Patent Publication Number: US-7710658-B2

Title: Zoom lens systems with wavefront coding

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims the benefit of priority to U.S. Provisional Patent Application No. 60/779,712 filed 6 Mar. 2006 and entitled “Zoom Lens Systems With Wavefront Coding”, which is expressly incorporated herein by reference in its entirety. 
    
    
     BACKGROUND OF THE INVENTION 
     Modern zoom lens systems differ from traditional zoom lens systems in that changes in the imaging system are not achieved through motion of optical elements along an optical axis. Rather, the optical properties of particular optical elements in modern zoom systems change through application of voltage, pressure, translation or rotations in planes that are not parallel to the optical axis. One example of the use of variable optical elements in a modern zoom lens system is replacing one lens in the traditional zoom lens system with a variable optical lens at the same physical location. 
     However, modern zoom lens systems that vary through changes in optical properties of one or more optical elements may introduce aberrations that act to limit imaging performance. These aberrations may worsen as the number of optical elements in the imaging system decreases. Aberrations that may limit the performance of these modern zoom systems include, for example, image curvature, chromatic aberration, spherical aberration, astigmatism, coma, and fabrication, assembly and temperature related aberrations. 
     Turning now to the drawings, wherein like components are indicated by like reference numbers throughout the various figures,  FIGS. 1 and 2  illustrate an example of a prior art, four group traditional zoom lens system. In this traditional zoom lens system, movement of optical elements is generally parallel to an optical axis of the system. In a configuration  10  of the traditional zoom lens system as shown in  FIG. 1 , a combination of first through fourth optical elements  12 ,  14 ,  16  and  18  having focal lengths f 1 , f 2 , f 3  and f 4 , respectively, is configured to form a sharp image at an image plane  20 . A dashed line indicates an optical axis  21  of configuration  10 . A marginal ray  22  entering configuration  10  of the traditional zoom lens system is focused at the intersection of image plane  20  and optical axis  21 . 
     Continuing to refer to  FIG. 1 , first and second optical elements  12  and  14  may generally be considered as controlling the magnification of the traditional zoom lens system in configuration  10 , while third and fourth optical elements  16  and  18  may generally be considered as controlling the location of image plane  20 . Optical element  14  may be referred to as a “variator lens” or a “variator,” defined as an optical subsystem that controls magnification. Optical element  16  may be called a “compensator,” defined as an optical subsystem that controls focus. In some cases, particularly in zoom lens systems formed of a small number of optical elements, a given subsystem may simultaneously act as a variator and a compensator. 
     In  FIG. 2 , in an alternative configuration  10 ′ of the traditional zoom lens system, second optical element  14 ′ (e.g., the variator) is moved away from first optical element  12  along optical axis  21  and towards image plane  20 , as indicated by an arrow  24 , so as to effect a magnification change. In order to keep image plane  20  at a fixed location relative to fourth optical element  18 , third optical element  16 ′ (e.g., the compensator) is also moved towards image plane  20  along optical axis  21 , as indicated by an arrow  26 . Through a combination of both of these motions, magnification of alternative configuration  10 ′ of the traditional zoom lens system is altered from that of configuration  10  shown in  FIG. 1 , while the image plane location remains fixed. In other words, a marginal ray  28 , which enters closer to the edge of first optical element  12  than marginal ray  22  of  FIG. 1 , may now focus at the intersection of image plane  20  with optical axis  21 . 
     As illustrated in  FIGS. 1 and 2 , the movement of the variator and the compensator in prior art, traditional zoom lens systems requires space along the optical axis. That is, no other optical element may be located within a space through which second optical element  14  must move to form configuration  10 ′, for example. A similar requirement for space is common to traditional zoom lens systems that require movement of optical elements along the optical axis; consequently, it may be difficult to reduce length of traditional zoom lens systems along the optical axis. 
     Modern zoom lens systems may reduce or eliminate the need for physical movement of optical elements along the optical axis, thus reducing overall length of the system as compared to traditional zoom lens systems. However, limitations of these modern zoom lens systems inhibit further improvements in imaging quality, size and cost. For example, currently available modern zoom lens systems require at least two actuated or variable elements to vary magnification and focus simultaneously. Also, certain modern zoom lens systems require an actuated system to control focus, which requires a certain number of elements to vary in order to keep the image in focus. 
     SUMMARY 
     The present disclosure provides a zoom lens system for imaging incoming rays over a range of ray angles. The incoming rays are characterized by at least phase. The zoom lens system includes an optical axis and is characterized by a plurality of modulation transfer functions (MTFs) corresponding at least to the range of ray angles. The zoom lens system includes an optical group disposed along the optical axis, including at least one variable optical element that has a variable focal length selectable between at least two distinct focal length values. The optical group also includes a wavefront coding element. The wavefront coding element alters at least the phase of the incoming rays, such that the plurality of MTFs corresponding to the range of ray angles, for each one of the two distinct focal length values, are less sensitive to misfocus-like aberrations than the same zoom lens system without the wavefront coding element. 
     In one embodiment, a method for use in a zoom lens system images incoming rays over a range of ray angles. The incoming rays include at least phase. The zoom lens system includes an optical axis and at least one variable optical element that has a variable focal length selectable between at least two distinct focal length values. The zoom lens system is characterized by a plurality of modulation transfer functions (MTFs) corresponding at least to the range of ray angles and the two distinct focal length values. The method includes modifying the phase of the incoming rays, such that the plurality of MTFs corresponding to the range of ray angles, for each one of the at least two distinct focal length values, are substantially similar in shape and in magnitude. 
     In one embodiment, a zoom lens system includes an optical axis. The zoom lens system also includes an optical group disposed along the optical axis. The optical axis, in turn, includes at least one variable optical element exhibiting a variable focal length selectable between at least two distinct focal length values and a wavefront coding (WFC) element. The at least one variable optical element is not translatable along the optical axis. The optical group is also characterized by a plurality of modulation transfer function (MTFs) corresponding to the range of ray angles and the at least two distinct focal length values, and the variable optical element and the WFC element are configured to cooperate with each other such that the plurality of MTFs are substantially similar in shape and in magnitude. 
    
    
     
