Abstract:
The present invention relates generally to stereoscopic displays, and more particularly, but not exclusively, to stereoscopic displays with addressable focus cues. A stereoscopic display system with the addressable focus cues, comprises: a microdisplay for providing a virtual image for display to a user; a reflective active optical element configured to provide a variable optical power; a relay lens disposed along an optical path between the microdisplay and the active optical element, the relay lens positioned therebetween such that the microdisplay and the active optical element are disposed at conjugate planes of the relay lens; a beamsplitter disposed along the optical path between the microdisplay and the active optical element at an orientation to receive optical radiation from the active optical element; and a see-through eyepiece comprising a selected surface configured to receive optical radiation from the beamsplitter and reflect the received radiation to an exit pupil of the system to provide a virtual display path, the selected surface also configured to receive optical radiation from a source other than the microdisplay and to transmit such optical radiation to the exit pupil to provide a see-through optical path.

Description:
RELATED APPLICATIONS 
       [0001]    This present application claims the benefit of priority of U.S. Provisional Application No. 61/795,500, filed on Oct. 18, 2012, the entire contents of which application(s) are incorporated herein by reference. 
     
    
     GOVERNMENT LICENSE RIGHTS 
       [0002]    This invention was made with government support under IIS0915035 awarded by NSF. The government has certain rights in the invention. 
     
    
     FIELD OF THE INVENTION 
       [0003]    The present invention relates generally to stereoscopic displays, and more particularly, but not exclusively, to stereoscopic displays with addressable focus cues. 
       BACKGROUND OF THE INVENTION 
       [0004]    Conventional stereoscopic 3D displays create the illusion of depth based on binocular disparities, rendering 3D scenes from a single pair of 2D perspective images at a fixed distance to the viewer. Therefore conventional stereoscopic displays force an unnatural decoupling of the accommodation and convergence cues, which may contribute to various visual artifacts in stereoscopic displays, such as distortion in perceived depth, diplopic vision, visual discomfort, and fatigue. Many approaches have been proposed which may overcome the drawbacks of conventional stereoscopic displays, including volumetric displays, holographic displays, and multi-focal-plane displays. However, a need exists to develop an optical see-through stereoscopic display which solves the fundamental accommodation-convergence problems and also renders large volumes of continuous 3D scene at high image quality and flickering-free speed. 
         [0005]    A stereoscopic display with addressable focus cues is one of the most promising approaches to solving the fundamental accommodation-convergence problems of providing the capability of rendering correct or near-correct focus cues for virtual 3D objects. Unlike traditional stereoscopic displays, stereoscopic displays with addressable focus cues enable the ability to either dynamically vary the focal distance of the virtual display through an active optical element in accordance to the viewer&#39;s region of interest, known as vari-focal display mode, or present multiple focal planes at a flickering-free speed through with no need to track a viewer&#39;s region of interest, known as multi-focal display mode. For instance, multi-focal-plane displays present perspective 2D images at multiple carefully placed, discrete focal distances along the visual axis. These discrete focal planes sample a 3D scene volume into multiple zones, and objects within a zone are rendered by the corresponding pair of adjacent focal planes, as shown in  FIG. 1 . Therefore multi-focal-plane displays are able to render correct or near-correct focus cues for virtual objects at different depths. Contrary to multi-viewpoint displays such as holographic displays and volumetric displays, the multi-focal-plane display is a fixed-viewpoint display. By restricting the viewing position, multi-focal-plane display systems only need to display a small number of viewpoints. Also multi-focal-point displays can preserve disparity, occlusion and perspective in conventional 2D displays as well as rendering viewing-dependent lighting effects such as specular reflection and shading. In practice, the implementations of multi-focal-plane displays can be categorized into two categories: spatially multiplexed or temporally multiplexed. In a spatial-multiplexed system, multi-focal capability is achieved by stacking multiple 2D displays. In the alternative and more elegant, time-multiplexed system, the focal distances of images from a single 2D display are fast switched by an active optical element in synchronization with the frame rendering of multiple focal planes. In general, a multi-focal plane display can be readily adapted for use in a vari-focal mode without much change to the optics layout. 
         [0006]    In addition, progress has recently been made in the field of stereoscopic displays that can be head-mounted and that have addressable focal planes for improved depth perception but require substantially less computational power than existing methods, as reflected in commonly owned U.S. Patent Application Publication. No. 2011/0075257, the contents of which are incorporated herein by reference. However, a need still exists for optical imaging systems which can provide enhanced imaging performance in stereoscopic displays with addressable focus cues. 
       SUMMARY OF THE INVENTION 
       [0007]    In one of its aspects, the present invention may provide a virtual display system with addressable focus cues comprising a microdisplay for providing a virtual image for display to a user. A reflective active optical element, configured to provide a variable optical power, may also be provided. A relay lens may be disposed along an optical path between the microdisplay and the active optical element, with the relay lens positioned therebetween such that the microdisplay and active optical element are disposed at conjugate planes of the relay lens. A beamsplitter may be disposed along the optical path between the microdisplay and the active optical element at an orientation to receive optical radiation from the active optical element. In addition, a see-through eyepiece may be provided which includes a selected surface configured to receive optical radiation from the beamsplitter and reflect the received radiation to an exit pupil of the system to provide a virtual display path. The selected surface may also be configured to receive optical radiation from a source other than the microdisplay (such as the real world) and to transmit such optical radiation to the exit pupil to provide a see-through optical path. The eyepiece may include a freeform prism shape, and, in particular, may include a first surface configured to receive and refract optical radiation from the beamsplitter and may include a second surface configured to receive the refracted optical radiation from the first surface, with the second surface configured to reflect the optical radiation to the selected surface of the eyepiece. The second surface may be configured to total internally reflect the optical radiation, and one or more of the surfaces of the eyepiece may comprise a rotationally asymmetric surface. 
         [0008]    In another of its aspects, the present invention may provide a virtual display system with addressable focus cues comprising a microdisplay for providing a virtual image for display to a user, and an eyepiece comprising a reflective optical element configured to reflect optical radiation from the microdisplay to an exit pupil of the system. A relay lens, comprising a refractive active optical element configured to provide a variable optical power, may be disposed along an optical path between the microdisplay and the eyepiece to relay an image from the microdisplay to the eyepiece. The relay lens may include first and second lens groups disposed along the optical path with the active optical element located between the first and second lens groups. In addition, a beamsplitter may be disposed along the optical path between the microdisplay and the eyepiece, with the beamsplitter configured to receive and transmit optical radiation from a source other than the microdisplay (such as the real world) to the exit pupil to provide a see-through optical path. The eyepiece may comprise a spherical mirror, and the system may be telecentric in the microdisplay. The system may also have an f-number less than 3. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0009]    The foregoing summary and the following detailed description of exemplary embodiments of the present invention may be further understood when read in conjunction with the appended drawings, in which: 
           [0010]      FIG. 1  schematically illustrates 3D objects rendered by multiple focal planes relative to a viewer; 
           [0011]      FIG. 2  schematically illustrates the unfolded optical path of an exemplary display system in accordance with the present invention; 
           [0012]      FIG. 3A  schematically illustrates a 2D layout of virtual display optics of an exemplary display system in accordance with the present invention; 
           [0013]      FIG. 3B  schematically illustrates a 2D layout of the exemplary display system of  FIG. 3A  but having a field lens comprising two optical elements rather than a singlet; 
           [0014]      FIG. 3C  schematically illustrates the free-form eyepiece and compensator of  FIGS. 3A-3B  showing the see-through optical path; 
           [0015]      FIG. 3D  schematically illustrates the free-form eyepiece and compensator of  FIGS. 3A-3B  showing both the see-through and display paths; 
           [0016]      FIGS. 4A-4E  illustrate the polychromatic MTF through the display path for the virtual display system of  FIG. 3B ; 
           [0017]      FIG. 4F  illustrates a distortion grid through the display path for the virtual display system of  FIG. 3B ; 
           [0018]      FIGS. 5A-5E  illustrate the polychromatic MTF through the see-through path for the virtual display system of  FIG. 3B ; 
           [0019]      FIG. 5F  illustrates a distortion grid through the see-through path for the virtual display system of  FIG. 3B ; 
           [0020]      FIG. 6A  schematically illustrates a 3D layout of the free-form eyepiece with relay optics of the display system of  FIG. 3B ; 
           [0021]      FIG. 6B  schematically illustrates a 3D Solidworks model of the assembled free-form eyepiece and compensator of the display system of  FIG. 6A ; 
           [0022]      FIG. 7A  illustrates a depth-fused 6-focal-plane 3D scene of 40 degrees of field of view and 3 diopters of depth, captured through the eyepiece for a prototype built according to the design of  FIG. 3B ; 
           [0023]      FIGS. 7B-7C  illustrate 6-focal-plane 3D scenes captured by a camera focused at 2 m and 30 cm, respectively, for a prototype built according to the design of  FIG. 3B ; 
           [0024]      FIG. 8A  illustrates a retinal image MTF as a function of accommodations in a dual-focal-plane display, with two focal planes placed at 1.2D and 1.8D, respectively, and with a luminance ratio of 1:1; 
           [0025]      FIG. 8B  illustrates a retinal image contrast as a function of accommodation showing the contrast gradient for different spatial frequencies; 
           [0026]      FIGS. 9A-9B  illustrate that the spatial frequency of a transition point decreases when the focal plane separation increases and when the eye pupil size increases, respectively; 
           [0027]      FIGS. 10A ,  10 B schematically illustrate 2D and 3D layouts, respectively, of the virtual display optics of a further exemplary display system in accordance with the present invention; 
           [0028]      FIG. 11  schematically illustrates a 2D layout and element descriptions of the relay lens group of  FIG. 10A  along with an optional display illumination path; and 
           [0029]      FIGS. 12A ,  12 B illustrate polychromatic MTF and field curves, respectively, of the system of  FIGS. 10A-11 . 
       
