ) 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       d 
                       1 
                       2 
                     
                   
                   + 
                   
                     
                       d 
                       1 
                     
                     . 
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 32 
               
             
           
         
       
     
   
   Substituting Eq. 31 into Eq. 29, gives and M as constant, 
   
     
       
         
           
             
               
                 M 
                 = 
                 
                   - 
                   
                     
                       
                         s 
                         i2 
                       
                       
                         d 
                         1 
                       
                     
                     . 
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 33 
               
             
           
         
       
     
   
   An object separation constant, δs o1    580 , having the dimension of length, can be defined as the derivative of s o1    510  with respect to k, i.e., 
   
     
       
         
           
             
               
                 
                   
                     δ 
                     
                       S 
                       
                         o 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                     
                   
                   ≡ 
                   
                     
                       ∂ 
                       
                         s 
                         
                           o 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                           ⁢ 
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                     
                     
                       ∂ 
                       k 
                     
                   
                 
                 = 
                 
                   - 
                   
                     
                       
                         d 
                         1 
                         2 
                       
                       
                         Δ 
                         m 
                       
                     
                     . 
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 34 
               
             
           
         
       
     
   
   An initial object plane s o1(0)    590 , i.e., the value of s o1(k)  for k=0, can be expressed as 
   
     
       
         
           
             
               
                 
                   δ 
                   
                     o 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       ( 
                       0 
                       ) 
                     
                   
                 
                 = 
                 
                   
                     d 
                     1 
                   
                   - 
                   
                     
                       Nd 
                       1 
                       2 
                     
                     
                       f 
                       2 
                       0 
                     
                   
                   + 
                   
                     
                       
                         d 
                         1 
                         2 
                       
                       
                         s 
                         
                           i 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           2 
                         
                       
                     
                     . 
                   
                 
               
             
             
               
                 Eq 
                 . 
                 
                     
                 
                 ⁢ 
                 35 
               
             
           
         
       
     
   
   Substituting Eqs. 34 and 35 into Eq. 32 gives 
                       s     o   ⁢           ⁢   1       ⁡     (   k   )       ⁢     |       k   =   0     ,       1   ⁢   …   ⁢           ⁢     2   N       -   1           =       s     o   ⁢           ⁢   1   ⁢           ⁢     (   0   )         +       δ     S     o   ⁢           ⁢   1         ⁢     k   .                 Eq   .           ⁢   36               
and in expanded form
   s   o1 ε{( s   o1(0) ), ( s   o1(0)   +δs   o1 ) . . . ( s   o1(0)   +δs   o1 [2 N −1])}.  Eq. 37 
   From the above discussion, it can be seen that the present embodiment of the invention is similar to a telescope with quantized or “digital” zoom control of the focal length. In particular, system  400  is similar to a Galilean telescope, wherein: F 1    420  and F 2    440  are similar to the field lens and ocular lens, respectively; wherein d 1    450  is similar to the distance separating F 1    420  and F 2    440 ; and wherein M, s o1    510  and s i2    560  are similar to the transverse magnification, object distance and image distance, respectively. However, the present embodiment of the invention has the following distinctive properties: s o1    510  is selectable from a set of quantized locations relative to the location of F 1    420 ; the relative locations of F 1    420  and F 2    440  may be fixed such that d 1    450  may have a constant value for all object distances in the set of s o1 ; s i2    560  and M may both have constant values for all object distances in the set of s o1 ; and the system and its components may be solid state, i.e., comprise no moving parts. 
   Digital Camera Lens System 
   In cases where s i2  has a negative value, the image formed by the system is a virtual image and the system can function similarly to simple two-lens telescope. In this fashion, remote objects may be viewable by the human eye and may not require additional optical elements for viewing such as oculars. However, when s i2  has a positive value, the image formed by the system is a real image and the system may function as an imaging system such as a camera. For example, the system may function as a camera wherein an image sensor (sensor) may be positioned a distance s i2  from F 2 . Such sensors may include, without limitation, CCD arrays, CMOS image sensors or sensor arrays, artificial retinas or conventional photographic film. In this fashion, information, pertaining to the image of an object located at a distance s o1  from F 1 , may be captured by the sensor. 
   Additional optical elements may be in incorporated in all embodiments of the invention in fashions similar to those used commonly in telescopes, cameras and other imaging and non-imaging systems or in other ways that will be understood by those skilled in the art. Examples of such additional optical elements may include without limitation oculars, field lenses, and apertures, stops, partially- or fully-reflective mirrors, prisms, gratings, lenses, and lens complexes. 
   Digital Projector Lens System 
   In another embodiment of the present invention, the system may be configured to function as an image projector. For example, with 2-lens image projectors, generally, an object is located at a distance, s o1 , from an object lens, L 1 ; an image is formed at a distance, s i2 , from an image lens L 2 ; and L 1  and L 2  are separated by a distance d 1 . Similarly to telescopes, for image projectors it is sometimes desirable for both the separation distance between the two optical elements d 1  and the magnification M to have constant values. In the previous preferred embodiment of a digital telescope lens system, s i2  was held constant while s o1  was variable. However, for the present embodiment of a digital projector lens system, it may be desirable that s i2 , be variable while s o1  is held constant. To accomplish this functionality, the previous embodiment of the invention is utilized, however, the system may now be flipped relative to the positions of the object and the image. In this fashion, the F 2 , may be left constant and F 1 , may now be variable and expressed as a function of the variable k, and given a form similar to that described above and in Eq. 12. 
   Digital Microscope Lens System 
   The above discussions described embodiments of the invention that utilize digital focus lens systems for purposes that include the controlling of the position of an image without requiring changes in the magnification of the image, and while simultaneously allowing the system to be solid state. Similarly, however, it may also sometimes be desirable for the system to exhibit properties such as allowing the control of the magnification of an image without requiring changes in the location of the image, and while simultaneously allowing the system to be solid state. 
   For example, for microscopes in general, and specifically for 3-lens microscopes, an object may be located at a distance, s o1 , from a first module. A second module may be located at a first distance d 1  from the first module. A third module may be located at a second distance d 2  from the second module. An image may then be formed at an image distance s i3  from the third module. In the present embodiment of a 3-lens microscope, it may be desirable for d 1 , d 2 , s o1 , and s i3  to all have generally constant values, while it may also be desirable that the magnification, M, may be variable. 
   In the previous embodiments, M and d 1  were held constant while either s o1  or s i2  were variable. However, for the present embodiment of a combinatorial optical microscope, the magnification M will now be variable while s o1 , s i3 , d 1  and d 2 , will be held generally constant. 
   One way to accomplish this functionality incorporates the previous embodiment of the invention, a digital projector lens system and a third module having variable focal power. As in the previous embodiment, the focal length of the second module F 2  may be left constant. Further, the first module may incorporate switchable elements as previously described. In this fashion the focal length of the first module F 1  may be variable and expressed as a function of k, and given the form similar to that described in Eq. 12. 
   A solution may be found for which the first derivative of s i3  with respect to k is equal to zero, i.e., 
   