       BRIEF DESCRIPTION OF DRAWINGS 
       The present disclosure may be understood by reference to the following detailed description taken in conjunction with the drawings briefly described below. It is noted that, for purposes of illustrative clarity, certain elements in the drawings may not be drawn to scale. 
         FIG. 1  is a diagrammatic illustration of one configuration of a prior art, traditional zoom lens system including traditional optics. 
         FIG. 2  is a diagrammatic illustration of an alternative configuration of the prior art, traditional zoom lens system of  FIG. 1 . 
         FIG. 3  is a diagrammatic illustration of one configuration of a prior art, modern zoom lens system including modern, variable optics. 
         FIG. 4  is a diagrammatic illustration of an alternative configuration of the prior art, modern zoom lens system of  FIG. 3 . 
         FIG. 5  is a diagrammatic illustration of one configuration of another example of a prior art, modern zoom lens system including modern, variable optics. 
         FIG. 6  is a diagrammatic illustration of an alternative configuration of the prior art, modern zoom lens system of  FIG. 5 . 
         FIG. 7  is a diagrammatic illustration of a prior art, traditional two lens imaging system, shown here to illustrate the curvature of the image plane in a case where the two lenses exhibit unequal focal lengths and indices of refraction. 
         FIG. 8  is a diagrammatic illustration of an alternative, prior art, traditional two lens imaging system, shown here to illustrate the flattening of the image plane in cases where the two lenses are selected to exhibit equal indices of refraction and focal lengths that are the negative of each other. 
         FIGS. 9 and 10  are diagrammatic illustrations of still another prior art, traditional two lens imaging system, shown here to illustrate the variation in chromatic aberration exhibited by such a system depending on lens parameter selections. 
         FIGS. 11-13  are diagrammatic illustrations of one embodiment of a zoom lens system that utilizes a combination of variable optics with a wavefront coding compensator. 
         FIGS. 14 and 15  are diagrammatic illustrations of another embodiment of a zoom lens system that utilizes a liquid lens variator in combination with a wavefront coding compensator. 
         FIGS. 16 and 17  are diagrammatic illustrations of still another embodiment of a zoom lens system that utilizes a liquid crystal variator in combination with a wavefront coding compensator. 
         FIGS. 18 and 19  are diagrammatic illustrations of an optical arrangement for providing variable optical power by the use of a slidable optical element configuration, in accord with an embodiment. 
         FIGS. 20 and 21  are diagrammatic illustrations of another embodiment of a zoom lens system that utilizes the optical arrangement of  FIGS. 18 and 19  in combination with a wavefront coding compensator. 
         FIGS. 22 and 23  are diagrammatic illustrations of yet another embodiment of a zoom lens system that utilizes a plano/aspheric sliding variator in combination with a wavefront coding compensator. 
         FIGS. 24 and 25  are diagrammatic illustrations of another embodiment of a zoom lens system that utilizes an aspheric/aspheric sliding variator in combination with a wavefront coding compensator. 
         FIG. 26  is a diagrammatic illustration of another embodiment of a zoom lens system that utilizes a rotatable variator in combination with a wavefront coding compensator. 
         FIGS. 27 and 28  are diagrammatic illustrations of another embodiment of a zoom lens system that utilizes a sliding single group variator in combination with a wavefront coding compensator. 
         FIGS. 29 and 30  are diagrammatic illustrations of another embodiment of an improved zoom lens system that utilizes a sliding aperture variator in combination with a wavefront coding compensator. 
         FIGS. 31 and 32  are diagrammatic illustrations of two configurations of a two-group zoom lens system with wavefront coding, in accordance with an embodiment. 
         FIGS. 33-36  are graphical plots of ray intercept curves corresponding to the configurations shown in  FIGS. 31 and 32  but calculated without including the effects of wavefront coding and signal processing. 
         FIG. 37  is a graphical plot of calculated modulation transfer functions corresponding to on- and off-axis rays imaged through the configurations shown in  FIGS. 31 and 32  but without including the effects of wavefront coding and signal processing. 
         FIGS. 38 and 39  are graphical plots of calculated modulation transfer functions as a function of focus shift corresponding to on-axis and off-axis rays imaged through the configurations of  FIGS. 31 and 32 , but not including wavefront coding and signal processing, for a specific spatial frequency value. 
         FIG. 40  is a graphical plot of calculated modulation transfer functions corresponding to on- and off-axis rays imaged through the configurations of  FIGS. 31 and 32 , this time including effects of wavefront coding and signal processing. 
         FIGS. 41 and 42  are graphical plots of calculated modulation transfer functions as a function of focus shift corresponding to on-axis and off-axis rays imaged through the configurations of  FIGS. 31 and 32  for a specific spatial frequency value, this time including the effects of wavefront coding. 
         FIG. 43  is a 3-D mesh representation of calculated linear filter applied in the signal processing used to calculate the graphical plot in  FIG. 40 . 
         FIGS. 44 and 45  are diagrammatic illustrations of two configurations of a three-group zoom lens system with wavefront coding, in accordance with an embodiment. 
         FIGS. 46-49  are graphical plots of ray intercept curves corresponding to the configurations of  FIGS. 44 and 45  but calculated without including effects of wavefront coding and signal processing. 
         FIGS. 50 and 51  are graphical plots of calculated modulation transfer functions corresponding to on- and off-axis rays imaged through the configurations of  FIGS. 44 and 45  but without including effects of wavefront coding and signal processing. 
         FIGS. 52 and 53  are graphical plots of calculated modulation transfer functions as a function of focus shift corresponding to on-axis and off-axis rays imaged through the configurations of  FIGS. 44 and 45 , but not including wavefront coding and signal processing, for a specific spatial frequency value. 
         FIGS. 54 and 55  are graphical plots of calculated modulation transfer functions corresponding to on- and off-axis rays imaged through the configurations of  FIGS. 44 and 45 , this time including effects of wavefront coding but before signal processing. 
         FIGS. 56 and 57  are graphical plots of calculated modulation transfer functions as a function of focus shift corresponding to on-axis and off-axis rays imaged through the configurations of  FIGS. 44 and 45  for a specific spatial frequency value, this time including the effects of wavefront coding. 
         FIGS. 58-69  are calculated point spread functions corresponding to on- and off-axis rays imaged through the configurations of  FIGS. 44 and 45  without and with the effects of wavefront coding and signal processing. 
         FIG. 70  is a 3-D mesh representation of calculated linear filter applied in the signal processing used to calculate the results of  FIGS. 54-55  and  66 - 69 . 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     In the present disclosure, “zoom lens system” and “zoom imaging system” are used interchangeably, and “variable optical element” is intended to encompass optical elements with optical properties (such as, but not limited to, focal length, transmittance, and refractive index) that are modifiable by using techniques such as (but not limited to) application of voltage and/or pressure to one or more of the optical elements, and translation and/or rotation of one or more of the optical elements. 
     The use of certain aspheric optics and signal processing may provide improvements to modern zoom lens systems by mitigating certain limitations. The present disclosure concerns the use of certain aspheric optics to improve the performance, cost, and size of modern zoom lens imaging systems. Such optics and signal processing of the detected blurred images, may reduce or eliminate the effects of certain aberrations. Systems utilizing such aspheric optics and signal processing for wavefront coding are, for example, described in U.S. Pat. No. 5,748,371 (hereinafter, the &#39;371 patent), U.S. Pat. No. 6,873,733 (hereinafter, the &#39;733 patent), U.S. Pat. No. 6,842,297 (hereinafter, the &#39;297 patent), U.S. Pat. No. 6,911,638 (hereinafter, the &#39;638 patent), and U.S. Pat. No. 6,940,649 (hereinafter, the &#39;649 patent), each of which is incorporated herein by reference. The addition of wavefront coding to zoom lens systems may eliminate need for an actuated system to control focus, thereby further reducing the number of elements that need to be varied and, consequently, also reducing the cost and size of such zoom lens systems. It is desirable to achieve a robust zoom lens system with high reliability, low cost, reduced mechanical tolerances, reduced power consumption and reduced sensitivity to environmental factors, such as thermally induced variations and chromatic dependence. 
     As a simple example of a modern zoom imaging system, consider a two group imaging system illustrated in  FIG. 3  and  FIG. 4 . This imaging system has two configurations shown by the two figures. In a configuration  100  shown in  FIG. 3 , an object  110  is imaged by optical groups  111  and  112  onto an image plane  120 . Each one of optical groups  111  and  112  includes, for example, one or more of refractive elements, diffractive elements, holographic elements, and variable optics. A detector (not shown) is located at image plane  120  to detect the imaged object. When object  110  is located on an optical axis  121  of the imaging system (indicated by a dashed line), light rays from object  110  that pass between optical groups  111  and  112  are substantially parallel, as shown. In this case, a focal length f 1  of optical group  111  is equal to a distance D 1  (indicated by a double-headed arrow) between object  110  and a first principal plane  125  of optical group  111 . Similarly, a focal length f 2  of optical group  112  is equal to a distance D 2  between image plane  120  and a second principal plane  127  of optical group  112 . 
       FIG. 4  shows an alternative configuration  100 ′ of the two group imaging system. In configuration  100 ′ of  FIG. 4 , object  110 ′ is at a different position relative to an optical group  111 ′ than the position of object  110  relative to optical group  111  in  FIG. 3 . The position of object  110 ′ requires a change in focal length of optical group  111 ′ in order to clearly image object  110 ′ onto image plane  120  if optical group  112  is the same in  FIG. 4  as in  FIG. 3 . In this case, a focal length f 1 ′ of optical group  111 ′ should be equal to a distance D 1 ′ between object  110 ′ and first principal plane  125  of optical group  111 ′. 
     Still referring to  FIG. 3  and  FIG. 4 , marginal rays from objects  110  and  110 ′, respectively, are denoted as  122  and  122 ′. Marginal rays at the image plane in the configurations of  FIGS. 3 and 4  are denoted as  123 . An overall magnification of the two group imaging system is given by the ratio of a marginal ray angle at the object (θ obj ) to a marginal ray angle at the image plane (θ im ) as measured from optical axis  121 . In comparing configurations  100  and  100 ′, the marginal ray angle at the object in configuration  100 ′ increases to θ′ obj  from θ obj  while the marginal ray angle at the image θ im  remains fixed. As a result, magnification in configuration  100 ′ is greater than that of configuration  100 . That is, by changing one of two optical groups and changing object distance, magnification of the two group imaging system changes without changing location of an image plane. 
     In another situation shown in  FIGS. 5 and 6 , object location does not change. In a configuration  150  of a zoom imaging system of  FIG. 5 , optical groups  151  and  152  image object  110 . Optical group  151  acts as a variator, controlling optical power (e.g., magnification). Optical group  152  acts as a compensator, controlling image plane location (e.g., focus).  FIG. 6  shows an alternative configuration  150 ′ of the same zoom imaging system shown in  FIG. 5 , wherein the properties of optical groups  151 ′ and  152 ′ are different from those of optical groups  151  and  152  of  FIG. 5 , such that marginal ray angle θ obj  at the object remains unchanged while marginal ray angle θ′ im  at the image reduces from θ im  of configuration  150 . In other words, in configuration  150 ′ of  FIG. 6 , object  110  is at the same location relative to optical group  151 ′ as object  110  relative to optical group  151  in  FIG. 5 , but optical group  151 ′ yields a reduced focal length, as indicated by the fact that marginal rays to the right of optical group  151 ′ converge (as compared to being parallel, as shown in configuration  150  of  FIG. 5 ). Thus, optical group  152 ′ of  FIG. 6  exhibits an increased focal length (or, alternatively, decreased optical power) in order to keep the location of image plane  120  fixed relative to the location of optical group  152 ′ (as compared to a location of image plane  120  relative to optical group  152  in  FIG. 5 ). As a result, the combination of variator and compensator in configuration  150 ′ results in a focused image at image plane  120  with an increased overall magnification given by the ratio of θ obj  over θ′ im  as compared to a like ratio for the configuration shown in  FIG. 5 . That is, the overall magnification of configuration  150 ′ of  FIG. 6  increases over that of configuration  150  shown in  FIG. 5  because angle θ′ im  of marginal ray  123 ′ at image plane  120  decreases over the angle θ im  of marginal ray  123 , while the angle θ obj  of marginal ray  122  from the object remains constant. Thus, with a change of focal length of both the variator (i.e., f 1  to f 1 ′) and the compensator (i.e., f 2  to f 2 ′), the magnification of a focused image of an object at a fixed distance changed. Notice that optical groups  151 ′ and  152 ′ did not change in physical location along optical axis  121  relative to object  110  and image plane  120  in  FIG. 6 , as compared to optical groups  151  and  152  in  FIG. 5 , in order to effect the change in magnification. 
       FIGS. 3-6  thus illustrate that changes to focal lengths of certain components in a zoom imaging system may be used to change magnification of the zoom imaging system while simultaneously keeping a resulting image in focus. The use of changes in focal length of optical elements to yield changes in magnification exemplifies the advantage of variation in modern optical elements by modifying, for instance, voltage, pressure, translation or rotation. However, the resulting focal length changes of the zoom imaging system may also affect the quality of the images produced. Optical aberrations that may degrade image quality are also affected by changes in the focal lengths. In practical examples of these zoom imaging systems, certain fundamental optical aberrations vary when focal length of the components within the zoom imaging system change. In particular, two types of fundamental optical aberrations, i.e., field or image curvature and chromatic aberration, may act as the main limitation of the performance of these zoom imaging systems. 
       FIGS. 7 and 8  illustrate one way in which changes in the focal length of optical elements in an exemplary zoom imaging system influences these fundamental optical aberrations.  FIGS. 7 and 8  show two configurations of a traditional, two lens imaging system. The principles used to describe this system also apply to general, multi-element cases. A configuration  170  of the two lens imaging system as shown in  FIG. 7  includes two optical elements  171  and  172  with respective indices of refraction n 1  and n 2 . As indicated by a plurality of rays  180 , an image formed by these two elements lies on a curved image surface  175 . An approximation of the curvature of image surface  175  may be determined by adding an inverse focal length multiplied by index of refraction of optical elements  171  and  172 , or generally: 
               Curvature   =     -       ∑   i     ⁢     1       n   i     ⁢     f   i               ,         
where i is an integer corresponding to the optical elements, n i  is the index of refraction of the i-th optical element, and f i  is the focal length of the i-th optical element. Curvature of image surface  175  is undesirable in certain imaging applications.
 
     An alternative configuration  170 ′ of  FIG. 8  describes a particular configuration of a two lens imaging system in which indices of refraction of optical elements  171 ′ and  172 ′ are equal (i.e., n 1 =n 2 ) and focal lengths f 1 ′ and f 2 ′ of optical elements  171 ′ and  172 ′ are the negative of each other (i.e., f 1 ′=−f 2 ′). In this configuration, a curvature of an image plane  175 ′ may be considered to be essentially zero. Configuration  170 ′ of  FIG. 8  exhibits a particular, effective focal length determined by actual focal lengths of component elements and an element separation d (indicated by a double-headed arrow). If element separation d is fixed, any change in focal length of either one of optical elements  171 ′ and  172 ′ from the relationship f 1 =−f 2  will result in curvature of image plane  175 ′. Consequently, image quality of the two lens imaging system will decrease and will be a function of magnification or zoom position of the imaging system. 
     It is notable that traditional zoom lens systems such as shown in  FIG. 1  require movement of optical elements along optical axis  21  without changing focal length of the individual optical elements and, therefore, do not incur a change in image curvature. An approximate curvature of image field in such traditional zoom lens systems is generally unchanged as the magnification or zoom position is varied. However, change in image curvature may occur in currently available modern zoom lens systems that do not utilize translation of optical elements along the optical axis, such as the systems illustrated in  FIGS. 3-6 . 
     Another phenomenon encountered in modern zoom lens systems that do not utilize translation of optical elements along an optical axis is variation in chromatic aberration. Change in focal length of optical elements, without translating the optical elements along the optical axis, generally changes chromatic aberration exhibited by a system, limiting system performance. 
       FIGS. 9 and 10  illustrate a simple example of a change in chromatic aberration with change in focal length of the optical elements. Although  FIGS. 9 and 10  show two configurations of a two element lens, the principle involved is applicable to imaging systems with a plurality of elements that change in focal length. Referring first to  FIG. 9 , a configuration  190  is a two element lens system with optical elements  191  and  192  in close proximity (e.g., distance between optical elements  191  and  192  is essentially zero). Optical elements  191  and  192  exhibit different focal lengths and, in general, different Abbe or V numbers. As is well known, the V number of a particular optical material describes a change in index of refraction as a function of wavelength for that optical material. The parameters of optical elements  191  and  192  are generally chosen such that a monochromatic image forms at a best focus image plane; however, due to chromatic dependence of the refractive index value of a given material (i.e., variation in refractive index with wavelength), a location of best focus will be a function of wavelength used to form the image. This effect is commonly called chromatic aberration. In  FIG. 9 , light  193  includes red illumination  195  and blue illumination  197 . A best focused image formed with red illumination  195  may be at a red image plane  196 , while a best focused image formed with blue illumination  197  may be at a different, blue image plane  198 . Since index of refraction of a given material is generally a function of wavelength, focal length of individual optical elements may also be functions of wavelength. A change Δf in effective focal length for a given set of optical elements with wavelength may be approximated by a sum of focal length of each optical element divided by its respective V number: 
     
       
         
           
             
               Δ 
               ⁢ 
               
                   
               
               ⁢ 
               f 
             
             = 
             
               
                 ( 
                 
                   
                     
                       f 
                       1 
                     
                     
                       V 
                       1 
                     
                   
                   + 
                   
                     
                       f 
                       2 
                     
                     
                       V 
                       2 
                     
                   
                 