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
       [0030]    Referring now to the figures, wherein like elements are numbered alike throughout, in accordance with one aspect of the present invention,  FIG. 2  schematically illustrates the first-order unfolded optical path of an exemplary optical system which is particularly suited for providing high imaging quality in depth-fused multi-focal-plane stereoscopic displays with addressable focus cues.  FIGS. 3A ,  3 B schematically illustrate particular designs according to the layout of  FIG. 2 , with a first optical system  100  having a single field lens  18 ,  FIG. 3A , and an alternative system  200  having a two-element  17 ,  19  field lens  18 ,  FIG. 3B . (While a single set of optics for a single viewer&#39;s eye is illustrated, it is understood that in a final stereoscopic device, two such sets of optics will be provided, one for each eye.) 
         [0031]    A relevant feature of the designs is the inclusion of a relay lens group  20  which relays the image from a microdisplay, such as a digital micro-mirror device (DMD)  60 , to the front of an eyepiece  12 . The relay lens group  20  may include a traditional, non-deformable lens  21  and a reflective active optical element, such as a deformable membrane mirror  80 ,  FIG. 2 . The relay lens  21  may include a pair of doublets  22 ,  24  and lens  26 ,  FIGS. 3A ,  3 B. The deformable membrane mirror device (DMMD)  80  may be placed at the focal plane of the relay lens  21  and may serve as the system stop. Together the lens  21 ,  FIG. 2 , (or lenses  22 ,  24 ,  26 ,  FIGS. 3A ,  3 B) and the DMMD  80  may provide a folded double-telecentric system particularly suited to depth-fused multi-focal-plane stereoscopic displays. An advantage of designing a double-telecentric relay  20  is that the change of optical power on the DMMD  80  changes only the location of the intermediate image without changing the image magnification, so that the field of view of the system and the angular resolution in eye space remain constant, and so that the corresponding pixels on multiple focal planes overlap with each other one-to-one. Thus, the designs of  FIGS. 3A ,  3 B are well suited for a depth fusing technique without the need for correcting misaligned multiple focal images otherwise resulting from different image magnifications. These benefits may be further understood by analysis of the focus cue and accommodation range. 
         [0032]    The focus cue or accommodation range, ΔD accommodation , which implies the depth range of the 3D volume the system can render, is determined by 
         [0000]    
       
         
           
             
               
                 Δ 
                  
                 
                     
                 
                  
                 
                   D 
                   accommodation 
                 
               
               = 
               
                 
                   
                     Φ 
                     eye 
                     2 
                   
                   
                     Φ 
                     1 
                     2 
                   
                 
                 × 
                 Δ 
                  
                 
                     
                 
                  
                 
                   Φ 
                   DMMD 
                 
               
             
             , 
           
         
       
     
         [0000]    where Φ eye  is the power of the eyepiece  12 , Φ 1  is the power of the relay lens  21 , and ΔΦ DMMD  denotes the range of power by which the deformable mirror  80  can change. The above equation gives and the relationship between the focal length of the relay lens  20  and the eyepiece  12 . Since the eyepiece  12  will relay the system stop, i.e., the DMMD  80 , to form an exit pupil, the ratio between the size of the exit pupil D xp  and the size of the DMMD  80 , is fixed once the desired accommodation range is determined: 
         [0000]    
       
         
           
             
               D 
               XP 
             
             = 
             
               
                 
                   D 
                   DMMD 
                 
                 
                   
                     f 
                     1 
                   
                   / 
                   
                     f 
                     eye 
                   
                 
               
               . 
             
           
         
       
     
         [0033]    One drawback of the folded double-telecentric relay design is that it provides no magnification to the image of the DMD display  60 . Therefore to get a desired system field of view, a field lens  18  may be added in front of the DMD display  60  to magnify the image. (The display  60  may be an emissive display, or maybe a reflective display that is illuminated through an illumination path.) The magnification introduced by the field lens  18  is 
         [0000]    
       
         
           
             m 
             = 
             
               
                 1 
                 
                   
                     
                       z 
                       0 
                     
                     
                       f 
                       field 
                     
                   
                   - 
                   1 
                 
               
               . 
             
           
         
       
     
         [0000]    The system half field of view is then: 
         [0000]    
       
         
           
             HFOV 
             = 
             
               
                 
                   y 
                   DMD 
                   ′ 
                 
                 
                   f 
                   eye 
                 
               
               = 
               
                 
                   
                     
                       y 
                       DMD 
                     
                     × 
                     m 
                   
                   
                     f 
                     eye 
                   
                 
                 . 
               
             
           
         
       
     
         [0034]    Based on the design goal, device specifications, and mechanical considerations, Table 1 lists the first-order system specifications for the designs of  FIGS. 3A ,  3 B. 
         [0000]    
       
         
               
             
               
               
             
               
               
               
             
               
               
             
               
               
               
             
           
               
                 TABLE 1 
               
               
                   
               
               
                 First-order system specifications for virtual display. 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 Microdisplay 
                 0.7″ DMD, XGA resolution, 14 μm pixel 
               
             
          
           
               
                 Deformable mirror defocus range 
                 0~1.2  
                 diopters 
               
               
                 Accommodation Range 
                 0~3  
                 diopters 
               
             
          
           
               
                 Field of view 
                 40° (32.5° H × 24.5° V) 
               
             
          
           
               
                 Relay lens focal length, f 1   
                 43.9  
                 mm 
               
               
                 Eyepiece focal length, f eye   
                 27.8  
                 mm 
               
               
                 Exit pupil diameter 
                 6  
                 mm 
               
               
                   
               
             
          
         
       
     
       Free Form Eyepiece and Compensator 
       [0035]    The optical see-through capabilities of the systems  100 ,  200  may be realized by using beamsplitters to fold the virtual display optics (e.g., DMD  60 , field lens  18 , relay lens group  20 ) out of the way. However, given that the eyepiece  12  in the present design has a short focal length, it was very difficult to design the system conventionally. 
         [0036]    In the instant exemplary design, a more elegant solution was pursued. As shown in  FIGS. 3C ,  3 D, the eyepiece  12  was designed as a wedge-shaped free-form plastic lens. The free-form prism eyepiece  12  may include three rotationally asymmetric surfaces, labeled as S 1 , S 2 , and S 3 , respectively. Considering the virtual display path,  FIG. 3D , a ray from the intermediate image of the display  60  is first refracted by the surface S 3 . After two consecutive reflections by surfaces S 1  and S 2 , the ray is transmitted through the surface S 1  and reaches the exit (eye) pupil of the system. The surface S 1  desirably satisfies the condition of total internal reflection for all rays reflected on the surface S 1 . The surface S 2  of the eyepiece  12  may be coated as a half mirror in order to facilitate the optical see-through capability. A free-form compensator  14 , which may include two rotationally asymmetric surfaces, S 2  and S 4 , may be cemented to the eyepiece  12  to compensate for the aberrations and distortions introduced to the rays from the real-world scene when the two pieces  12 ,  14  are combined together,  FIG. 3C . In addition, an optional cylindrical lens  13  may be included with the free-form compensator  14  to help minimize aberrations and distortion for the see-through path. 
         [0037]    To achieve the desired optical performance, MTF values were selected to evaluate the overall image sharpness. Since the virtual display system  100  was designed backwards from the eyepiece  12  to the display  60 , the goal was to have MTF values no less than 20% at spatial frequency of 36 lp/mm on the display  60 , which is the cut-off frequency for pixel size of 14 μm. The human eye has an angular resolution of 1 arcmin. Therefore the compensator  14  was optimized such that the MTF values at 30 cycles/degree are greater than 0.2 to minimize degradation of the real-world scene. Another important optical performance factor of the system  100 ,  200  was image distortion. In conventional system, distortion is regular and can be compensated easily electronically or computationally. However in systems with off-axis freeform optics, the distortion can be very large and irregular. Therefore the design of the systems  100 ,  200  should have tight constraints on distortion sampled over the entire FOV. The distortion is especially important for the see-through optical path, because it can change the sizes and shapes of objects seen through the see-through view, thus greatly affecting 3D perception. 
       Design and Optimization Procedures 
       [0038]    The design of the systems  100 ,  200  involved two steps, the virtual display path and the optical see-through path. For the virtual display path, the freeform eyepiece  12  was setup with the relay lens  20  and field lens  18  in CodeV and optimized together. The display performance was balanced for fields sampled across the 40-degrees of FOV and across the desired accommodation range of 3 diopters. After the virtual display optimization was finished, the freeform eyepiece  12  was setup alone with the compensator  14  and the compensator&#39;s back surface S 4  was optimized for see-through performance. The see-through performance was optimized for 60 degrees of field of view while emphasizing the central 40 degrees. A progressive optimization strategy was adopted in both steps by gradually increasing the number of variable surface coefficients of the freeform surfaces as the system performance improved. 
         [0039]    In the final design, the freeform eyepiece and compensator surfaces S 1 , S 2 , S 3 , S 4  were described by XY-polynomials to the 10 th  order and prototyped by single-point diamond turning on PMMA. In the system  200  of  FIG. 3B , one of the field lenses elements, element  17 , was optimized, and a diffractive optical feature was added to correct for chromatic aberrations introduced by the freeform eyepiece  12 . All other lenses  19 ,  22 ,  24 ,  26  are all off-the-shelf components to reduce the prototyping costs. 
         [0040]    The polychromatic MTF values of the virtual display, evaluated for a 3 mm pupil, are greater than 0.2 at 36 lp/mm across the 40 degrees of field of view with a central field value of 0.5,  FIGS. 4A-4E . The virtual display also shows minimal distortion,  FIG. 4F . The polychromatic MTF values of the see-through optical path, evaluated for 3 mm pupil, are greater than 0.4 at 30 cycles/degree across the 40 degrees of field of view,  FIGS. 5A-5E . The distortion of the see-through scene was also very well corrected,  FIG. 5F . The prescription for the particular design of  FIG. 3B  is provided as follows. 
       System Prescription for Virtual Display Path 
       [0041]    In Table 2, surfaces #2-#4 specify the free-form eyepiece  12 . Surface #2 and #4 represent the same physical surface and is also denoted as eyepiece surface S 1 . Surface #3 is also denoted as eyepiece surface S 2 , and Surface #5 is also denoted as eyepiece surface S 3 . Surfaces #8-#15 and surfaces #17-#24 are the same group of relay lenses  22 ,  24 ,  26  modeled in double path. The deformable mirror  80  is modeled at Surface #16. Surfaces #25-#26 model the beam splitter  16  at 45 degrees. Surfaces #27-#28 represent the field lens element  17 , and Surfaces #29-#30 represent the field lens element  19 . 
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
             