     
       
         
           
             
               ∂ 
               
                 s 
                 i3 
               
             
             
               ∂ 
               k 
             
           
           = 
           0. 
         
       
     
   
   In this fashion, the distance of the image to the third module will be a constant and independent of the state of variable k. One possible solution to the above condition exists for the case when the focal power of at least one of the modules is continuously variable between two specified values of focal power. The magnification, M, (or, transverse magnification, M T ) of this three-module system may also be similar to the magnification of a common three-lens system, similar to the previous discussion of M for a two-module system, 
   
     
       
         
           
             M 
             ≡ 
             
               M 
               T 
             
           
           = 
           
             
               
                 - 
                 
                   s 
                   
                     i 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                   
                 
               
               
                 s 
                 
                   o 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   1 
                 
               
             
             · 
             
               
                 - 
                 
                   s 
                   
                     i 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     2 
                   
                 
               
               
                 s 
                 
                   o 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   2 
                 
               
             
             · 
             
               
                 
                   - 
                   
                     s 
                     
                       i 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       3 
                     
                   
                 
                 
                   s 
                   
                     o 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     3 
                   
                 
               
               . 
             
           
         
       
     
   
   The desired functionality of M being a variable function of k, M(k), can be achieved for a system utilizing a module, for example, the third module, the focal length, F 3 , of which is a variable function of  k . Ways to achieve this functionality include without limitation the use of electro-optic or liquid crystal lenses or other variable or switchable optical elements that have generally continuously variable focal power. For example, for a material having an r 33  or other electro-optic coefficient, such as lithium niobate, may be polished in the form of a lens. Transparent electrodes, such as indium tin oxide, may be deposited on the surfaces of the lens. An electric field may then be applied from one electrode to the other, across the thickness of the lens. Due to the electro-optic coefficient of the material of the lens, the index of the lens will be a function of the strength of the applied electric field. In this fashion, the focal length of the lens will also be a function of the applied electric field. Similarly, liquid crystal (LC) lenses and gratings, such as modal LCs and LC lenses similar to LC blazed-grating beam deflectors based on nematic LC cells in parallel homogeneous alignment can provide variable focusing of the above form. 
   It will be understood by those skilled in the art of optics that many additional optical elements, components, complexes, etc., may be included in the present invention; those additional elements have been omitted from this discussion for simplicity. 
   While the above is a complete description of the preferred embodiment of the present invention, it is possible to use various alternatives, modifications and equivalents. Therefore, the scope of the present invention should be determined not with reference to the above description but should, instead, be determined with reference to the appended claims, along with their full scope of equivalents. The appended claims are not to be interpreted as including means-plus-function limitations, unless such a limitation is explicitly recited in a given claim using the phrase “means for.”