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     An alternative configuration  190 ′ shown  FIG. 10  illustrates a case where focal lengths and V numbers of optical elements  191 ′ and  192 ′ are selected such that a ratio of focal lengths to V numbers of optical elements  191 ′ and  192 ′ are negatives of each other (i.e., f 1 ′/V 1 ′=−f 2 ′/V 2 ′). For this choice of parameters, the change Δf in effective focal length of the set of optical elements, with wavelength (over a wavelength region where the respective V numbers are valid), is approximately zero. Thus, an effective focal length of the set of optical elements may be made approximately independent of wavelength through selection of focal lengths and V numbers, so that both red illumination  195 ′ and blue illumination  197 ′ focus at image plane  198 ′. However, changing focal lengths of the optical elements in configuration  190 ′ (e.g., to change overall magnification of arrangement  190 ′) in unequal proportion may make Δf nonzero, and therefore may reintroduce chromatic aberration such that both red illumination  195 ′ and blue illumination  197 ′ no longer focus at image plane  198 ′. In other words, modern zoom lens systems based on variation of focal lengths of one or more optical elements therein may exhibit variation in chromatic aberration when magnification changes, and provide accordingly reduced image quality. 
     The image degradation issues illustrated in  FIGS. 7 through 10 , as well as other aberrations, may be ameliorated by utilizing wavefront coding, as now described. 
     The zoom systems described in the present disclosure image incoming rays over a range of ray angles. These incoming rays are characterized by at least phase that form the wavefront imaged by the zoom system. Each zoom lens system operates with an optical axis and is characterized by a plurality of modulation transfer functions (MTFs) corresponding at least to the range of ray angles. In each zoom lens system, an optical group is disposed along the optical axis, and includes at least one variable optical element that has a variable focal length selectable between at least two distinct focal length values. The optical group also includes a wavefront coding element. The wavefront coding element alters at least the phase of the incoming rays, such that the plurality of MTFs corresponding to the range of ray angles, for each one of the two distinct focal length values, are less sensitive to misfocus-like aberrations than the same zoom lens system without the wavefront coding element. The MTF corresponding to each of the ray angles, for each one of the at least two distinct focal length values, is substantially similar in shape and in magnitude. 
     One embodiment of a zoom lens system with wavefront coding is illustrated in  FIGS. 11 and 12 . Referring first to  FIG. 11  in conjunction with  FIG. 5 , in a configuration  200 , object  110  is imaged by optical group  151 , acting as a variator, and a wavefront coding (WFC) compensator  202 . In configuration  200  (as in configuration  150  of  FIG. 5 ), light from object  110 , with marginal rays  122 , travels through optical group  151  toward WFC compensator  202 . Light traveling through optical group  151  includes a wavefront as well as marginal rays that are parallel to optical axis  121 . WFC compensator  202  then codes the wavefront of light incident thereon such that a blurred image is formed on a detector  210 . An electronic representation of the blurred image at detector  210  is directed to a digital signal processor (DSP)  215  that forms a final image  220 , which is substantially insensitive to focus related aberrations. DSP  215  may be configured, for example, to remove the blur in the blurred image and/or to format the final image in a suitable manner for a particular task. For example, DSP  215  may format final image  200  for machine or for human viewing. 
     Referring now to  FIG. 12  in conjunction with  FIG. 6 , a configuration  200 ′ is shown, in which focal length of the variator is altered from that shown in configuration  200 ; that is, optical group  151 ′ exhibits an altered focal length f 1 ′ and magnification of the zoom lens system is changed from that shown in configuration  200 . As a result, light emerging to the right of optical group  151 ′ includes a wavefront as well as marginal rays that are not parallel to optical axis  121 , as shown. A WFC compensator  202 ′ also acts to code the wavefront of light incident thereon such that a blurred image forms on detector  210 . An electronic representation of the blurred image is further processed by DSP  215  to form a final image  220 ′ that is essentially insensitive to focus related aberrations. 
     A difference between optical group  151  and optical group  151 ′ is that this same optical group includes a variable optical element that permits change of focal length from f 1  to f 1 ′. A difference between WFC compensator  202  and WFC compensator  202 ′ is that this same optical element may be configured to be variable so as to enable at least coarse adjustment of focus by, for instance, altering the focal length exhibited by the WFC compensator from f 2 , as in configuration  200 , to f′ 2  in configuration  200 ′. Thus, in configurations  200  and  200 ′ shown in  FIGS. 11 and 12 , respectively, both the variator and compensator are variable in order to enable adjustment of both magnification and best focus position. Wavefront coding is cooperatively used to reduce effects of focus related aberrations, such as chromatic aberration as illustrated in  FIGS. 9 and 10 . 
     Another embodiment of a zoom lens system with wavefront coding is shown in  FIG. 13 . Referring to  FIG. 13  in conjunction with  FIGS. 11 and 12 , in a configuration  200 ″, the focal length of the variator is again changed from that shown in configuration  200  of  FIG. 11 ; that is, optical group  151 ″ exhibits an altered focal length f 1 ″ and magnification of the zoom lens system is changed from that shown in configuration  200 . However, unlike configuration  200 ′ of  FIG. 12 , WFC compensator  202  remains unchanged. Consequently, while marginal rays transmitted through WFC compensator  202  in configuration  200 ″ are different from marginal rays  204  of configuration  200  or marginal rays  204 ′ of configuration  200 ′, a focus change that is needed to keep a final image  220 ″ sharp and clear may be accomplished by the combination of fixed WFC compensator  202 , detector  210 , and a digital signal processor (DSP)  215 ″. DSP  215 ″ is, for example, programmed to perform signal processing according to the particular configuration of the zoom system. In one embodiment, DSP  215 ″ acquires electronic feedback about the component parameters of optical group  151 ″ and WFC compensator  202 ; in another embodiment, DSP  215 ″ automatically estimates a configuration or parameters of the zoom lens system from detected images. 
     The examples illustrated in  FIGS. 11 through 13  thus describe two classes of zoom lens systems including wavefront coding: in one class, aberrations resulting from optical element configurations are controlled by utilizing aspheric optics and signal processing of the detected images; in the other class, at least one fewer variable optical element is used to form sharp images while changing magnification, without sacrificing image quality. A focus shift that might be accomplished with the inclusion of one or more additional variable optical element is in effect provided by a fixed WFC compensator ( FIG. 13 ) and signal processing of the resulting images. Nonetheless, both classes of such zoom lens systems may be incorporated into a single zoom lens system; that is, the WFC compensator and signal processing may reduce certain effects caused by changing focus and by image-degrading aberrations of specific lens configurations of the zoom lens system. Therefore, rather than eliminating one or more variable optical elements, a zoom system with wavefront coding may utilize one or more variable optical elements that change in a more coarse fashion (e.g., with less precise control) than would be required without a WFC compensator and signal processing of the images. 
     Specific types of optical elements whose optical characteristics may be changed by variation of parameters such as, but not limited to, voltage, pressure, translation and rotation are described in the context of improved zoom lens systems immediately hereinafter. 
       FIGS. 14 and 15  show a liquid lens utilized as a variator. There are at least two types of liquid lenses available today. One type, currently commercialized by Varioptic Company of Lyon, France, changes shape in accordance with changes in voltage. Another type, available from Rhevision Technology, Inc. of San Diego, Calif., changes shape in accordance with changes in pressure. These liquid lenses are similar (as compared to fixed optics) in that an optical element changes in physical shape, resulting in variation of optical power without requiring movement of the optical element along an optical axis, as in traditional zoom lens systems. However, zoom lens systems constructed with such liquid lenses, even if the liquid lenses exhibit ideal behavior, still suffer from image curvature and/or chromatic aberration as discussed in reference to  FIGS. 7 through 10 . Other aberrations such as spherical aberration, coma, astigmatism, temperature related aberrations, and form errors may also limit image performance of such systems. The inclusion of wavefront coding into liquid lens zoom lens systems may alleviate such errors so as to produce images with improved image quality. 
       FIG. 14  shows one configuration  250  of a liquid lens zoom lens system including wavefront coding. In configuration  250 , a liquid lens  251  acts as a variator, changing the magnification of the zoom lens system. Light transmitted through liquid lens  251  includes a wavefront as well as marginal rays that are substantially parallel to optical axis  121 . A WFC compensator  252  codes the wavefront so that a blurred image forms at detector  210 . WFC compensator  252  may be a fixed element or, alternatively, may include a variable optical element, such as a liquid lens optical element, for controlling a location of an image plane (e.g., an ideal location of detector  210 ). DSP  215  then transforms data representative of the blurred image from detector  210  into a final image  260  that is suitable for human and/or machine viewing, and is substantially insensitive to focus related aberrations. 
     Turning now to  FIG. 15 , an alternative configuration  250 ′ includes a liquid lens  251 ′, which, as compared to liquid lens  251  of configuration  250 , alters magnification of the resulting zoom lens system and alters light transmitted through liquid lens  251 ′ such that marginal rays are not parallel to optical axis  121 , as shown. WFC compensator  252 ′, which may include fixed and/or variable elements, again codes a wavefront so that a blurred image forms at detector  210 . DSP  215 ′ makes the final image suitable for human and/or machine viewing, and insensitive to focus related aberrations. WFC compensators  252  and  252 ′ of configurations  250  and  250 ′, respectively, may be configured, for example, to change in response to changes in the liquid lenses  251  and  251 ′, respectively. DSP  215  and  215 ′ may then perform processing that does or, alternatively, does not depend on the configuration of liquid lenses  251  and  251 ′ and/or WFC compensators  252  and  252 ′. By including WFC compensator  252  or  252 ′ and DSP  215  or  215 ′, one of the variable optical elements in the zoom lens system may be eliminated, simplified or require less precise actuation and/or variation while still alleviating the effects of the aberrations present in the zoom lens system configurations without wavefront coding. 