           
               
                 TABLE 2 
               
             
             
               
                   
               
               
                 System prescription for virtual display path. 
               
             
          
           
               
                 Ele- 
                   
                   
                   
                   
                   
                   
               
               
                 ment 
                   
                   
                   
                   
                   
                   
               
               
                 num- 
                   
                   
                   
                   
                   
                   
               
               
                 ber 
                 Sur- 
                   
                   
                   
                   
                   
               
               
                 used in 
                 face 
                 Surface 
                 Y  
                 Thick- 
                   
                 Refract 
               
               
                 figures 
                 No. 
                 Type 
                 Radius 
                 ness 
                 Material 
                 Mode 
               
               
                   
               
             
          
           
               
                   
                  1  
                 Sphere 
                 Infinity 
                    0.000 
                   
                 Refract 
               
               
                   
                 (Stop) 
                   
                   
                   
                   
                   
               
               
                 12, S1 
                  2 
                 XY Poly 
                 −185.496 
                    0.000 
                 PMMA 
                 Refract 
               
               
                 12, S2 
                  3 
                 XY Poly 
                 −67.446 
                    0.000 
                 PMMA 
                 Reflect 
               
               
                 12, S1 
                  4 
                 XY Poly 
                 −185.496 
                    0.000 
                 PMMA 
                 Reflect 
               
               
                 12, S3 
                  5 
                 XY Poly 
                 −830.046 
                    0.000 
                   
                 Refract 
               
               
                   
                  6 
                 Sphere 
                 Infinity 
                    0.000 
                   
                 Refract 
               
               
                   
                  7 
                 Sphere 
                 Infinity 
                   53.933 
                   
                 Refract 
               
               
                 24 
                  8 
                 Sphere 
                 435.850 
                    4.000 
                 NSF10 
                 Refract 
               
               
                   
                  9 
                 Sphere 
                 36.730 
                   12.070 
                 NBAF10 
                 Refract 
               
               
                   
                 10 
                 Sphere 
                 −53.760 
                   18.079 
                   
                 Refract 
               
               
                 22 
                 11 
                 Sphere 
                 53.760 
                   12.070 
                 NBAF10 
                 Refract 
               
               
                   
                 12 
                 Sphere 
                 −36.730 
                    4.000 
                 NSF10 
                 Refract 
               
               
                   
                 13 
                 Sphere 
                 −435.850 
                   19.826 
                   
                 Refract 
               
               
                 26 
                 14 
                 Sphere 
                 Infinity 
                    2.000 
                 NBK7 
                 Refract 
               
               
                   
                 15 
                 Sphere 
                 38.900 
                    3.502 
                   
                 Refract 
               
               
                 80 
                 16 
                 Sphere 
                 −4000.000 
                  −3.502 
                   
                 Reflect 
               
               
                 26 
                 17 
                 Sphere 
                 38.900 
                  −2.000 
                 NBK7 
                 Refract 
               
               
                   
                 18 
                 Sphere 
                 Infinity 
                 −19.826 
                   
                 Refract 
               
               
                 22 
                 19 
                 Sphere 
                 −435.850 
                  −4.000 
                 NSF10 
                 Refract 
               
               
                   
                 20 
                 Sphere 
                 −36.730 
                 −12.070 
                 NBAF10 
                 Refract 
               
               
                   
                 21 
                 Sphere 
                 53.760 
                 −18.079 
                   
                 Refract 
               
               
                 24 
                 22 
                 Sphere 
                 −53.760 
                 −12.070 
                 NBAF10 
                 Refract 
               
               
                   
                 23 
                 Sphere 
                 36.730 
                  −4.000 
                 NSF10 
                 Refract 
               
               
                   
                 24 
                 Sphere 
                 435.850 
                 −23.000 
                   
                 Refract 
               
               
                 16 
                 25 
                 Sphere 
                 Infinity 
                  −1.600 
                 471400.6541 
                 Refract 
               
               
                   
                 26 
                 Sphere 
                 Infinity 
                 −10.513 
                   
                 Refract 
               
               
                 19 
                 27 
                 Sphere 
                 −46.700 
                  −6.500 
                 NBK7 
                 Refract 
               
               
                   
                 28 
                 Sphere 
                 Infinity 
                  −1.896 
                   
                 Refract 
               
               
                 17 
                 29 
                 Asphere 
                 −102.223 
                  −2.800 
                 PMMA 
                 Refract 
               
               
                   
                 30 
                 Asphere 
                 −61.641 
                  −7.655 
                   
                 Refract 
               
               
                   
               
             
          
         
       
     
       System Prescription for Optical See-Through Path 
       [0042]    In Table 3 surfaces #2 and #3 are eyepiece surfaces S 1  and S 3 , modeled the same as in the virtual display path. Surfaces #4, #5 specify the free-form compensator  14 . Surface #4 is, it an exact replica of Surface #3 (eyepiece surface S 3 ). 
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
               
             
               
               
               
               
               
               
               
               
             
           
               
                 TABLE 3 
               
             
             
               
                   
               
               
                 System prescription for see-through path. 
               
             
          
           
               
                 Ele- 
                   
                   
                   
                   
                   
                   
                   
               
               
                 ment 
                   
                   
                   
                   
                   
                   
                   
               
               
                 num- 
                   
                   
                   
                   
                   
                   
                   
               
               
                 ber 
                   
                   
                   
                   
                   
                   
                   
               
               
                 used  
                   
                   
                   
                   
                   
                   
                   
               
               
                 in 
                 Sur- 
                   
                   
                   
                   
                   
                   
               
               
                 fig- 
                 face 
                 Surface 
                 Y  
                 X 
                 Thick- 
                 Ma- 
                 Refract 
               
               
                 ures 
                 No. 
                 Type 
                 Radius 
                 Radius 
                 ness 
                 terial 
                 Mode 
               
               
                   
               
             
          
           
               
                   
                 1  
                 Sphere 
                 Infinity 
                 Infinity 
                  0.000 
                   
                 Refract 
               
               
                   
                 (Stop) 
                   
                   
                   
                   
                   
                   
               
               
                 12, S1 
                 2 
                 XY Poly 
                 −185.496 
                 −185.496 
                  0.000 
                 PMMA 
                 Refract 
               
               
                 12, S3 
                 3 
                 XY Poly 
                 −67.446 
                 −67.446 
                  0.000 
                 PMMA 
                 Refract 
               
               
                 14, S2 
                 4 
                 XY Poly 
                 −67.446 
                 −67.446 
                  0.000 
                 PMMA 
                 Refract 
               
               
                 14, S4 
                 5 
                 XY Poly 
                 −87.790 
                 −87.790 
                 10.000 
                   
                 Refract 
               
               
                 13 
                 6 
                 Cylin- 
                 Infinity 
                 −103.400 
                  6.5 
                 NBK7 
                 Refract 
               
               
                   
                   
                 drical 
                   
                   
                   
                   
                   
               
               
                 13 
                 7 
                 Sphere 
                 Infinity 
                 Infinity 
                  0.000 
                   
                 Refract 
               
               
                   
               
             
          
         
       
     
         [0043]    As used in the system prescription Tables, e.g., Table 2 or Table 3, the term “XY Poly” refers to a surface which may be represented by the equation 
         [0000]    
       
         
           
             z 
             = 
             
               
                 
                   cr 
                   2 
                 
                 
                   1 
                   + 
                   
                     
                       1 
                       - 
                       
                         
                           ( 
                           
                             1 
                             + 
                             k 
                           
                           ) 
                         
                          
                         
                           c 
                           2 
                         
                          
                         
                           r 
                           
                             2 
                              
                             
                                 
                             
                           
                         
                       
                     
                   
                 
               
               + 
               
                 
                   ∑ 
                   
                     j 
                     = 
                     2 
                   
                   66 
                 
                  
                 
                   
                     C 
                     j 
                   
                    
                   
                     x 
                     m 
                   
                    
                   
                     y 
                     n 
                   
                 
               
             
           
         
       
       
         
           
             
               j 
               = 
               
                 
                   
                     
                       
                         ( 
                         
                           m 
                           + 
                           n 
                         
                         ) 
                       
                       2 
                     
                     + 
                     m 
                     + 
                     
                       3 
                        
                       n 
                     
                   
                   2 
                 
                 + 
                 1 
               
             
             , 
           
         
       
     
         [0000]    where z is the sag of the free-form surface measured along the z-axis of a local x, y, z coordinate system, c is the vertex curvature (CUY), r is the radial distance, k is the conic constant, and C, is the coefficient for x m y n . The term “Asphere” in the Tables refers to an aspherical surface which may be represented by the equation 
         [0000]    
       
         
           
             
               z 
               = 
               
                 
                   
                     cr 
                     2 
                   
                   
                     1 
                     + 
                     
                       
                         1 
                         - 
                         
                           
                             ( 
                             
                               1 
                               + 
                               k 
                             
                             ) 
                           
                            
                           
                             c 
                             2 
                           
                            
                           
                             r 
                             2 
                           
                         
                       
                     
                   
                 
                 + 
                 
                   Ar 
                   4 
                 
                 + 
                 
                   Br 
                   6 
                 
                 + 
                 
                   Cr 
                   8 
                 
                 + 
                 
                   Dr 
                   10 
                 
                 + 
                 
                   Er 
                   12 
                 
               
             
             , 
           
         
       
     
         [0000]    where z is the sag of the surface measured along the z-axis of a local x, y, z coordinate system, c is the vertex curvature, r is the radial distance, k is the conic constant, A through E are the 4th, 6th, 8th, 10th and 12th order deformation coefficients, respectively. 
         [0000]    
       
         
               
             
               
               
               
               
             
           
               
                 TABLE 4 
               
               
                   
               
               
                 Optical surface prescription of Surface #2 and #4 of Table 2. 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 Y Radius 
                 −1.854965E+02 
                 X**2 * Y**5 
                 −1.505674E−10 
               
               
                 Conic Constant 
                 −2.497467E+01 
                 X * Y**6 
                  0.000000E+00 
               
               
                 X 
                  0.000000E+00  
                 Y**7 
                 −4.419392E−11 
               
               
                 Y 
                  0.000000E+00 
                 X**8 
                  4.236650E−10 
               
               
                 X**2 
                 −2.331157E−03 
                 X**7 * Y 
                  0.000000E+00 
               
               
                 X * Y 
                  0.000000E+00 
                 X**6 * Y**2 
                 −1.079269E−10 
               
               
                 Y**2 
                  6.691726E−04  
                 X**5 * Y**3 
                  0.000000E+00 
               
               
                 X**3 
                  0.000000E+00  
                 X**4 * Y**4 
                 −1.678245E−10 
               
               
                 X**2 * Y 
                 −1.066279E−04 
                 X**3 * Y**5 
                  0.000000E+00 
               
               
                 X Y**2 
                  0.000000E+00  
                 X**2 * Y**6 
                  2.198604E−12 
               
               
                 Y**3 
                 −2.956368E−05 
                 X * Y**7 
                  0.000000E+00 
               
               
                 X**4 
                 −1.554280E−06 
                 Y**8 
                 −2.415118E−12 
               
               
                 X**3 * Y 
                  0.000000E+00  
                 X**9 
                  0.000000E+00 
               
               
                 X**2 * Y**2 
                  1.107189E−06 
                 X**8 * Y 
                  4.113054E−12 
               
               
                 X * Y**3 
                  0.000000E+00 
                 X**7 * Y**2 
                  0.000000E+00 
               
               
                 Y**4 
                  1.579876E−07 
                 X**6 * Y**3 
                 −1.805964E−12 
               
               
                 X**5 
                  0.000000E+00  
                 X**5 * Y**4 
                  0.000000E+00 
               
               
                 X**4 * Y 
                  1.789364E−07 
                 X**4 * Y**5 
                  9.480632E−13 
               
               
                 X**3 * Y**2 
                  0.000000E+00 
                 X**3 * Y**6 
                  0.000000E+00 
               
               
                 X**2 * Y**3 
                 −2.609879E−07 
                 X**2 * Y**7 
                  2.891726E−13 
               
               
                 X * Y**4 
                  0.000000E+00 
                 X * Y**8 
                  0.000000E+00 
               
               
                 Y**5 
                 −6.129549E−10 
                 Y**9 
                 −2.962804E−14 
               
               
                 X**6 
                 −3.316779E−08 
                 X**10 
                 −6.030361E−13 
               
               
                 X**5 * Y 
                  0.000000E+00  
                 X**9 * Y 
                  0.000000E+00 
               
               
                 X**4 * Y**2 
                  9.498635E−09  
                 X**8 * Y**2 
                 −7.368710E−13 
               
               
                 X**3 * Y**3 
                  0.000000E+00 
                 X**7 * Y**3 
                  0.000000E+00 
               
               
                 X**2 * Y**4 
                  9.042084E−09 
                 X**6 * Y**4 
                  9.567750E−13 
               
               
                 X * Y**5 
                  0.000000E+00 
                 X**5 * Y**5 
                  0.000000E+00 
               
               
                 Y**6 
                 −4.013470E−10 
                 X**4 * Y**6 
                  4.280494E−14 
               
               
                 X**7 
                  0.000000E+00 
                 X**3 * Y**7 
                  0.000000E+00 
               
               
                 X**6 * Y 
                 −8.112755E−10 
                 X**2 * Y**8 
                 −7.143578E−15 
               
               
                 X**5 * Y**2 
                  0.000000E+00  
                 X * Y**9 
                  0.000000E+00 
               
               
                 X**4 * Y**3 
                  1.251040E−09 
                 Y**10 
                  3.858414E−15 
               
               
                 X**3 * Y**4 
                  0.000000E+00 
                 N-Radius 
                  1.000000E+00 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE 5 
               
             
             
               
                   
               
               
                 Decenter of Surface #2 and #4 of Table 2, relative to Surface #1 of Table 2. 
               
             
          
           
               
                 Y DECENTER 
                 Z DECENTER 
                 ALPHA TILT 
               
               
                   
               
               
                 6.775E+00 
                 2.773E+01 
                 7.711E+00 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
             
           
               
                 TABLE 6 
               
               
                   
               
               
                 Optical surface prescription of Surface #3 of Table 2. 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 Y Radius 
                 −6.744597E+01 
                 X**2 * Y**5 
                 −3.464751E−11 
               
               
                 Conic Constant 
                 −1.258507E+00 
                 X * Y**6 
                  0.000000E+00 
               
               
                 X 
                  0.000000E+00 
                 Y**7 
                 −8.246179E−12 
               
               
                 Y 
                  0.000000E+00 
                 X**8 
                 −2.087865E−11 
               
               
                 X**2 
                 −1.300207E−03 
                 X**7 * Y 
                  0.000000E+00 
               
               
                 X * Y 
                  0.000000E+00 
                 X**6 * Y**2 
                  2.845323E−11 
               
               
                 Y**2 
                  4.658585E−04  
                 X**5 * Y**3 
                  0.000000E+00 
               
               
                 X**3 
                  0.000000E+00  
                 X**4 * Y**4 
                 −5.043398E−12 
               
               
                 X**2 * Y 
                 −1.758475E−05 
                 X**3 * Y**5 
                  0.000000E+00 
               
               
                 X Y**2 
                  0.000000E+00  
                 X**2 * Y**6 
                  2.142939E−14 
               
               
                 Y**3 
                 −1.684923E−06 
                 X * Y**7 
                  0.000000E+00 
               
               
                 X**4 
                 −1.463720E−06 
                 Y**8 
                  1.607499E−12 
               
               
                 X**3 * Y 
                  0.000000E+00 
                 X**9 
                  0.000000E+00 
               
               
                 X**2 * Y**2 
                 −1.108359E−06 
                 X**8 * Y 
                 −1.922597E−12 
               
               
                 X * Y**3 
                  0.000000E+00 
                 X**7 * Y**2 
                  0.000000E+00 
               
               
                 Y**4 
                 −1.098749E−07 
                 X**6 * Y**3 
                  1.100072E−13 
               
               
                 X**5 
                  0.000000E+00  
                 X**5 * Y**4 
                  0.000000E+00 
               
               
                 X**4 * Y 
                 −7.146353E−08 
                 X**4 * Y**5 
                 −4.806130E−14 
               
               
                 X**3 * Y**2 
                  0.000000E+00  
                 X**3 * Y**6 
                  0.000000E+00 
               
               
                 X**2 * Y**3 
                 −1.150619E−08 
                 X**2 * Y**7 
                 −2.913177E−14 
               
               
                 X * Y**4 
                  0.000000E+00 
                 X * Y**8 
                  0.000000E+00 
               
               
                 Y**5 
                  5.911371E−09  
                 Y**9 
                  9.703717E−14 
               
               
                 X**6 
                 −5.406591E−10 
                 X**10 
                  2.032150E−13 
               
               
                 X**5 * Y 
                  0.000000E+00  
                 X**9 * Y 
                  0.000000E+00 
               
               
                 X**4 * Y**2 
                 −1.767107E−09 
                 X**8 * Y**2 
                 −1.037107E−13 
               
               
                 X**3 * Y**3 
                  0.000000E+00  
                 X**7 * Y**3 
                  0.000000E+00 
               
               
                 X**2 * Y**4 
                 −7.415334E−10 
                 X**6 * Y**4 
                  3.602862E−14 
               
               
                 X * Y**5 
                  0.000000E+00 
                 X**5 * Y**5 
                  0.000000E+00 
               
               
                 Y**6 
                 −5.442400E−10 
                 X**4 * Y**6 
                 −8.831469E−15 
               
               
                 X**7 
                  0.000000E+00 
                 X**3 * Y**7 
                  0.000000E+00 
               
               
                 X**6 * Y 
                  6.463414E−10  
                 X**2 * Y**8 
                  2.178095E−15 
               
               
                 X**5 * Y**2 
                  0.000000E+00  
                 X * Y**9 
                  0.000000E+00 
               
               
                 X**4 * Y**3 
                  1.421597E−10 
                 Y**10 
                  1.784074E−15 
               
               
                 X**3 * Y**4 
                  0.000000E+00 
                 N-Radius 
                  1.000000E+00 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE 7 
               