       FIGS. 16 and 17  show similar configurations as in  FIGS. 14 and 15  except that liquid crystal variators  281 ,  281 ′ are utilized instead of liquid lens variators  251  and  251 ′. Liquid crystal optical components, such as those described in U.S. Patent Application Publication No. 2005/0018127 A1, entitled “Electrically variable focus polymer-stabilized liquid crystal lens” (hereinafter, the &#39;127 application), use voltage or other means to change optical characteristics of a liquid crystal optical element. For example, an effective focal length of a liquid crystal optical element can be controlled by an applied voltage, thereby making the element potentially suitable for use in a modern zoom lens system. Other liquid crystal lenses are described by Ye et al. in “Liquid-crystal lens with a focal length that is variable in a wide range,” Applied Optics, vol. 43, no. 35 (2004), pp. 6407-6412, and by Okada et al. in U.S. Pat. No. 4,904,063 entitled “Liquid crystal lenses having a Fresnel lens.” Use of liquid crystal variators in the zoom lens system may still potentially cause the type of aberrations discussed above in reference to  FIGS. 7-10 , and thus limit performance. For example, any of image curvature, chromatic aberration, spherical aberration, astigmatism, coma, temperature related aberrations and general form errors resulting from the inclusion of a liquid crystal variator may reduce imaging performance. Improvements to these types of systems, through the inclusion of WFC compensators  282 ,  282 ′ and signal processing  215 , as explained below, are in general similar to those shown in  FIGS. 14 and 15 . Fewer variable optical elements, lower electrical power consumption, shorter system length, and lower cost are all improvements that may be achieved in the system shown in  FIGS. 16 and 17  over corresponding systems without WFC compensators  282 ,  282 ′. 
     Considering the configurations of  FIGS. 16 and 17  in detail,  FIG. 16  shows a configuration  280  with a liquid crystal lens  281  acting as a variator for changing magnification of the resulting zoom lens system. In configuration  280 , light transmitted through liquid crystal lens  281  includes a wavefront as well as marginal rays that are substantially parallel to optical axis  121 . A WFC compensator  282  codes the wavefront and, subsequently, forms a blurred image at detector  210 . WFC compensator  282  may be a fixed or a variable optical element—such as another liquid crystal lens, translating optics and/or other optical element—for controlling a location of the image plane (i.e., an ideal location of detector  210 ). DSP  215  then transforms an electronic representation of the blurred image from detector  210  into a final image  290  that is suitable for human and/or machine viewing and that is essentially insensitive to focus related aberrations. 
     In  FIG. 17 , an alternative configuration  280 ′ includes a liquid crystal lens  281 ′, which is modified compared to liquid crystal lens  281  so as to alter magnification of the resulting zoom lens system as well as to alter light transmitted through liquid crystal lens  281 ′ such that marginal rays transmitted therethrough are no longer parallel to optical axis  121 . WFC compensator  282 ′, again either including fixed or variable elements, codes a wavefront of light incident thereon and forms a blurred image at detector  210 . DSP  215 ′ makes final image  290 ′ suitable for a human and/or machine viewing and essentially insensitive to focus related aberrations. 
     In one embodiment, optical components used in a zoom system include fixed elements (e.g., elements with fixed optical properties, but not necessarily fixed positions) that are movable relative to the system in directions that may not be along the optical axis, thereby reducing an amount of length required to form the system.  FIGS. 18 and 19  show an example of one movable optical arrangement for varying optical power. An optical arrangement  300  in  FIG. 18  includes two optical elements configured such that an effective optical power, or focal length, of the arrangement can change between at least two values through a sliding motion of one of the optical elements. In optical arrangement  300 , a fixed and stationary optical element  302  is disposed near an aperture  304  and a positive lens  305 . Stationary optical element  302  includes, for example, two portions: a first portion  306  that exhibits optical power due to a positive curvature surface shape; and a second portion  308  that is essentially a plano or flat surface shape with no optical power, as shown. Optical arrangement  300  also includes a transverse sliding optical element  310  that is configured to be slidable between at least two positions. Sliding optical element  310  has three portions: a first portion  312  that includes a negative curvature; a second portion  314  that is a plano surface with substantially no curvature or optical power; and a third portion  316  that exhibits a positive curvature, as shown. In  FIG. 18 , sliding optical element  310  is shown in a first position, in which first portion  306  and second portion  308  of stationary optical element  302  are aligned with first portion  312  and second portion  314 , respectively, of element  310 , as shown. The combination of aperture  304 , stationary optical element  302  and first and second portions  312  and  314  of sliding optical element  310  provide no optical power on light transmitted therethrough; that is, an optical effect of the negative curvature of portion  312  cancels that of the positive curvature of portion  306 , while each of portions  308  and  314  have no effect. 
       FIG. 19  shows an optical arrangement  300 ′ that includes elements of optical arrangement  300 , but with optical element  310  at an alternative position as compared to its position within optical arrangement  300 . In optical arrangement  300 ′, optical element  310  has been moved by a distance S, which is approximately ½ the diameter of aperture  304 , in a direction transverse to the optical axis of the system (wherein an optical axis is generally defined as being perpendicular to the plane of aperture  304 ) to a second position, relative to its position within optical arrangement  300 . When sliding optical element  310  is in the second position shown in  FIG. 19 , portion  306  of stationary optical element  302  aligns with portion  314  of sliding optical element  310 , and portion  308  of stationary optical element  302  aligns with portion  316  of sliding optical element  310 . As a result, light transmitted through the combination of aperture  304 , stationary optical element  302  and sliding optical element  310  in the second position encounters positive optical power. 
     In other words, the combination of stationary optical element  302  and sliding optical element  310 , in combination with aperture  304 , yield two different values of optical power depending on the sliding motion of sliding optical element  310 . Arrangement in the first position as shown in optical arrangement  300  in  FIG. 18  results in an effective optical power of zero. When sliding optical element  310  is shifted to the second position as shown in  FIG. 19 , the effective curvature of the elements provides positive optical power. Thus, by sliding optical element  310  perpendicularly with respect to the optical axis, effective optical power may be varied from zero to a pre-determined, non-zero value. A thin lens equivalent representation for the configurations of shown in  FIGS. 18 and 19  is a plano/plano element with zero optical power for optical arrangement  300 , and a plano/convex element with positive optical power for optical arrangement  300 ′. In general, optical arrangement  300  may be configured to exhibit a range of optical powers, where the zero optical power configuration of  FIG. 18  is a special case. 
     The aforedescribed sliding optical arrangement may be used to advantage in a zoom system using wavefront coding, as shown in  FIGS. 20 and 21 . In the configurations shown in  FIGS. 20 and 21 , the sliding optical arrangement of  FIGS. 18 and 19  is used as a variator. A configuration  400  of  FIG. 20  corresponds to a zoom lens system in which a position of sliding optical element  310  results in low optical power for optical arrangement  300 . In this case, a WFC compensator  402 , either fixed or dynamic, cooperates with detector  210  and DSP  215  such that a resulting final image  410  of object  110  is essentially insensitive to focus related aberrations. Such aberrations may be, for example, a result of a particular optical configuration of the system, where sufficient aberration control is not possible with a small number of optical elements used in the system or, alternatively, are produced by induced change in optical characteristics of optical arrangement  300  acting as a variator. An alternative configuration  400 ′, as shown in  FIG. 21 , illustrates a situation where a position of sliding optical element  310  provides optical arrangement  300 ′ with a larger optical power than that provided by optical arrangement  300 . 
       FIGS. 22 and 23  illustrate the use of plano/aspheric optical elements in a sliding variator configuration such that the optical power provided by the variator changes continuously with the relative position of the plano/aspheric optical elements. That is, while optical arrangements  300  and  300 ′ provides the selection between two values of optical power by selecting the relative location of sliding optical element  310  between two possible positions, the use of plano/aspheric optical elements provides a continuous range of optical power (e.g., the continuous range is obtained by translating a sliding optical element in a direction perpendicular to optical axis  121 ). 
     Referring first to  FIG. 22  in conjunction with  FIG. 20 , optical arrangement  300  in configuration  400  of  FIG. 20  is replaced, in a configuration  420  of a zoom lens system shown in  FIG. 22 , with an optical arrangement  430 . Optical arrangement  430  includes aperture  304 , as in optical arrangement  300 , but also includes plano/aspheric, first and second optical elements  432  and  434 , which are slidable with respect to each other in directions perpendicular to optical axis  121 . Turning briefly to  FIG. 23 , an alternative configuration  420 ′ of the zoom lens system includes an optical arrangement  430 ′ in which first and second sliding optical elements  432  and  434  have been moved with respect to each other, as compared to their positions in configuration  430 . As a result, optical arrangement  430 ′ provides a different value of optical power as compared to that provided by optical arrangement  430 . Configurations  420  and  420 ′ further include a WFC compensator  442 , detector  210  and DSP  215  that produce a final image  450 . WFC compensator  442  is configured to function in a similar manner to aforedescribed WFC compensators shown, for example, in  FIGS. 11-17  and  20 - 21  and, furthermore, may be customized to cooperate with optical arrangement  430 . 
     The lens configuration shown as optical arrangements  430  and  430 ′ is commonly referred to as an Alvarez lens (see, for example, U.S. Pat. No. 3,305,294). An aspheric surface shape of first and second optical elements of an Alvarez lens may be expressed as a cubic described by the expression:
 
height( y )=α· y   3 ,  (1)
 
     where y is a vertical dimension in the plane of the paper in  FIGS. 22 and 23 , height(y) is the height of the aspheric surface of each sliding optical element as measured from the opposing, plano surface, and α is a constant parameter. 
     The combination of optical elements  432  and  434  as shown in  FIGS. 22 and 23  is effectively equivalent to a combined optical element with second order phase or optical power that approximately varies with an amount of relative shift in position between elements  432  and  434 . That is, the combined phase of elements  432  and  434  may be expressed as:
 
phase( z )=height( y +Δ)−height( y −Δ)=α(6 Δy   2 +2Δ 2 ),  (2)
 
     where z is the dimension along optical axis  121  and Δ is a relative sliding distance between elements  432  and  434 , as shown in  FIG. 23 . As may be seen in Eq. (2), there are two phase terms related to the combination of the first and second sliding optical elements, namely a second order term of y 2  and a constant term. The constant term adds an aberration to the zoom lens system called piston that depends on Δ and, ideally, has substantially no effect on the resulting image. The second order term provides optical power depending on Δ. Variation of constant parameter α acts to increase or decrease the sensitivity of resulting optical power to the relative movement of elements  432  and  434 . Therefore, by shifting elements  432  and  434  with respect to each other, the effective optical power of the combination may be varied. 
     The Alvarez lens (i.e., optical arrangements  430  and  430 ′) in the zoom lens system illustrated in  FIGS. 22 and 23  has both first and second sliding optical elements  432  and  434  moving in opposite directions in a plane perpendicular to optical axis  121  in order to change the optical power provided by the Alvarez lens. If only one of the sliding optical elements is moved, e.g., just second sliding optical element  434 , a linear phase shift results from the shifted combination of the two elements. This phase shift acts to spatially displace the image as a function of the element displacement Δ. This phase shift may be removed if a more complicated form of slidable optical elements is used. For example, a second order term that provides an approximation of optical power, may also be included, so that a combination of shifted optical power terms results in a linear phase shift, depending on the amount of optical power and the element displacement. The phase shift from the optical power term can cancel the phase shift from the cubic term resulting in a combination of elements where only a single element needs to be shifted to yield an Alvarez lens with continuously variable amount of optical power. 
       FIGS. 24 and 25  illustrate a variation of continuously variable sliding optical elements acting as a variator in which only one element slides, but both surfaces of each element are aspheric. Referring to  FIG. 24  in conjunction with  FIG. 20 , optical arrangement  300  in configuration  400  of  FIG. 20  has been replaced, in configuration  460  of a zoom lens system in  FIG. 24 , with an optical arrangement  470 . Optical arrangement  470  includes aperture  304 , as in optical arrangement  300 , but also includes a first aspheric/aspheric element  472  and a sliding, second aspheric/aspheric element  474 . Element  472  includes a front surface  475  and a back surface  476 , both of which have aspheric surface profiles. Element  474  includes a front surface  477  and a back surface  478 , both of which also have aspheric surface profiles. In the example illustrated in  FIGS. 24 and 25 , elements  472  and  474  include optical power on the outer surfaces of the elements (e.g., those surfaces that do not face each other, as shown) and a cubic surface profile on the inner surfaces. That is, surface  475  of element  472  has a negative optical power, and surface  476  of element  472  has a cubic surface profile. Similarly, surface  477  of element  474  has a cubic surface profile, corresponding to the cubic surface profile of second surface  476  of element  472 , and surface  478  of element  474  has a positive optical power. 
     Continuing to refer to  FIG. 24 , in optical arrangement  470  element  472  is configured to remain stationary with respect to aperture  304 . Alternatively, element  472  may also be configured to be slidable with respect to aperture  304 , for example, in a direction perpendicular to optical axis  121 . Element  474  is slidable with respect to aperture  304  and element  472  such that optical arrangement  470  can provide a continuous variation in optical power. Spatially shifting element  474  with respect to element  472  results in changing the effective optical power of optical arrangement  470 . That is, the combination of elements  472  and  474  may be configured to yield a continuous range of optical power with the sliding motion of element  474 , keep optical axis  121  centered with respect to detector  210 , and provide additional optical degrees of freedom for minimizing aberrations. Configuration  460  further includes a WFC compensator  482 , detector  210  and DSP  215  that produce a final image  490 . WFC compensator  482  functions in a similar manner to aforedescribed WFC compensators shown, for example, in  FIGS. 11-17  and  20 - 23  and, furthermore, may be customized so as to be compatible with the specific characteristics of optical arrangement  470 . 
     The slidable configuration of element  474  is illustrated by referring to  FIG. 25  in conjunction with  FIG. 24 . Optical arrangement  470  of  FIG. 24  is shown with elements  472  and  474  completely overlapping each other; that is, they are approximately centered with respect to each other along optical axis  121 . In optical arrangement  470 ′ of  FIG. 25 , element  474  has moved downward with respect to element  472  by a displacement Δ. In  FIG. 24 , optical arrangement  470  provides a minimum optical power configuration, and in  FIG. 25 , optical arrangement  470 ′ provides a larger optical power configuration. 
     Aspheric optical elements may also be configured so that rotating one aspheric element with respect to another causes a change in optical power. One example of this type of rotational element is described in U.S. Pat. No. 4,650,292 to Baker et al. (hereinafter, the &#39;292 patent). 
       FIG. 26  illustrates rotation of an element to effect a change in optical power of an optical arrangement containing two elements. A configuration  500  of a zoom lens system includes an optical arrangement  510 . Referring now to  FIG. 26  in conjunction with  FIGS. 18 and 19 , optical arrangement  510  includes stationary optical element  302 ′ disposed near aperture  304 , as shown in optical arrangements  300 ,  300 ′ of  FIGS. 18 and 19 . Optical element  302 ′ includes a portion that provides negative optical power and a piano portion which does not provide optical power. Optical arrangement  510  further includes a second, rotatable optical element  514  that is rotatable about a rotation axis  516  in either a counterclockwise direction  518  (as indicated by an arrow) or a clockwise direction  518 ′. Element  514  includes an aspheric surface  519  and a piano surface  520 . Alternatively, element  514  may be an aspheric/aspheric element, like elements  472  and  474  shown in  FIG. 24 . Surface curvatures of elements  302 ′ and  514 , and a location of axis  516 , are selected such that when element  514  rotates about axis  516 , optical arrangement  510  provides a variable degree of optical power. Thus, like earlier-described optical arrangements including slidable optical elements, optical arrangement  510  includes movement of an optical element in a plane perpendicular to optical axis  121  so as to provide variable optical power without requiring additional length along the optical axis, as in traditional zoom lens systems. Configuration  500  further includes a WFC compensator  522 , detector  210  and DSP  215  to produce a final image  530 . WFC compensator  522  is configured to function in a similar manner to aforedescribed WFC compensators shown, for example, in  FIGS. 11-17  and  20 - 25  and, furthermore, may be customized to cooperate with optical arrangement  510 . 
       FIGS. 27 and 28  illustrate yet another variation of a sliding element used in a variator of an improved zoom lens. In the case illustrated in  FIGS. 27 and 28 , a single sliding aspheric element  570  changes optical power, without the use of a second optical element. That is, in contrast to the configurations shown in  FIGS. 20-26 , in which at least one of two complementary elements moves with respect to another, configurations  550  and  550 ′ of  FIGS. 27 and 28 , respectively, include only a single, movable optical element  570 . 
     Referring first to  FIG. 27 , optical arrangement  560  includes aperture  304  and a movable optical element  570 , which is translatable with respect to aperture  304 . By a continuous or discrete movement of movable optical element  570 , different portions of element  570 , corresponding to different optical power, are illuminated through aperture  304  such that optical arrangement  560  provides varying optical power depending on a relative position of element  570  with respect to aperture  304 . While  FIG. 27  shows element  570  as including an aspheric surface  572  and a plano surface  574 , element  570  may be configured with other surface contours, such as an aspheric/aspheric combination or a plano/aspheric combination, to achieve a desired optical power variation. 
       FIG. 28  shows configuration  550 ′ including an optical arrangement  560 ′, in which a movable optical element  570 ′ has been slid downward in the plane of the diagram such that a different portion of the aspheric/plano surface contour is illuminated through aperture  304 . While element  570  is shown to be translatable (e.g., as a linear slide) in  FIGS. 27 and 28 , a movable optical element may also be moved in other ways to provide optical power variation (such as, for example, by rotation about a rotation axis, as shown in  FIG. 26 ). Configurations  550  and  550 ′ further include WFC compensators  582  and  582 ′, respectively, detector  210  and DSP  215  to produce final images  590  and  590 ′, respectively, as shown. WFC compensators  582  and  582 ′ are configured to function in a similar manner to aforedescribed WFC compensators shown, for example, in  FIGS. 11-17  and  20 - 26  and, furthermore, may be customized to cooperate with the particular characteristics of optical arrangements  560  and  560 ′. 
     Yet another variation to a sliding optical element for use in zoom systems involves a sliding aperture and aspheric optical element used as a variator. A sliding aperture, such as that described by Togino in U.S. Pat. No. 6,603,608, moves with respect to an aspheric optical element such that only a certain section of the aspheric optical element is illuminated at a time. This movement of the aperture, in turn, results in a discrete or continuously variable optical power for the aperture/aspheric optic combination. The combination of the sliding aperture and aspheric optical element effectively results in an optical arrangement providing varying optical power. 
       FIGS. 29 and 30  illustrate an example of an improved zoom lens system including such a sliding aperture device. A configuration  600  shown in  FIG. 29  includes an optical arrangement  602 , which in turn includes a aperture  604  and an aspheric optical element  606 . By a continuous or discrete movement of aperture  604 , different portions of element  606 , corresponding to different optical powers, are illuminated through aperture  604  such that optical arrangement  602  provides variable optical power depending on a relative position of aperture  604  with respect to element  606 . In configuration  600 , both aperture  604  and element  606  are shown to be centered with respect to optical axis  121  in the plane of the figure, such that optical arrangement  604  provides a first value of optical power to light transmitted therethrough.  FIG. 30  shows a configuration  600 ′, in which aperture  604  in an optical arrangement  602 ′ has been moved (relative to its position in optical arrangement  602 ,  FIG. 29 ) so as to illuminate a different portion of element  606 . In particular, aperture  604  is shown to have been moved upward in the plane of the figure with respect to optical axis  121  such that light traveling through optical arrangement  602 ′ experiences a different, second value of optical power. 
     Continuing to refer to  FIGS. 29 and 30 , while element  606  is shown as including an aspheric surface  608  and a plano surface  610 , an aspheric optical element may alternatively be configured with other surface configurations such as, but not limited to, previously described aspheric/aspheric combinations or plano/aspheric combinations so as to cooperate with aperture  604  to achieve a desired optical power variation. Also, aperture  604  may be slidable in a transverse direction or rotatable about a rotation axis in the plane perpendicular to the optical axis in a manner analogous to the movement of the movable optical elements shown in, for example,  FIGS. 18-28 . Configurations  600  and  600 ′ further include WFC compensators  612  and  612 ′, respectively, detector  210  and DSP  215  to produce final images  620  and  620 ′, respectively. WFC compensators  600  and  600 ′ are configured to function in a similar manner to aforedescribed WFC compensators shown, for example, in  FIGS. 11-17  and  20 - 28 , and furthermore, may be customized to cooperate with the particular characteristics of optical arrangements  602  and  602 ′. 
     Referring again to  FIGS. 11-17  and  20 - 30 , a range of incoming ray angles (i.e., rays from object  110  that are within θ obj ) is imaged by each of the zoom systems; and a wavefront coding element modifies phase of a wavefront represented by the rays such that MTFs over the range of ray angles are similar in magnitude and shape, making the zoom system less sensitive to misfocus like aberrations (as compared to the same zoom system without wavefront coding). 
       FIGS. 31 and 32  illustrate another example of a zoom lens system in accordance with the present disclosure.  FIG. 31  shows a configuration  700  of a two-group zoom lens system including a first optical group  702  (with a focal length f 1 ) and a second optical group  704  (with a focal length f 2 ). Second optical group  704  includes variable optical element  706  and a WFC element  708 . First and second optical groups  702  and  704  are aligned with an optical axis  722 . Configuration  700  is shown as a wide-angle system, configured to receive both on-axis rays  725  and off-axis rays  727 , as shown. First and second optical groups  702  and  704  are configured to image both on-axis and off-axis rays  725  and  727  onto detector  210 . Image data generated by detector  210  is processed at DSP  215  to produce a final image  720 . First and second optical groups  702  and  704  may include a plurality of optical elements including, but not limited to, refractive, diffractive and holographic elements. WFC element  708  may be separate from variable optical element  706 , formed on a surface of element  706  or, alternatively, integrally formed therewith. 
       FIG. 32  shows a configuration  700 ′ that includes a second optical group  704 ′. In configuration  700 ′, variable optical element  706  is modified to form variable optical element  706 ′ having a focal length f 2 ′ that is different from focal length f 2  of element  706 ,  FIG. 31 . The focal length change may be achieved, for instance, by one of the aforedescribed implementations of variable lenses. Optical group  704 ′ also has a WFC element  708 ′ with characteristics that may be the same as, or different from, characteristics of WFC element  708  of optical group  704 . Optical group  704 ′ is also closer along optical axis  722  to first optical group  702  (as indicated by an arrow  730 ) as compared to the positions of optical groups  704  and  702  in configuration  700 . Configuration  700 ′ is suitable for use as a telephoto system which accepts on-axis and nearly on-axis rays  725 ′ (indicated by a dashed ellipse), which are imaged onto detector  210  and processed at DSP  215  so as to form a final image  720 ′. In other words, configurations  700  and  700 ′ illustrate wide-angle and telephoto states, respectively, of a zoom lens system which combines translation of optical group  706 ,  706 ′ along optical axis  722  and focal length variation of variable optical group  704 ,  704 ′. 
     As described above, specific characteristics of WFC element  708  may be varied as well. For example, WFC element  708  may be implemented using an adaptive optics element or a spatial light modulator such that a phase variation effected by WFC element  708  may be varied depending on a configuration of second optical group  704  and/or  704 ′. Also, signal processing performed by DSP  215  in configuration  700 ′ may or may not be the same as that performed in configuration  700 ; signal processing in configuration  700 ′ may be modified to accommodate the changes in second optical group  704 ′. 
     By simultaneously effecting the translation and focal length variation of at least one optical group in the zoom lens system, the zoom lens system illustrated in  FIG. 31  and  FIG. 32  may achieve a range of characteristics, from wide-angle to telephoto, using less movement of optical groups along an optical axis than would be required by a traditional zoom lens system alone, and less focal length variation than would be required by a modern zoom lens system alone. Moreover, WFC element  708  may further provide aberration compensation that is not achievable by traditional or modern zoom lens system configurations. It is noted that any of the previously described improved zoom lens systems, as shown, for example, in  FIGS. 11-17  and  20 - 30 , may be modified to simultaneously implement translation and focal length variation, as exemplified in  FIGS. 31 and 32 , by equipping these abovedescribed systems with a translation mechanism to supplement the focal length variation already shown. 
     Configurations  700  and  700 ′ as illustrated in  FIGS. 31 and 32 , and equivalent configurations without WFC element  708 , were numerically modeled using the following exemplary characteristics. In configuration  700 , the effective focal length exhibited by the combination of groups  702  and  704  is 2.7 mm. Focal length f 1  of group  702  is assumed to be −6.62 mm, and focal length f 2  of group  704  is 3.41 mm. A spacing between principal planes of groups  702  and  704  is 4.33 mm. In configuration  700 ′, group  702  remains stationary and focal length f 1  does not change (still −6.62 mm), while focal length f 2 ′ of group  704 ′ changes to 4.04 mm. Spacing between principal planes of groups  702  and  704 ′ is 1.71 mm, such that an effective focal length exhibited by the combination of groups  702  and  704 ′ is 5.4 mm. A wavelength of light rays is assumed to be 0.55 microns, and detector  210  is assumed to include pixels that are 4 microns square. 
     A specific prescription of the various optical groups is derived from the well-known sag equation: 
                     z   =         cr   2       1   +       1   -       (     1   +   k     )     ⁢     c   2     ⁢     r   2               +       α   1     ⁢     r   2       +       α   2     ⁢     r   4       +       α   3     ⁢     r   6       +       α   4     ⁢     r   8       +       α   5     ⁢     r   10       +       α   6     ⁢     r   12       +       α   7     ⁢     r   14       +       α   8     ⁢     r   16           ,           (   3   )               
where z=sag of a surface, c=surface curvature, k=conic constant, r=radial distance from vertex and α n =aspheric constants. The surfaces are defined as seen by a ray approaching the zoom lens system from the left side of the paper in configurations  700  and  700 ′ of  FIGS. 31 and 32 . Prescriptions used in the numerical modeling of configurations  700  and  700 ′ are given by:
 
     
       
         
           
               
               
               
               
             
               
                   
                 TABLE 1 
               
             
            
               
                   
                   
               
               
                   
                   
                 Second optical group 
                 Second optical group 
               
               
                   
                   
                 704 
                 704′ 
               
               
                   
                 First optical group 
                 Thickness = 1.0 mm 
                 Thickness = 1.0 mm 
               
               
                   
                 702 
                 Back focal distance = 
                 Back focal distance = 
               
               
                   
                 Thickness = 1.0 mm 
                 4.522 mm 
                 7.116 mm 
               
            
           
           
               
               
               
               
               
               
               
            
               
                   
                 Surface 1 
                 Surface 2 
                 Surface 1 
                 Surface 2 
                 Surface 1 
                 Surface 2 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                 c 
                 −5.681E−3 
                 −5.681E−3 
                  1.377E−1 
                 −2.439E−1 
                 1.608E−1 
                 −1.601E−1  
               
               
                 k 
                 7.492 
                 −1.070E+2 
                 −1.045E+2 
                 9.029 
                 1.598E−1 
                 −2.461 
               
               
                 a 2   
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 a 4   
                 −8.973E−3 
                 −1.715E−3 
                  4.458E−3 
                 0 
                 −3.635E−4  
                 1.925E−3 
               
               
                 a 6   
                 −1.965E−3 
                  1.304E−3 
                 −6.332E−3 
                 0 
                 1.050E−3 
                 2.438E−4 
               
               
                 a 8   
                  5.367E−3 
                  1.579E−3 
                 −2.339E−2 
                 0 
                 −7.584E−4  
                 9.016E−5 
               
               
                 a 10   
                 −1.512E−3 
                 −1.298E−3 
                 −4.608E−2 
                 0 
                 1.121E−3 
                 4.987E−4 
               