               
                   
               
               
                 Decenter of Surface #3 of Table 2 relative to Surface #1 of Table 2. 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 Y DECENTER 
                 Z DECENTER 
                 ALPHA TILT 
               
               
                 1.329E+01 
                 4.321E+01 
                 −8.856E+00 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
             
           
               
                 TABLE 8 
               
               
                   
               
               
                 Optical surface prescription of Surface #5 of Table 2. 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 Y Radius 
                 −8.300457E+02 
                 X**2 * Y**5 
                  4.051880E−08 
               
               
                 Conic Constant 
                 −9.675799E+00 
                 X * Y**6 
                  0.000000E+00 
               
               
                 X 
                  0.000000E+00 
                 Y**7 
                 −3.973293E−09 
               
               
                 Y 
                  0.000000E+00  
                 X**8 
                 −1.881791E−10 
               
               
                 X**2 
                 −1.798206E−04 
                 X**7 * Y 
                  0.000000E+00 
               
               
                 X * Y 
                  0.000000E+00 
                 X**6 * Y**2 
                  5.519986E−09 
               
               
                 Y**2 
                 −2.606383E−03 
                 X**5 * Y**3 
                  0.000000E+00 
               
               
                 X**3 
                  0.000000E+00  
                 X**4 * Y**4 
                  3.822268E−09 
               
               
                 X**2 * Y 
                 −7.767146E−05 
                 X**3 * Y**5 
                  0.000000E+00 
               
               
                 X Y**2 
                  0.000000E+00 
                 X**2 * Y**6 
                 −3.024448E−09 
               
               
                 Y**3 
                 −8.958581E−05 
                 X * Y**7 
                  0.000000E+00 
               
               
                 X**4 
                  1.978414E−05 
                 Y**8 
                  2.673713E−11 
               
               
                 X**3 * Y 
                  0.000000E+00 
                 X**9 
                  0.000000E+00 
               
               
                 X**2 * Y**2 
                  2.081156E−05  
                 X**8 * Y 
                  1.006915E−10 
               
               
                 X * Y**3 
                  0.000000E+00  
                 X**7 * Y**2 
                  0.000000E+00 
               
               
                 Y**4 
                 −1.073001E−06 
                 X**6 * Y**3 
                 −2.945084E−10 
               
               
                 X**5 
                  0.000000E+00  
                 X**5 * Y**4 
                  0.000000E+00 
               
               
                 X**4 * Y 
                  2.585164E−07 
                 X**4 * Y**5 
                  5.958040E−10 
               
               
                 X**3 * Y**2 
                  0.000000E+00  
                 X**3 * Y**6 
                  0.000000E+00 
               
               
                 X**2 * Y**3 
                 −2.752516E−06 
                 X**2 * Y**7 
                 −3.211903E−10 
               
               
                 X * Y**4 
                  0.000000E+00 
                 X * Y**8 
                  0.000000E+00 
               
               
                 Y**5 
                 −1.470053E−06 
                 Y**9 
                  2.296303E−11 
               
               
                 X**6 
                 −1.116386E−07 
                 X**10 
                  5.221834E−12 
               
               
                 X**5 * Y 
                  0.000000E+00 
                 X**9 * Y 
                  0.000000E+00 
               
               
                 X**4 * Y**2 
                 −3.501439E−07 
                 X**8 * Y**2 
                  1.135044E−11 
               
               
                 X**3 * Y**3 
                  0.000000E+00 
                 X**7 * Y**3 
                  0.000000E+00 
               
               
                 X**2 * Y**4 
                  1.324057E−07 
                 X**6 * Y**4 
                 −1.050621E−10 
               
               
                 X * Y**5 
                  0.000000E+00 
                 X**5 * Y**5 
                  0.000000E+00 
               
               
                 Y**6 
                 −9.038017E−08 
                 X**4 * Y**6 
                  5.624902E−11 
               
               
                 X**7 
                  0.000000E+00  
                 X**3 * Y**7 
                  0.000000E+00 
               
               
                 X**6 * Y 
                  3.397174E−10 
                 X**2 * Y**8 
                  5.369592E−12 
               
               
                 X**5 * Y**2 
                  0.000000E+00  
                 X * Y**9 
                  0.000000E+00 
               
               
                 X**4 * Y**3 
                 −1.873966E−08 
                 Y**10 
                  2.497657E−12 
               
               
                 X**3 * Y**4 
                  0.000000E+00  
                 N-Radius 
                  1.000000E+00 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE 9 
               
             
             
               
                   
               
               
                 Decenter of Surface #5 of Table 2 relative to Surface #1 of Table 2. 
               
             
          
           
               
                 Y DECENTER 
                 Z DECENTER 
                 ALPHA TILT 
               
               
                   
               
               
                 .427E+01 
                 3.347E+01 
                 7.230E+01 
               
               
                   
               
             
          
         
       
     
         [0044]    Turning to the prescription of the second field lens element  17 , both surfaces of the field lens element  17  are aspheric surfaces. Additionally, Surface #29 (Table 2) of field lens element  17  has a kinoform diffractive optical feature which may be represented according to the following equation 
         [0000]      φ= Ar   2   +Br   4   +Cr   6   +Dr   8   +Er   10 ,
 
         [0000]    where φ is the phase function of the diffractive element, r is the radial distance, A through E are the 4th, 6th, 8th, 10th and 12th order phase coefficients, respectively. The surface prescriptions of second field lens element  17  are provide in Table 10-Table 12. 
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE 10 
               
               
                   
               
               
                 Surface Prescription for Surface #29 of Table 2. 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 Y Radius 
                   
                 1.022230E+02 
               
               
                 Conic Constant  
                 (K) 
                 1.091191E+01 
               
               
                  4th Order Coefficient  
                 (A) 
                 4.372314E−06 
               
               
                  6th Order Coefficient  
                 (B) 
                 −6.940740E−08 
               
               
                  8th Order Coefficient  
                 (C) 
                 8.588869E−11 
               
               
                 10th Order Coefficient  
                 (D) 
                 2.348571E−14 
               
               
                 12th Order Coefficient  
                 (E) 
                 −1.463306E−16 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
             
               
               
               
               
             
           
               
                 TABLE 11 
               
               
                   
               
               
                 Diffractive Optical Element Phase  
               
               
                 Data for Surface #29 of Table 2. 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 Construction Wavelength (nm) 
                 525 
               
             
          
           
               
                   
                 R**2  
                 (HCO C1) 
                 −1.295858E−03 
               
               
                   
                 R**4  
                 (HCO C2) 
                 −3.879339E−07 
               
               
                   
                 R**6  
                 (HCO C3) 
                 8.494999E−09 
               
               
                   
                 R**8  
                 (HCO C4) 
                 −1.771348E−13 
               
               
                   
                 R**10 
                 (HCO C5) 
                 −3.584229E−15 
               
               
                   
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
             
           
               
                 TABLE 12 
               
               
                   
               
               
                 Surface Prescription for Surface #30 of Table 2. 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 Y Radius 
                   
                 6.164108E+01 
               
               
                   
                 Conic Constant 
                 (K) 
                 9.828641E+00 
               
               
                   
                  4th Order Coefficient 
                 (A) 
                 5.898651E−05 
               
               
                   
                  6th Order Coefficient 
                 (B) 
                 −2.951081E−07 
               
               
                   
                  8th Order Coefficient 
                 (C) 
                 −3.440910E−10 
               
               
                   
                 10th Order Coefficient 
                 (D) 
                 1.785109E−13 
               
               
                   
                 12th Order Coefficient 
                 (E) 
                 2.803121E−15 
               
               
                   
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
             
           
               
                 TABLE 13 
               
               
                   
               
               
                 Optical surface prescription of Surface #5 of Table 3. 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 Y Radius 
                 −8.779024E+01 
                 X**2 * Y**5 
                 −8.011955E−11 
               
               
                 Conic  
                 −7.055198E+00 
                 X * Y**6 
                   0.000000E+00 
               
               
                 Constant 
                   
                   
                   
               
               