               
                   
               
            
           
         
       
     
     In order to illustrate the effects of WFC element  708 , attention is first directed to  FIGS. 33-36 , which illustrate pairs of ray intercept curves corresponding to optical arrangements equivalent to configurations  700  and  700 ′ as shown in  FIGS. 31 and 32 , but without WFC element  708 . In all of  FIGS. 33-36  and  FIGS. 46-49 , axis “EY” corresponds to a spatial Y-axis in an image plane (e.g., a location of detector  210 ), axis “PY” corresponds to a spatial Y-axis in a pupil plane (e.g., a first surface of first optical group  702  that is encountered by incoming rays), axis “EX” corresponds to a spatial X-axis in the image plane, and axis “PX” corresponds to a spatial X-axis in the pupil plane. 
     As is known in the art, ray intercept curves may indicate a degree of focus-related aberrations present in a given system. A ray intercept curve is calculated by plotting, for a given ray position on a pupil plane, a corresponding position on an image plane to which that ray images. For example, for a perfectly focused system, the x-axis and y-axis ray intercept curves should be straight, horizontal lines along the EY=0 and EX=0 axes. Deviations from the ideal horizontal line indicate the presence of a variety of aberrations in the system. For example, a tilted, linear, ray intercept curve (i.e., a straight line with a non-zero slope) indicates misfocus; that is, a linearly increasing image plane height with pupil position. Also, if a y-plot and an x-plot exhibit different slopes, then the ray intercept curves indicate astigmatism. Furthermore, if the slope of a ray intercept curve changes as a function of field angle, then that ray intercept curve indicates field curvature. Additionally, if a ray intercept curve is a third order curve, that ray intercept curve indicates spherical aberration. Thus, ray intercept curves may indicate a variety of aberrations present in a given optical system. 
       FIG. 33  shows a y-plot  750  and an x-plot  752  with a y-axis ray intercept curve  754  and an x-axis ray intercept curve  756  corresponding to on-axis rays  725  going through configuration  700  but without WFC element  708 . Similarly,  FIG. 34  shows a y-plot  760  and an x-plot  762  with a y-axis ray intercept curve  764  and an x-axis ray intercept curve  766  corresponding to off-axis rays  727  being imaged through configuration  700  but without WFC element  708 .  FIG. 35  shows a y-plot  770  and an x-plot  772  with a y-axis ray intercept curve  774  and an x-axis ray intercept curve  776  corresponding to an on-axis portion of rays  725 ′ going through configuration  700 ′ but without WFC element  708 .  FIG. 36  shows a y-plot  780  and an x-plot  782  with a y-axis ray intercept curve  784  and an x-axis ray intercept curve  786  corresponding to an off-axis portion of rays  725 ′ going through configuration  700 ′ but without WFC element  708 . 
     Addressing each of  FIGS. 33-36  separately, y-axis and x-axis ray intercept curves  754  and  756  shown in  FIG. 33  are linear curves with substantially constant slopes, thus indicating misfocus. In  FIG. 34 , since y-axis ray intercept curve  764  and x-axis ray intercept curve  766  exhibit different slopes, they indicate the presence of astigmatism in addition to misfocus. In  FIGS. 35 and 36 , ray intercept curves  774  and  776  are substantially linear but have opposite slopes from curves  760  and  766 , indicating that configuration  700 ′ exhibits an opposite misfocus from configuration  700 . That is, while the misfocus indicated in  FIGS. 33 and 34  may be partially corrected by, for example, tilting an image plane (i.e., the location of detector  210 ) in configuration  700 , such movement of the image plane will worsen the misfocus exhibited by configuration  700 ′. In other words, correction of the misfocus indicated in  FIGS. 33-36  would require, for instance, moving the image plane as a function of configuration, which is undesirable. 
     To further illustrate the non-ideal performance of configurations equivalent to configurations  700  and  700 ′ but without WFC element  708 , calculated modulation transfer functions (MTFs) of these configurations are shown in  FIG. 37 . A plot  800  includes a plurality of MTF curves corresponding to on-axis and off-axis rays in configurations  700  and  700 ′ without WFC element  708 . The vertical axis of plot  800  corresponds to MTF magnitude, and the horizontal axis of plot  800  indicates normalized spatial frequency parameter. The maximum spatial frequency of one corresponds to 1 over (2 times the pixel size). The ideal MTF is a horizontal line at 0.5 magnitude. While having a cutoff value for a normalized spatial frequency parameter (i.e., a normalized spatial frequency parameter or spatial frequency at which MTF falls below 0.5) is acceptable for many applications, drastic variation in the MTF curves between different configurations is considered to be non-ideal because that characteristic would indicate that the system would exhibit a large variation in performance between different configurations; that is, in this case, certain configurations may perform better than others. Uniform performance (i.e., similar MTF curves) for different configurations is generally preferable. 
     Continuing to refer to  FIG. 37 , a first MTF group  810  includes MTF curves for on-axis and off-axis rays  725  and  727  in  FIG. 31 , and a second MTF group  820  includes MTF curves for on-axis and off-axis portions of rays  725 ′ in  FIG. 32 . In particular, a first on-axis MTF curve  812  is an MTF curve for on-axis rays  725 , a first off-axis MTF curve  814  is an MTF curve for off-axis rays  727 , a second on-axis MTF curve  822  is an MTF curve for an on-axis portion of rays  725 ′, and a second off-axis MTF  824  is an MTF curve for an off-axis portion of rays  725 ′. As may be seen in plot  800 , first and second MTF groups  810  and  820 , corresponding to configurations equivalent to configurations  700  and  700 ′ respectively, but without WFC element  708 , are significantly different from each other as well as within the groups themselves for on-axis and off-axis rays. Furthermore, MTF curve  824  is significantly lower than MTF curve  822  and first MTF group  810 , thus indicating degraded performance for the off-axis portion of rays  725 ′ shown in configuration  700 ′. 
     Still another indication of the non-ideal performance of configurations  700  and  700 ′ without WFC element  708  is shown in  FIGS. 38 and 39 .  FIGS. 38 and 39 , in conjunction with  FIGS. 31 and 32 , show MTF curves for on-axis and off-axis rays in configurations  700  and  700 ′ without WFC element  708  for a specific spatial frequency value (75 line-pairs/mm in the examples shown in these figures). In  FIGS. 38 and 39 , the vertical axis corresponds to a magnitude of a modulus of the optical transfer function (OTF); that is, the MTF, and the horizontal axis corresponds to a focus shift in millimeters, where a focus shift of zero corresponds to perfect focus at an image plane (e.g., a location of detector  210 ). A plot  850  in  FIG. 38  includes a first group of MTF curves  852  corresponding to on-axis rays  725  and off-axis rays  727  in configuration  700  of  FIG. 31 , but without WFC element  708 . A peak height of one MTF curve is labeled as h MTF , and a width of the corresponding peak at a value of h MTF /2 (e.g., a full width at half maximum, or FWHM, of the curve) is labeled as  856 . FWHM  856 , which is less than 0.2 mm in the example shown in  FIG. 38 , may increase when a WFC element is utilized in configuration  700 , as discussed in connection with  FIG. 41  below. 
     As may be seen in plot  850 , peaks of group of MTF curves  852  are located to the right of a line of ideal focus  854  at zero focus shift. Similarly, a plot  860  in  FIG. 39  includes a second group of MTF curves  862  corresponding to on-axis and off-axis portions of rays  725 ′ in configuration  700 ′ of  FIG. 32 , but without WFC element  708 . In plot  860 , the peaks of group of MTF curves  862  are located to the left of line of ideal focus  854 . While adjustments may be made to a system represented by configurations  700 ,  700 ′ without WFC element  708 , (e.g., a location of detector  210 , and thereby an image plane, may be moved closer or farther from first and second optical groups  702  and  704  or  704 ′) such that the image plane location is at the peaks of either first group of MTF curves  852  or second group of MTF curves  862 , adjusting the system to improve the performance of one configuration worsens the performance of the other configuration, and vice versa. In other words, it is not possible to select one location for detector  210  that achieves good performance in both of configurations  700  and  700 ′. 
     The inclusion of WFC element  708 , in combination with DSP  215 , in configurations  700  and  700 ′ of  FIGS. 31 and 32  may improve performance in the zoom lens system, as now described. 
     For purposes of the numerical modeling, WFC element  708  is accounted for by adding an extra element before the first surface of second optical groups  704  and  704 ′. One particular WFC element  708  is simulated as having a front surface expressed as: 
                     z   WFC     =         a   3     ⁢       (       x   3     +     y   3       )       r   0   3         +       a   5     ⁢       (       x   5     +     y   5       )       r   0   5         +       a   7     ⁢       (       x   7     +     y   7       )       r   0   7         +       a   9     ⁢       (       x   9     +     y   9       )       r   0   9                   (   4   )               
where x and y are spatial variables in a plane perpendicular to the optical axis, a 3 =1.418·10 −3 , a 5 =−0.5766·10 −3 , a 7 =1.388·10 −3 , a 9 =7.88·10 −3 , and r 0 =0.42 mm. Other configurations of WFC element  708  are possible.
 
       FIG. 40  shows a plot  900  of MTFs for on-axis and off-axis rays imaged through configurations  700  and  700 ′, this time with the effects of WFC element  708  taken into account in the numerical modeling. A first MTF group  910  includes MTF curves for on-axis and off-axis rays  725  and  727  in  FIG. 31  as well as on-axis and off-axis portions of rays  725 ′ in  FIG. 32 , thus corresponding to both wide-angle configuration  700  and telephoto configuration  700 ′, without processing by DSP  215 . As may be seen in plot  900 , individual MTF curves within first MTF group  910  are quite similar to each other. That is, comparing the substantially similar MTF curves of MTF group  910  to the earlier described MTF groups  810  and  820  of  FIG. 37  indicates that the performance uniformity for different ray angles within each configuration as well as in different configurations is improved when WFC element  708  is utilized. 
     Still referring to  FIG. 40 , plot  900  also includes a second MTF group  920  corresponding to MTF curves for on-axis and off-axis rays  725  and  727  imaged through configuration  700  as well as on-axis and off-axis portions of rays  725 ′ imaged through configuration  700 ′, with processing by DSP  215 . Details of DSP  215  will be discussed at an appropriate juncture in the discussion below. As may be seen by examination of second MTF group  920 , an overall magnitude of MTF curves within second MTF group  920  is increased over that of curves within first MTF group  910 , while preserving the uniformity of the MTF performance (indicated by the substantially similar magnitude and shape of the curves in MTF group  920 ). Furthermore, the MTF curves within second MTF group  920  are, on the whole, greater than 0.5 in magnitude across the normalized spatial frequency parameter range. In other words, when WFC element  708  and processing by DSP  215  are included, both configurations  700  and  700 ′ achieve improved performance in the zoom lens system of the present disclosure. 
     In another illustration of performance of configurations  700  and  700 ′ including WFC element  708 , but without signal processing,  FIGS. 41 and 42  show MTF curves for on-axis and off-axis rays in configurations  700  and  700 ′ for a specific spatial frequency value (75 lp/mm, as in  FIGS. 38 and 39 ). Again, perfect focus at the image plane corresponds to a focus shift of zero. A plot  950  in  FIG. 41  includes a first group of MTF curves  952  corresponding to on-axis and off-axis rays in configuration  700  of  FIG. 31 , including WFC element  708 , but without processing by DSP  215 . A peak height of one MTF curve is labeled as h MTF , and a FWHM of the corresponding peak (e.g., at a value of h MTF /2) is labeled as  956 . A plot  960  in  FIG. 42  includes a second group of MTF curves  962  corresponding to on-axis and off-axis rays in configuration  700 ′ of  FIG. 32 , with WFC element  708  but without processing by DSP  215 . It may be seen that individual MTF curves within each of first and second groups of MTF curves  952  and  962 , respectively, are quite similar to each other within the respective groups. This characteristic indicates uniform performance across configurations as well as across ray angles. 
     By comparing  FIGS. 41 and 42  with aforedescribed  FIGS. 38 and 39 , it may be seen that the peaks of both first and second groups of MTF curves  952  and  962  have been flattened and broadened, such that line of ideal focus  854  intersects both groups of MTF curves  952  and  962  at points not far from the peak MTF magnitudes (in fact, the MTF curves included within second group of MTF curves  962  are so similar that they virtually overlap each other). That is, plots  950  and  960  indicate that there is a broad range of settings in which high MTF values may be obtained without having to actually move the image plane. FWHM  956  is seen as being about 0.6 mm in  FIG. 41 , as compared to corresponding FWHM  856  in  FIG. 38 , which is less than 0.2 mm. It may be seen in  FIG. 41  that other MTF curves of group  952  have similar widths that are all greater than corresponding peak widths of MTF curves of group  852 ,  FIG. 38 . It may similarly be seen, by comparing  FIG. 39  to  FIG. 42 , that corresponding peak widths (i.e., FWHM) of MTF curves of group  962  are wider than corresponding peak widths of MTF curves of group  862 . Thus, the increase in width of their MTF curves with respect to misfocus, over a range of ray angles, makes both configurations  700  and  700 ′ with WFC element  708  less sensitive to misfocus and/or to misfocus-like aberrations over the range of ray angles imaged by detector  210 , than corresponding configurations  700  and  700 ′ without WFC element  708 . That is, the zoom lens system that utilizes WFC element  708  has a broader MTF curve, as indicated by FWHM, at least one spatial frequency and over a range of focus shift, that is wider than an MTF curve formed by the corresponding system at the one spatial frequency over a range of ray angles imaged by the system and at any focal length of the system, than a corresponding zoom lens system without WFC element  708 . 
     The specifics of the algorithm applied by DSP  215  are shown in  FIG. 43 .  FIG. 43  shows a mesh rendering of a linear filter used to generate the results shown in  FIG. 40 .  FIG. 43  shows a 3-D plot  1000  including a linear filter  1010 . The specific values of each point in the mesh of linear filter  1010  are shown in TABLE 2, below. It is noted that a sum of all of the values in TABLE 2 is one. Linear filter  1010  is applied by DSP  215  as a 2-dimensional, linear convolution to image data received from detector  210  in order to generate MTF curves  920  shown in  FIG. 40 . 
     
       
         
           
               
               
               
               
               
               
               
               
               
               
               
             
               
                 TABLE 2 
               
               
                   
               
             
            
               
                 −0.0371 
                 0.0652 
                 −0.0112 
                 −0.0086 
                 0.0218 
                 −0.0648 
                 0.0093 
                 −0.0324 
                 −0.0352 
                 0.0605 
                 −0.0233 
               
               
                 0.0627 
                 −0.0291 
                 −0.0473 
                 0.0856 
                 0.0378 
                 −0.1345 
                 −0.0507 
                 0.0795 
                 −0.0026 
                 −0.0544 
                 0.0484 
               
               
                 −0.0099 
                 −0.0503 
                 0.1322 
                 −0.1261 
                 −0.0368 
                 0.1261 
                 0.0542 
                 −0.1366 
                 0.0811 
                 −0.0131 
                 −0.0051 
               
               
                 −0.0098 
                 0.0849 
                 −0.1235 
                 0.0524 
                 0.2273 
                 −0.4149 
                 0.0481 
                 0.0927 
                 −0.0949 
                 0.0496 
                 −0.0345 
               
               
                 0.0174 
                 0.0413 
                 −0.0280 
                 0.2361 
                 −0.0822 
                 −0.4170 
                 −0.3459 
                 0.0926 
                 −0.0127 
                 0.0146 
                 −0.0036 
               
               
                 −0.0571 
                 −0.1331 
                 0.1077 
                 −0.4259 
                 −0.3827 
                 1.9235 
                 0.5541 
                 −0.1689 
                 0.0711 
                 −0.0478 
                 0.0243 
               
               
                 0.0117 
                 −0.0508 
                 0.0537 
                 0.0410 
                 −0.3391 
                 0.5601 
                 −0.1348 
                 −0.0103 
                 0.0548 
                 −0.0323 
                 0.0301 
               
               
                 −0.0327 
                 0.0773 
                 −0.1347 
                 0.0956 
                 0.0885 
                 −0.1692 
                 −0.0090 
                 0.1306 
                 −0.1013 
                 0.0457 
                 −0.0233 
               
               
                 −0.0351 
                 −0.0003 
                 0.0785 
                 −0.0956 
                 −0.0108 
                 0.0720 
                 0.0536 
                 −0.1012 
                 0.0349 
                 0.0288 
                 −0.0363 
               
               
                 0.0595 
                 −0.0556 
                 −0.0113 
                 0.0504 
                 0.0129 
                 −0.0484 
                 −0.0313 
                 0.0459 
                 0.0284 
                 −0.0654 
                 0.0428 
               
               
                 −0.0220 
                 0.0487 
                 −0.0064 
                 −0.0350 
                 −0.0026 
                 0.0252 
                 0.0297 
                 −0.0240 
                 −0.0361 
                 0.0431 
                 0.0008 
               
               
                   
               
            
           
         
       
     
       FIGS. 44-70  illustrate a numerical modeling example of a three-group zoom lens system.  FIGS. 44 and 45  illustrate two different configurations of a three-group zoom lens system in accordance with the present disclosure.  FIG. 44  shows a configuration  1100  including a first optical group  1102  (with a focal length f 1 ), a second optical group  1104  (with a focal length f 2 ) and a third optical group  1106  (with a focal length f 3 ). Optical groups  1102  and  1104  may include one or more optical elements. Optical group  1106  includes optics  1108  and a WFC element  1110  that may be formed adjacent to or juxtaposed with optics  1108 , or integrally formed therewith. In analogy to  FIG. 31 , configuration  1100  is a wide-angle system configured to accept both on-axis rays  725  and off-axis rays  727  (which encompass a range of incoming rays imaged by system zoom system  1100 ) and image these rays through first, second and third optical groups onto detector  210 . Image data generated by detector  210  is directed to DSP  215  where the data is processed to form a final image  1120 . Optical groups  1102 ,  1104  and  1106  all align along an optical axis  1122 . 
       FIG. 45  shows a configuration  1100 ′, in which the three optical groups remain at the relative positions shown in configuration  1100  (e.g., without translation of optical groups along optical axis  1122 ). In configuration  1100 ′, second optical group  1104 ′ now exhibits a focal length f 2 ′ and a third optical group  1106 ′ now includes modified optics  1108 ′ so as to result in a focal length f 3 ′, such that configuration  1100 ′ functions as a telephoto system. After detection at detector  210  and signal processing at DSP  215 , a final image  1120 ′ results. WFC element  1110  and/or DSP  215  may be identical between configurations  1100  and  1100 ′, or may be modified to accommodate changes in the system due to the focal length variations described. 
     In general, the use of three groups of optical elements helps to control certain fixed aberrations of a zoom lens system. While first optical group  1102  is shown in  FIGS. 44 and 45  to be a stationary, non-variable optical group, the positions of the three optical groups may be changed such that, for example, the stationary optical group is the second or the third group encountered by incident light rays. 
     Configurations  1100  and  1100 ′ as illustrated in  FIGS. 44 and 45 , and equivalent configurations without WFC element  1110 , were numerically modeled using the following exemplary characteristics. A 0.75 mm image height is assumed. An effective focal length of a combination of optical groups  1102 ,  1104  and  1106 , respectively, is 4.8 mm for (wide angle) configuration  1100  14.2 mm for (telephoto) configuration  100 ′. A wavelength of light rays is assumed to be 0.55 microns, and detector  210  is assumed to include pixels that are 4 microns square. In configuration  1100 , focal length f 1  of optical group  1102  is assumed as −14.86 mm, focal length f 2  of optical group  1104  is assumed as 23.91 mm, and focal length f 3  of optical group  1106  is assumed as 6.55 mm. In configuration  1100 ′, focal length f 1  of first optical group  1102  remains −14.86 mm, focal length f 2 ′ of second optical group  1104  is assumed as 6.03 mm, and focal length f 3 ′ of third optical group  1106  is assumed as −4.94 mm. A specific prescription of the various optical groups is again derived from Equation (3) with parameters as shown in TABLES 3A and 3B: 
     
       
         
           
               
               
               
               
             
               
                   
                 TABLE 3A 
               
             
            
               
                   
                   
               
               
                   
                 First optical group 
                 Second optical group 
                 Second optical group 
               
               
                   
                 1102 
                 1104 
                 1104′ 
               
               
                   
                 Thickness = 0.5 mm 
                 Thickness = 1.7 mm 
                 Thickness = 1.7 mm 
               
               
                   
                 Distance to second optical 
                 Distance to third optical 
                 Distance to third optical 
               
               
                   
                 group = 4.98 mm 
                 group = 4.9 mm 
                 group = 4.9 mm 
               
            
           
           
               
               
               
               
               
               
               
            
               
                   
                 Surface 1 
                 Surface 2 
                 Surface 1 
                 Surface 2 
                 Surface 1 
                 Surface 2 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
               
               
            
               
                 c 
                 −5.681E−3 
                 −5.681E− 
                 −2.179E−2 
                 −2.179E−2 
                  1.077E−1 
                 −1.007E−1 
               
               
                 k 
                 4.620 
                 4.620 
                 −8.234E+2 
                 −8.234E+2 
                 −4.885E−1 
                 −2.606E−1 
               
               
                 a 2   
                 0 
                 0 
                 0 
                 0 
                 0 
                 0 
               
               
                 a 4   
                  7.702E−4 
                  7.702E−4 
                  1.586E−3 
                  1.345E−3 
                 −5.454E−5 
                  6.371E−4 
               
               
                 a 6   
                 −2.625E−5 
                 −2.625E− 
                  5.677E−5 
                  2.290E−4 
                 −3.866E−6 
                  3.696E−7 
               
               
                 a 8   
                 −6.316E−6 
                 −6.316E− 
                  5.890E−6 
                 −1.037E−5 
                  4.178E−7 
                 −1.085E−6 
               
               
                 a 10   
                  3.273E−7 
                  3.273E−7 
                 −2.898E−6 
                  1.298E−6 
                 −1.027E−7 
                 −2.028E−8 
               
               
                   
               
            
           
         
       
     
     
       
         
           
               
               
               
             
               
                   
                 TABLE 3B 
               
             
            
               
                   
                   
               
               
                   
                 Third optical group 1104 
                 Third optical group 1104′ 
               
               
                   
                 Thickness = 1.0 mm 
                 Thickness = 1.7 mm 
               
               
                   
                 Back focal distance = 6.27 mm 
                 Back focal distance = 6.27 mm 
               
            
           
           
               
               
               
               
               
            
               
                   
                 Surface 1 
                 Surface 2 
                 Surface 1 
                 Surface 2 
               
               
                   
                   
               
            
           
           
               
               
               
               
               
            
               
                 c 
                  9.788E−2 
                 −9.632E−2 
                 −1.225E−1 
                  1.227E−1 
               
               
                 k 
                 7.303 
                  4.602E−2 
                 −2.659E+1 
                  3.609E+1 
               
               
                 a 2   
                 0 
                 0 
                 0 
                 0 
               
               
                 a 4   
                 −5.420E−4 
                 −7.039E−4 
                  1.744E−2 
                  1.710E−2 
               
               
                 a 6   
                 −2.814E−3 
                 −2.151E−4 
                 −6.794E−4 
                 −2.511E−3 
               
               
                 a 8   
                 −2.829E−3 
                  5.410E−4 
                 −6.362E−3 
                 −2.393E−4 
               
               
                 a 10   
                  1.261E−2 
                  9.668E−5 
                  3.676E−3 
                 −6.028E−3 
               
               
                   
               
            
           
         
       
     