                 X 
                   0.000000E+00 
                 Y**7 
                   3.606142E-11 
               
               
                 Y 
                   0.000000E+00 
                 X**8 
                   3.208020E-11 
               
               
                 X**2 
                 −3.191225E−03 
                 X**7 * Y 
                   0.000000E+00 
               
               
                 X * Y 
                   0.000000E+00 
                 X**6 * Y**2 
                 −2.180416E−11 
               
               
                 Y**2 
                   4.331992E−03 
                 X**5 * Y**3 
                   0.000000E+00 
               
               
                 X**3 
                   0.000000E+00 
                 X**4 * Y**4 
                 −3.616135E−11 
               
               
                 X**2 * Y 
                 −9.609025E−05 
                 X**3 * Y**5 
                   0.000000E+00 
               
               
                 X Y**2 
                   0.000000E+00 
                 X**2 * Y**6 
                 −5.893434E−12 
               
               
                 Y**3 
                 −2.432809E−05 
                 X * Y**7 
                   0.000000E+00 
               
               
                 X**4 
                 −2.955089E−06 
                 Y**8 
                   3.081069E−12 
               
               
                 X**3 * Y 
                   0.000000E+00 
                 X**9 
                   0.000000E+00 
               
               
                 X**2 * Y**2 
                   2.096887E−07 
                 X**8 * Y 
                   1.267096E−12 
               
               
                 X * Y**3 
                   0.000000E+00 
                 X**7 * Y**2 
                   0.000000E+00 
               
               
                 Y**4 
                 −9.184356E−07 
                 X**6 * Y**3 
                 −1.848104E−12 
               
               
                 X**5 
                   0.000000E+00 
                 X**5 * Y**4 
                   0.000000E+00 
               
               
                 X**4 * Y 
                   3.707556E-08 
                 X**4 * Y**5 
                   5.208420E−14 
               
               
                 X**3 * Y**2 
                   0.000000E+00 
                 X**3 * Y**6 
                   0.000000E+00 
               
               
                 X**2 * Y**3 
                 −1.535357E−07 
                 X**2 * Y**7 
                   1.198597E-13 
               
               
                 X * Y**4 
                   0.000000E+00 
                 X * Y**8 
                   0.000000E+00 
               
               
                 Y**5 
                 −1.445904E−08 
                 Y**9 
                 −6.834914E−14 
               
               
                 X**6 
                 −4.440851E−09 
                 X**10 
                 −1.706677E−14 
               
               
                 X**5 * Y 
                   0.000000E+00 
                 X**9 * Y 
                   0.000000E+00 
               
               
                 X**4 * Y**2 
                   1.686424E−09 
                 X**8 * Y**2 
                 −1.614840E−14 
               
               
                 X**3 * Y**3 
                   0.000000E+00 
                 X**7 * Y**3 
                   0.000000E+00 
               
               
                 X**2 * Y**4 
                   6.770909E-09 
                 X**6 * Y**4 
                   8.739087E−14 
               
               
                 X * Y**5 
                   0.000000E+00 
                 X**5 * Y**5 
                   0.000000E+00 
               
               
                 Y**6 
                 −3.713094E−10 
                 X**4 * Y**6 
                   3.940903E−15 
               
               
                 X**7 
                   0.000000E+00 
                 X**3 * Y**7 
                   0.000000E+00 
               
               
                 X**6 * Y 
                 −1.316067E−10 
                 X**2 * Y**8 
                   5.435162E−15 
               
               
                 X**5 * Y**2 
                   0.000000E+00 
                 X * Y**9 
                   0.000000E+00 
               
               
                 X**4 * Y**3 
                   7.924387E−10 
                 Y**10 
                 −2.259169E−15 
               
               
                 X**3 * Y**4 
                   0.000000E+00 
                 N-Radius 
                   1.000000E+00 
               
               
                   
               
             
          
         
       
     
         [0000]                                  TABLE 14                   Decenter of Surface #5 relative to Surface #1 of Table 3.            Y DECENTER   Z DECENTER   ALPHA TILT               3.358E+00   4.900E+01   6.765E+00                    
Alternative Exemplary Design without Cylindrical Lens
 
         [0045]    In the designs of  FIGS. 3A ,  3 B above, an optional cylindrical lens  13  has been included with the free-form compensator  14  to help minimize aberrations and distortion. An alternative design is also provided without the cylindrical lens  13 , in which the virtual display path is the same as shown in  FIG. 3B  and Table 2. The only difference among the remaining surfaces of the see-through path in the absence of the cylindrical lens  13  is the eyepiece/compensator surface S 2  (Surface #5 in the optical see-through path of Table 3). In Table 15, Surfaces #2 and #3 are eyepiece surfaces S 1  and S 3 , modeled the same as in the virtual display path. Surfaces #4-5 describe the free-form compensator  14 . Surface #4 is a exact replica of Surface #3. 
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
             
           
               
                 TABLE 15 
               
             
             
               
                   
               
               
                 Alternative eyepiece optics prescription without cylindrical lens. 
               
             
          
           
               
                   
                 Surface  
                 Y  
                 X  
                   
                   
                 Refract  
               
               
                   
                 Type  
                 Radius 
                 Radius 
                 Thickness 
                 Material 
                 Mode 
               
               
                   
               
               
                 Object 
                 Sphere 
                 Infinity 
                 Infinity 
                 −666.700 
                   
                 Refract 
               
               
                 1  
                 Sphere 
                 Infinity 
                 Infinity 
                    0.000 
                   
                 Refract 
               
               
                 (Stop) 
                   
                   
                   
                   
                   
                   
               
               
                 2 
                 XY  
                 −185.496 
                 −185.496 
                    0.000 
                 PMMA 
                 Refract 
               
               
                   
                 Polynomial 
                   
                   
                   
                   
                   
               
               
                 3 
                 XY  
                  −67.446  
                  −67.446 
                    0.000 
                 PMMA 
                 Refract 
               
               
                   
                 Polynomial 
                   
                   
                   
                   
                   
               
               
                 4 
                 XY  
                  −67.446  
                  −67.446 
                    0.000 
                 PMMA 
                 Refract 
               
               
                   
                 Polynomial 
                   
                   
                   
                   
                   
               
               
                 5 
                 XY  
                 −492.346 
                 −492.346 
                    0.000 
                   
                 Refract 
               
               
                   
                 Polynomial 
                   
                   
                   
                   
                   
               
               
                 Image 
                 Sphere 
                 Infinity 
                 Infinity  
                    0.000 
                   
                 Refract 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
             
           
               
                 TABLE 16 
               
               
                   
               
               
                 Optical surface prescription of Surface #5 of Table 15. 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 Y Radius 
                 −4.923462E+02 
                 X**2 * Y**5 
                 −1.476237E−10 
               
               
                 Conic  
                   3.982960E+00 
                 X * Y**6 
                   0.000000E+00 
               
               
                 Constant 
                   
                   
                   
               
               
                 X 
                   0.000000E+00 
                 Y**7 
                   2.044705E−11 
               
               
                 Y 
                   0.000000E+00 
                 X**8 
                   2.971746E−11 
               
               
                 X**2 
                 −3.001720E−03 
                 X**7 * Y 
                   0.000000E+00 
               
               
                 X * Y 
                   0.000000E+00 
                 X**6 * Y**2 
                 −6.199724E−12 
               
               
                 Y**2 
                 −5.233825E−04 
                 X**5 * Y**3 
                   0.000000E+00 
               
               
                 X**3 
                   0.000000E+00 
                 X**4 * Y**4 
                 −2.279723E−11 
               
               
                 X**2 * Y 
                 −6.009699E-05 
                 X**3 * Y**5 
                   0.000000E+00 
               
               
                 X Y**2 
                   0.000000E+00 
                 X**2 * Y**6 
                 −1.041364E−12 
               
               
                 Y**3 
                 −2.244921E−05 
                 X * Y**7 
                   0.000000E+00 
               
               
                 X**4 
                 −6.379076E-07 
                 Y**8 
                   1.125487E−12 
               
               
                 X**3 * Y 
                   0.000000E+00 
                 X**9 
                   0.000000E+00 
               
               
                 X**2 * Y**2 
                   2.968752E-06 
                 X**8 * Y 
                   1.210373E−12 
               
               
                 X * Y**3 
                   0.000000E+00 
                 X**7 * Y**2 
                   0.000000E+00 
               
               
                 Y**4 
                   3.771516E−07 
                 X**6 * Y**3 
                 −1.331110E−12 
               
               
                 X**5 
                   0.000000E+00 
                 X**5 * Y**4 
                   0.000000E+00 
               
               
                 X**4 * Y 
                   5.359865E-08 
                 X**4 * Y**5 
                 −9.781602E−14 
               
               
                 X**3 * Y**2 
                   0.000000E+00 
                 X**3 * Y**6 
                   0.000000E+00 
               
               
                 X**2 * Y**3 
                 −1.965407E−07 
                 X**2 * Y**7 
                   4.515428E−13 
               
               
                 X * Y**4 
                   0.000000E+00 
                 X * Y**8 
                   0.000000E+00 
               
               
                 Y**5 
                 −7.301859E−09 
                 Y**9 
                 −5.050786E−14 
               
               
                 X**6 
                 −6.841269E−09 
                 X**10 
                 −1.058279E−14 
               
               
                 X**5 * Y 
                   0.000000E+00 
                 X**9 * Y 
                   0.000000E+00 
               
               
                 X**4 * Y**2 
                 −2.507411E−09 
                 X**8 * Y**2 
                 −2.975833E−14 
               
               
                 X**3 * Y**3 
                   0.000000E+00 
                 X**7 * Y**3 
                   0.000000E+00 
               
               
                 X**2 * Y**4 
                   4.627014E−10 
                 X**6 * Y**4 
                   6.309574E−14 
               
               
                 X * Y**5 
                   0.000000E+00 
                 X**5 * Y**5 
                   0.000000E+00 
               
               
                 Y**6 
                 −4.841692E−10 
                 X**4 * Y**6 
                   1.214005E−15 
               
               
                 X**7 
                   0.000000E+00 
                 X**3 * Y**7 
                   0.000000E+00 
               
               
                 X**6 * Y 
                 −3.343485E−10 
                 X**2 * Y**8 
                   1.180350E−14 
               
               
                 X**5 * Y**2 
                   0.000000E+00 
                 X * Y**9 
                   0.000000E+00 
               
               
                 X**4 * Y**3 
                   7.999315E−10 
                 Y**10 
                 −5.938353E−16 
               
               
                 X**3 * Y**4 
                   0.000000E+00 
                 N-Radius 
                   1.000000E+00 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
             