       FIGS. 46-53  illustrate non-ideal performance of the configurations shown in  FIGS. 44 and 45  if WFC element  1110  is not present.  FIGS. 46-49  show a series of ray intercept curves corresponding to configurations  1100  and  1100 ′ shown in  FIGS. 44 and 45 .  FIG. 46  shows a y-plot  1150  and an x-plot  1152  with a y-axis ray intercept curve  1154  and an x-axis ray intercept curve  1156  corresponding to on-axis rays  725  going through configuration  1100  but without WFC element  1110 . 
     Similarly,  FIG. 47  shows a y-plot  1160  and an x-plot  1162  with a y-axis ray intercept curve  1164  and an x-axis ray intercept curve  1166  corresponding to off-axis rays  727  being imaged through configuration  1100  but without WFC element  1110 .  FIG. 48  shows a y-plot  1170  and an x-plot  1172  with a y-axis ray intercept curve  1174  and an x-axis ray intercept curve  1176  corresponding to the on-axis portion of rays  725 ′ going through configuration  1100 ′ but without WFC element  1110 .  FIG. 49  shows a y-plot  1180  and an x-plot  1182  with a y-axis ray intercept curve  1184  and an x-axis ray intercept curve  1186  corresponding to the off-axis portion of rays  725 ′ going through configuration  1100 ′ but without WFC element  1110 . 
     Examining each of  FIGS. 46-49  in turn, y-axis and x-axis ray intercept curves  1154  and  1156  shown in  FIG. 46  are linear curves with substantially constant slopes, thus indicating misfocus. In  FIG. 47 , y-axis ray intercept curve  1164  and x-axis ray intercept curve  1166  are similarly linear. In  FIGS. 48 and 49 , the ray intercept curves are substantially linear but have opposite slopes from those shown in  FIGS. 46 and 47 , thereby indicating that configuration  1100 ′ exhibits an opposite misfocus from configuration  1100 . That is, while the misfocus indicated in  FIGS. 46 and 47  may be partially corrected by, for example, moving the image plane (i.e., the location of detector  210 ) in configuration  1100 , such movement of the image plane will worsen the misfocus exhibited by configuration  1100 ′. In other words, correction of the misfocus as shown in  FIGS. 46-49  would require, for instance, moving the image plane as a function of configuration, which is undesirable. 
       FIGS. 50 and 51  show calculated MTFs as a function of spatial frequency (in units of cycles (or line pairs) per millimeter) for the on-axis and off-axis rays in configurations  1100  and  1100 ′ without WFC elements  1110 . In  FIG. 50 , a plot  1200  includes a group of MTF curves  1210  corresponding to on- and off-axis rays imaged through configuration  1100  of  FIG. 44  but without WFC element  1110 . Similarly, a plot  1220  in  FIG. 51  includes another group of MTF curves  1230  corresponding to on- and off-axis rays imaged through configuration  1100 ′ of  FIG. 45 , again without WFC element  1110 . As shown in plots  1200  and  1220 , both groups of MTF curves  1210  and  1230  exhibit large drops and variations with increasing spatial frequency, indicating non-uniform performance as well as large misfocus within each configuration and between the two configurations. Furthermore, both groups of MTF curves  1210  and  1230  include spatial frequency values at which the MTF drops to essentially zero. These zeros in the MTF values are particularly undesirable because they indicate loss of image data. 
       FIGS. 52 and 53 , like  FIGS. 38 and 39 , show the MTF curves for on-axis and off-axis rays in configurations  1100  and  1100 ′ without wavefront coding for a specific spatial frequency value (75 lp/mm). In  FIGS. 52 and 53 , the vertical axis corresponds to the magnitude of the modulus of the OTF; that is, the MTF, and the horizontal axis corresponds to a focus shift in millimeters, where a focus shift of zero corresponds to perfect focus at the image plane (e.g., a location of detector  210 ). A plot  1250  in  FIG. 52  includes a first group of MTF curves  1252  corresponding to on-axis and off-axis rays in configuration  1100  of  FIG. 44 , but without WFC element  1110 . As may be seen in plot  1250 , peaks of first group of MTF curves  1252  are located to the right of a line of ideal focus  1254  at zero focus shift. Individual MTF curves within group of MTF curves  1252  vary widely in shape, indicating astigmatism and field curvature in addition to misfocus. A FWHM  1256  is shown for one of the curves, and can be seen to have a value less than 0.3 mm. A plot  1260  in  FIG. 53  includes a second group of MTF curves  1262  corresponding to on-axis and off-axis rays in configuration  1100 ′ of  FIG. 45 , but without WFC element  1110 . In plot  1260 , the peaks of second group of MTF curves  1262  are located to the left of line of ideal focus  1254 . Therefore, like configurations  700  and  700 ′ illustrated earlier in the context of  FIGS. 38 and 39 , plots  1250  and  1260  in  FIGS. 52 and 53 , respectively, indicate that it is not possible to select one location for detector  210  that achieves good performance in both configurations  1100  and  1100 ′. 
     In contrast with  FIG. 50  and  FIG. 51 ,  FIGS. 54 and 55  show MTF curves simulated for configurations  1100  and  1100 ′ including WFC element  1110 . For purposes of the numerical modeling, WFC element  1110  is accounted for by adding an extra element before the first surface of third optical groups  1106  and  1106 ′ according to the sag equation of Equation (4) with the following parameters: a 3 =−2.858·10 −3 , a 5 =−0.08·10 −3 , a 7 =−1.707·10 −3 , a 9 =3.426·10 −3 , and r 0 =0.60 mm. 
     Referring to  FIGS. 54 and 55  in conjunction with  FIGS. 50 and 51 ,  FIG. 54  shows a plot  1270  including a group of MTF curves  1272  for on-axis and off-axis rays imaged through configuration  1100  including WFC element  1110 , without processing by DSP  215 . As may be seen by comparing MTF curves  1272  with MTF curves  1210  shown in  FIG. 50 , MTF values are greater in MTF curves  1272  than for MTF curves  1210 , thus the MTF values are increased by the addition of WFC element  1110  in configuration  1100 . Similarly,  FIG. 55  shows a plot  1280  including a group of MTF curves  1282  for on- and off-axis rays imaged through configuration  1100 ′, again including WFC element  1110 , without processing by DSP  215 . It may be seen that MTF curves  1282  show improvement over MTFs  1230  of  FIG. 51  in terms of magnitude, uniformity in performance and lack of zeros. 
       FIGS. 56 and 57  show MTF curves for on-axis and off-axis rays in configurations  1100  and  1100 ′ for a specific spatial frequency value (75 lp/mm, as in  FIGS. 52 and 53 ). Again, perfect focus at the image plane corresponds to a focus shift of zero. A plot  1290  in  FIG. 56  includes a first group of MTF curves  1292  corresponding to on-axis and off-axis rays in configuration  1100  of  FIG. 44 , including WFC element  1110 , but without processing by DSP  215 . A FWHM  1296  is shown for one of the curves, and can be seen to have a value that cannot be measured utilizing plot  1290 , because the peak extends beyond the focus shift values shown in plot  1290 , but is at least 0.6 mm. A plot  1295  in  FIG. 57  includes a second group of MTF curves  1297  corresponding to on-axis and off-axis rays in configuration  1100 ′ of  FIG. 45 , with WFC element  1110  but without processing by DSP  215 . It may be seen that peaks of both first and second groups of MTF curves  1292  and  1297  have been flattened and broadened such that line of ideal focus  1294  intersects both groups of MTF curves at points not far from the peak MTF magnitudes. In other words, plots  1290  and  1295  indicate that there is a broad range of settings in which high MTF values may be obtained without having to actually move the image plane. Furthermore, the zoom lens system that utilizes WFC element  1110  has a broader MTF curve, as indicated by FWHM, at least one spatial frequency and over a range of focus shift, that is wider than an MTF curve formed by the corresponding system at the one spatial frequency over a range of ray angles imaged by the system and at any focal length of the system, than a corresponding zoom lens system without WFC element  1110 . 
     Referring again to  FIGS. 31-32 ,  44 - 45 , a range of incoming ray angles (e.g., indicated by on axis and off axis rays  725  and  727 , respectively) is imaged by each of the zoom systems; and a wavefront coding element modifies phase of a wavefront represented by the rays such that MTFs over the range of ray angles is similar in magnitude and shape, making the zoom system less sensitive to misfocus like aberrations (as compared to the same zoom system without wavefront coding). 
     In order to further illustrate the improvement in system performance obtained by the addition of WFC element  1110  and DSP  215  in the configurations shown in  FIGS. 44 and 45 , calculated evaluations of these configurations in terms of the point spread functions (PSFs) are shown in  FIGS. 58-69 .  FIGS. 58-61  respectively correspond to visualizations of a calculated PSF  1300  for on-axis rays imaged through configuration  1100 , a calculated PSF  1302  for off-axis rays imaged through configuration  1100 , a calculated PSF  1304  for on-axis rays image through configuration  1100 ′ and a calculated PSF  1306  for off-axis rays imaged through configuration  1100 ′ for a point object when WFC element  1110  is not included in the configurations and no processing is performed by DSP  215 . As may be seen in comparing PSFs  1300 ,  1302 ,  1304  and  1306 , the imaging performance of the three-group zoom lens system within and between configurations varies widely, as PSFs  1300 ,  1302 ,  1304  and  1306  are quite different from each other. 
       FIGS. 62-65 , respectively, show visualizations of a calculated PSF  1310  for on-axis rays imaged through configuration  1100 , a calculated PSF  1312  for off-axis rays imaged through configuration  1110 , a calculated PSF  1314  for on-axis rays image through configuration  1100 ′ and a calculated PSF  1316  for off-axis rays imaged through configuration  1100 ′ for a point object when WFC element  1110  is included in the configurations, but no processing is performed by DSP  215 . PSFs  1310 ,  1312 ,  1314  and  1316  are all quite similar to each other. While PSFs  1310 ,  1312 ,  1314  and  1316  are not perfect points, as would be ideal, they are all quite uniform and include only similar aberrations that spread out the PSFs in the 9-o&#39;clock and 12-o&#39;clock directions in these figures. Such aberrations may be corrected using a single linear filter, as indicated in  FIGS. 66-69 , which show the calculated PSFs when the effects of both WFC element  1110  and processing by DSP  215  are included in the calculations. 
       FIGS. 66-69 , respectively, correspond to on-axis rays imaged through configuration  1100 , off-axis rays imaged through configuration  1110 , on-axis rays imaged through configuration  1100 ′ and off-axis rays imaged through configuration  1100 ′ for a point object. As may be seen, calculated PSFs  1320 ,  1322 ,  1324  and  1326  are very close to uniform points covering only a few pixels. Therefore, by the addition of WFC element  1110  and subsequent signal processing by DSP  215 , uniform performance may be achieved by the three-group zoom lens system for a range of configurations, from wide angle to telephoto. 
       FIG. 70  illustrates a linear filter  1360  that is applied by DSP  215  in configurations  1100  and  1100 ′; filter  1360  is shown in mesh format in a plot  1350 . Filter  1360  is applied as a two-dimensional, linear convolution by DSP  215  in configurations  1100  and  1100 ′ to image data generated by detector  210  in order to generate the calculated PSFs shown in  FIGS. 66-69 . 
     Although each of the aforedescribed embodiments have been illustrated with various components having particular respective orientations, it should be understood that the present devices may take on different configurations with components located in different positions and mutual orientations and remain within the spirit and scope of the present disclosure. Furthermore, suitable equivalents may be used in place of, or in addition to, the various components. The function and use of such substitute or additional components may be familiar to those skilled in the art and are therefore regarded as falling within the scope of the present disclosure. For example, an optical photon sieve may be added into the zoom lens system of the present disclosure as part of one or more of the optical groups. Details regarding an optical photon sieve may be found, for example, in Andersen, “Large optical photon sieve,” Optics Letters, vol. 30, no. 22, November 2005, pp. 2976-2978. Such an optical photon sieve may act as a simple diffractive element within the zoom lens system to replace or complement one or more of the optical elements in the embodiments of the present disclosure as described above. Another possible modification includes the addition of a polymer dispersed or polymer stabilized liquid crystal (PDLC or PSLC) light modulating device in the optical path. Such a PDLC or PSLC device may act, for instance, as a binary or analog light valve to regulate the amount of light transmitted through the zoom lens system. Alternatively, the PDLC or PSLC may be patterned to provide additional light control, or may be integrated into a variable liquid crystal lens so as to enhance the variability of a zoom lens system including the variable LC lens. PDLCs and PSLCs are described, for example, in Drzaic, “Recent progress in dichroic polymer-dispersed liquid crystal materials,” Pure &amp; Appl. Chem., vol. 68, no. 7, pp. 1435-1440, 1996, and Doane et al., U.S. Pat. No. 5,691,795 issued 25 Nov. 1997. Other modern optical elements that act to change the wavefront of light may be suitably configured into the improved zoom lens systems as disclosed above. 
     Therefore, the present examples are to be considered as illustrative and not restrictive, and are not limited to the details given herein but may be modified within the scope of the appended claims. The following claims are intended to cover generic and specific features described herein, as well as statements of the scope of the present method and system, which, as a matter of language, might be said to fall therebetween.