           
               
                 TABLE 17 
               
               
                   
               
               
                 Decenter of Surface #5 of Table 15 
               
               
                 relative to Surface #1 of Table 15. 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 Y DECENTER 
                 Z DECENTER 
                 ALPHA TILT 
               
               
                 4.618E+00 
                 4.853E+01 
                 7.007E+00 
               
               
                   
               
             
          
         
       
     
       Prototype of System of FIG. 3B 
       [0046]    A prototype of the multi-focal-plane display system  200  of  FIG. 3B  was built with off-the-shelf lenses and customized optics, 3D views of which are provided in  FIGS. 6A ,  6 B. The system  200  was folded to avoid conflicting with the viewer&#39;s head. Custom electronics were also developed to control and synchronize the display images on the DMD  60  (LUXBEAM® 4500, VISITECH, Drammen, Norway.), the illumination of the LED (not shown), and focal-plane switching of the deformable mirror  80  (OKO® Technologies MMDM10-1-focus, Flexible Optical B.V., Rijswijk, Netherlands). 
         [0047]    A continuous 3D scene was rendered which was composed of a slanted planar object and a green floor grid, both extending from 0 to 2.5 diopters. The scene was decomposed onto 6 focal planes, placed at 3 diopter, 2.4 diopter, 1.8 diopter, 1.2 diopter, 0.6 diopter and 0 diopter, based on the targets&#39; depth values, and a depth-fusing technique was used to blend the 6 focal planes into a smooth continuum. The whole 3D scene was refreshed at about 60 Hz; thus, flickering was not visible.  FIG. 7A  shows the actual 6-focal-plane scene as seen through the system; the image was sharp and had very low distortion. Without special algorithms, the pixels on different focal planes overlap and fuse smoothly due to the constant field of view design. In addition, a camera lens with shallow depth of field was used and manually focused at different parts of the scene. In  FIG. 7B , the resolution target was displayed on the back wall, and the camera was focused at about 2 m. The near end of the floor grid was out of focus and the back grids as well as the logos were in sharp focus. In  FIG. 7C , the resolution target was displayed on the front focal plane, the camera was focused at 30 cm, and now the near scene was in focus and the content on the back was blurred. Thus, the prototype was demonstrated to be capable of rendering 6 or more focal planes of high quality, high resolution color images at a flickering-free speed. It also had very good optical see-through performance for augmented reality applications, and has the potentially to provide higher depth perception accuracy, higher stereoacuity, and lower user fatigue. 
         [0048]    Alternative Exemplary Tunable Lens 
         [0049]    In another of its aspects, the present invention provides an exemplary multi-focal-plane display system  300  combining high-speed display technologies, such as Liquid Crystal on Silicon (LCOS) and Ferroelectric Liquid Crystal On Silicon (FLCoS) and a high-speed active refractive optical element, such as an electrical tunable lens  380 . The Specific design is based on a 0.8″ WXGA LCOS/FLCOS display  360  and a 10 mm aperture electrical tunable lens  380  (Optotune EL-10-30, Optotune AG, Dietikon, Switzerland.)). The tunable lens  380  changes shape when electrical current flows through the lens  380  to produce a change in optical power. The Optotune lens  380  has a response time of about 2.5 ms and therefore potentially can be used in multi-focal-plane displays. 
         [0000]    
       
         
               
             
               
               
             
           
               
                 TABLE 18 
               
               
                   
               
               
                 Design specification for tunable lens system. 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                 Microdisplay 
                 0.8” LCOS/FLCOS WXGA, 14, μm pixel 
               
               
                 Tunable Lens 
                 10 mm aperture, 12 diopter, focus range 
               
               
                 Virtual Display Field of view 
                 50° (43.6° H x 26.4° V) 
               
               
                 Pupil size 
                 10 mm 
               
               
                 Eye Clearance 
                 20 mm (accommodates low-profile glasses) 
               
               
                 Image Quality (DMD space) 
                 MTF &gt; 0.2 at 36 lp/mm (5 mm pupil) 
               
               
                 Display Distortion 
                 &lt; 2.0 % 
               
               
                 Virtual Image Distance 
                 33 cm ~ Infinity 
               
               
                 f-number 
                 2.2 
               
               
                   
               
             
          
         
       
     
         [0050]    The final layout of the design is shown in  FIGS. 10A-11 . A relay lens group (lenses  302 ,  304 ,  306 ,  380 ,  308 ,  310 ) relays the image to a spherical mirror  318 , which acts as an eyepiece and forms a virtual image for a user. Optionally, the mirror  318  may be aspherical. A beamsplitter  316  is used to enable see-through capability. The tunable lens  380  may provide the system stop and the system may be telecentric to the microdisplay  360  due to the requirements of LCOS/FLCoS. Enough space is also given for an illumination beamsplitter  317 ,  FIG. 11 . The prescription of the system  300  is provided in Table 19-Table 26. (In Table 19, surfaces #9-12 model the Optotune electric tunable lens  380 .) The performance of the system is illustrated in  FIGS. 12A ,  12 B. 
         [0000]    
       
         
               
             
               
               
               
               
               
               
               
             
           
               
                 TABLE 19 
               
             
             
               
                   
               
               
                 Tunable lens system prescription. 
               
             
          
           
               
                 Ele- 
                   
                   
                   
                   
                   
                   
               
               
                 ment 
                   
                   
                   
                   
                   
                   
               
               
                 num- 
                   
                   
                   
                   
                   
                   
               
               
                 ber 
                 Sur- 
                   
                   
                   
                   
                   
               
               
                 used in 
                 face 
                 Surface 
                 Y  
                 Thick- 
                   
                 Refract 
               
               
                 figures 
                 No. 
                 Type 
                 Radius 
                 ness 
                 Material 
                 Mode 
               
               
                   
               
               
                   
                  1  
                 Sphere 
                 Infinity 
                   44.000 
                   
                 Refract 
               
               
                   
                 (Stop) 
                   
                   
                   
                   
                   
               
               
                   
                  2 
                 Sphere 
                 Infinity 
                 −24.000 
                   
                 Reflect 
               
               
                 318 
                  3 
                 Sphere 
                 68.000 
                   34.000 
                   
                 Reflect 
               
               
                 316 
                  4 
                 Sphere 
                 Infinity 
                   17.371 
                   
                 Refract 
               
               
                 310 
                  5 
                 Asphere 
                 −23.777 
                    6.000 
                 PMMA 
                 Refract 
               
               
                   
                  6 
                 Asphere 
                 363.193 
                    0.100 
                   
                 Refract 
               
               
                 308 
                  7 
                 Sphere 
                 39.587 
                    6.000 
                 NSF11 
                 Refract 
               
               
                   
                  8 
                 Sphere 
                 −119.109 
                    4.385 
                   
                 Refract 
               
               
                 380 
                  9 
                 Sphere 
                 Infinity 
                    0.500 
                 BK7 
                 Refract 
               
               
                 380 
                 10 
                 Sphere 
                 Infinity 
                    4.377 
                   
                 Refract 
               
               
                 380 
                 11 
                 Sphere 
                 30.270 
                    2.023 
                 ′OL1024′ 
                 Refract 
               
               
                 380 
                 12 
                 Sphere 
                 Infinity 
                    0.500 
                 BK7 
                 Refract 
               
               
                   
                 13 
                 Sphere 
                 Infinity 
                    3.724 
                   
                 Refract 
               
               
                 306 
                 14 
                 Asphere 
                 −24.004 
                    5.999 
                 Polystyrene 
                 Refract 
               
               
                   
                 15 
                 Asphere 
                 27.079 
                    0.251 
                   
                 Refract 
               
               
                 304 
                 16 
                 Sphere 
                 38.710 
                    5.944 
                 Zeonex 
                 Refract 
               
               
                   
                   
                   
                   
                   
                 ZE48R 
                   
               
               
                   
                 17 
                 Sphere 
                 −21.557 
                    5.631 
                   
                 Refract 
               
               
                 302 
                 18 
                 Asphere 
                 33.959 
                    9.698 
                 Zeonex 
                 Refract 
               
               
                   
                   
                   
                   
                   
                 ZE48R 
                   
               
               
                   
                 19 
                 Asphere 
                 −21.555 
                   29.000 
                   
                 Refract 
               
               
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
             
           
               
                 TABLE 20 
               
               
                   
               
               
                 Surface Prescription for Surface #5 of Table 19. 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 Y Radius 
                   
                 −23.777 
               
               
                   
                 Conic Constant 
                 (K) 
                 2.040996E+00 
               
               
                   
                  4th Order Coefficient 
                 (A) 
                 1.385342E−04 
               
               
                   
                  6th Order Coefficient 
                 (B) 
                 −1.022594E−06 
               
               
                   
                  8th Order Coefficient 
                 (C) 
                 8.784855E−09 
               
               
                   
                 10th Order Coefficient 
                 (D) 
                 −2.891372E−11 
               
               
                   
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
             
           
               
                 TABLE 21 
               
               
                   
               
               
                 Surface Prescription for Surface #6 of Table 19. 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 Y Radius 
                   
                 363.193 
               
               
                   
                 Conic Constant 
                 (K) 
                 −1.060606E+01 
               
               
                   
                  4th Order Coefficient 
                 (A) 
                 6.247531E−05 
               
               
                   
                  6th Order Coefficient 
                 (B) 
                 −8.622953E−07 
               
               
                   
                  8th Order Coefficient 
                 (C) 
                 9.037984E−09 
               
               
                   
                 10th Order Coefficient 
                 (D) 
                 −4.513968E-11 
               
               
                   
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
             
           
               
                 TABLE 22 
               
               
                   
               
               
                 Surface Prescription for Surface #14 of Table 19. 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 Y Radius 
                   
                 −24.004 
               
               
                   
                 Conic Constant 
                 (K) 
                 2.609562E+00 
               
               
                   
                  4th Order Coefficient 
                 (A) 
                 −1.053175E−04 
               
               
                   
                  6th Order Coefficient 
                 (B) 
                 3.126004E−07 
               
               
                   
                  8th Order Coefficient 
                 (C) 
                 −2.716200E−08 
               
               
                   
                 10th Order Coefficient 
                 (D) 
                 2.112687E−10 
               
               
                   
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
             
           
               
                 TABLE 23 
               
               
                   
               
               
                 Surface Prescription for Surface #15 of Table 19. 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 Y Radius 
                   
                 27.079 
               
               
                   
                 Conic Constant 
                 (K) 
                 −6.178694E+00 
               
               
                   
                  4th Order Coefficient 
                 (A) 
                 −1.075797E−05 
               
               
                   
                  6th Order Coefficient 
                 (B) 
                 −1.383769E−07 
               
               
                   
                  8th Order Coefficient 
                 (C) 
                 4.641779E−10 
               
               
                   
                 10th Order Coefficient 
                 (D) 
                 9.831856E−13 
               
               
                   
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
             
           
               
                 TABLE 24 
               
               
                   
               
               
                 Surface Prescription for Surface #16 of Table 19. 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 Construction  
                 (nm) 
                 589 
               
               
                   
                 Wavelength 
                   
                   
               
               
                   
                 R**2 
                 (HCO C1) 
                 −1.543448E−03 
               
               
                   
                 R**4 
                 (HCO C2) 
                 7.864956E−06 
               
               
                   
                 R**6 
                 (HCO C3) 
                 −1.080042E−07 
               
               
                   
                 R**8 
                 (HCO C4) 
                 1.272753E−09 
               
               
                   
                 R**10 
                 (HCO C5) 
                 −5.114979E−12 
               
               
                   
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
             
           
               
                 TABLE 25 
               
               
                   
               
               
                 Surface Prescription for Surface #18 of Table 19. 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 Y Radius 
                   
                 33.959 
               
               
                   
                 Conic Constant 
                 (K) 
                 2.310849E+00 
               
               
                   
                 4th Order Coefficient 
                 (A) 
                 4.222932E−06 
               
               
                   
                 6th Order Coefficient 
                 (B) 
                 −2.501786E−08 
               
               
                   
                 8th Order Coefficient 
                 (C) 
                 3.154900E−11 
               
               
                   
                 10th Order Coefficient 
                 (D) 
                 2.517705E−13 
               
               
                   
                   
               
             
          
         
       
     
         [0000]    
       
         
               
             
               
               
               
               
             
           
               
                 TABLE 26 
               
               
                   
               
               
                 Surface Prescription for Surface #19 of Table 19. 
               
               
                   
               
             
             
               
                   
               
             
          
           
               
                   
                 Y Radius 
                   
                 −21.555 
               
               
                   
                 Conic Constant 
                 (K) 
                 −1.347355E+00 
               
               
                   
                 4th Order Coefficient 
                 (A) 
                 1.944341E−05 
               
               
                   
                 6th Order Coefficient 
                 (B) 
                 3.600425E−08 
               
               
                   
                 8th Order Coefficient 
                 (C) 
                 −1.998220E−11 
               
               
                   
                 10th Order Coefficient 
                 (D) 
                 6.798072E−13 
               
               
                   
                   
               
             
          
         
       
     
       Overall Design Considerations 
       [0051]    In another of its aspects the present invention relates to new criteria for determining depth-fused display (DFD) system design parameters. The optical quality of a fused pixel in DFD displays is quantified by the point spread function (PSF) of its retinal image, or, equivalently, by the modulation transfer function (MTF) which is characterized by the ratio of the contrast modulation of the retinal image to that of a sinusoidal object on the 3D display. For instance, when the eye is accommodated at a rendered depth, z, the PSF of a fused pixel, PSF 12 , by two pixels on a pair of adjacent focal planes located at z 1  and z 2 , respectively, may be described as the weighted sum of the PSFs from the front and the back pixels as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         PSF 
                         12 
                       
                        
                       
                         ( 
                         z 
                         ) 
                       
                     
                     = 
                     
                       
                         
                           
                             w 
                             1 
                           
                            
                           
                             ( 
                             z 
                             ) 
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             
                               PSF 
                               1 
                             
                              
                             
                               ( 
                               
                                 z 
                                 , 
                                 
                                   z 
                                   1 
                                 
                               
                               ) 
                             
                           
                           
                             
                               
                                 ∫ 
                                 ∫ 
                               
                               
                                 x 
                                 , 
                                 y 
                               
                             
                              
                             
                               
                                 PSF 
                                 1 
                               
                                
                               
                                 ( 
                                 
                                   z 
                                   , 
                                   
                                     z 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                              
                             
                                
                               x 
                             
                              
                             
                                
                               y 
                             
                           
                         
                       
                       + 
                       
                         
                           
                             w 
                             2 
                           
                            
                           
                             ( 
                             z 
                             ) 
                           
                         
                          
                         
                           
                             
                               PSF 
                               2 
                             
                              
                             
                               ( 
                               
                                 z 
                                 , 
                                 
                                   z 
                                   2 
                                 
                               
                               ) 
                             
                           
                           
                             
                               
                                 ∫ 
                                 ∫ 
                               
                               
                                 x 
                                 , 
                                 y 
                               
                             
                              
                             
                               
                                 PSF 
                                 2 
                               
                                
                               
                                 ( 
                                 
                                   z 
                                   , 
                                   
                                     z 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                              
                             
                                
                               x 
                             
                              
                             
                                
                               y 
                             
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   1 
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         [0000]    where PSF 1 (z, z 1 ) and PSF 2 (z, z 2 ) are the point spread functions of the front and back pixels when the eye is accommodated at distance z. The PSFs in Eq. (1) are normalized so that front and back pixels have the same luminance before calculating the weighted sum. w 1  and w 2  are the depth-weighted fusing functions modulating the luminance of the front and back pixels and typically w 1 (z)+w 2 (z)=1 is enforced so that the total luminance of the fused image stays the same when the simulated depth changes. The MTF of the display can then be calculated via Fourier Transform of PSF 12 (z). 
         [0052]    An example of the MTF plots of simulated retinal images of a dual-focal-plane DFD display is shown in  FIG. 8A . In the simulation, the two focal planes were placed at 1.2 diopters and 1.8 diopters, respectively, and the luminance ratio between the two focal planes was 1:1, indicating that the fused pixel was being simulated at the dioptric midpoint of the front and back focal planes, i.e., 1.5 diopters. To concentrate on the effects of depth fusion, an eye model was selected with a 3 mm pupil, with all residual aberrations removed.  FIG. 8A  shows how the MTF of the retinal image changes as the eye accommodates at various positions between the two focal planes.  FIG. 8B  shows the contrast gradient as a function of eye accommodation distance for different spatial frequencies, and the peak contrast for each frequency was marked by a black square marker. A transition frequency of around 17 cycles/degree (cpd) is observed from both plots. Below that transition frequency, the MTF of the retinal image is maximized at the dioptric midpoint of 1.5 diopters, which is the simulated depth by the 1:1 luminance ratio in the dual-focal plane system. Furthermore, as the eye approaches the simulated depth from either the far or near focal planes, the MTF values increase smoothly, providing the appropriate contrast gradient required for driving the eye accommodation. For frequencies higher than 17 cpd, however, the contrast of the fused pixel is always highest when the eye is accommodated at or near the physical focal planes, meaning that the contrast gradient has the tendency to drive the accommodation away from the simulated pixel depth, therefore creating a conflict accommodation cue. 
         [0053]      FIGS. 9A ,  9 B show how the transition frequency varies as a function of focal plane separation and as a function of pupil size.  FIG. 9A  assumes a 3 mm eye pupil, and  FIG. 9B  assumes a constant focal plane separation of 0.6 diopters. The results suggest that the smaller the focal plane separation and the smaller the designed eye pupil size, the higher in frequency the transition point is. Therefore a critical criterion for designing a DFD display is that the focal plane separation and the display&#39;s working pupil size should be determined such that the contrast gradient reverse point is higher than the system&#39;s cut-off frequency to avoid presenting a conflicting accommodation cue to the viewer. For instance, a 0.6-diopter separation between adjacent focal planes can be considered adequate for a DFD display affording an angular resolution of 1.8 arc minutes per pixel (approximately a spatial frequency of 17 cpd) and luminance greater than 10 cd/m 2 . The stimulation of 10 cd/m 2  display luminance leads to approximately a 3 mm eye pupil diameter. A 0.45-diopter spacing or smaller would be desired for displays affording an angular resolution of 1 arc minute per pixel (i.e., 30 cpd). The smaller the angular resolution per pixel or the lower the image brightness, the smaller the required focal-plane separation would be. 
         [0054]    These and other advantages of the present invention will be apparent to those skilled in the art from the foregoing specification. Accordingly, it will be recognized by those skilled in the art that changes or modifications may be made to the above-described embodiments without departing from the broad inventive concepts of the invention. It should therefore be understood that this invention is not limited to the particular embodiments described herein, but is intended to include all changes and modifications that are within the scope and spirit of the invention as set forth in the claims.