Abstract:
A circular LED illumination lens for short throw lighting, for example, as part of a set of such devices installed on mullions in reach-in refrigerator cabinets, to uniformly light access across the rectangular door and shelves. The lens has an upper surface with a cavity for the LED, an upper surface the shape of a toroid, generated by an elliptical arc, that serves to magnify the light rays from the LED in an outboard direction, and the minor axis tilted about 17 degrees relatives the center axis of the LED which serves to direct the rays at the center of the shelves. The upper surface also preferably includes a spherical dimple to direct light away from the center axis.

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application claims the benefit of U.S. Provisional Application No. 61/606,580, filed Mar. 5, 2012. 
    
    
     BACKGROUND 
     Light emitting diodes (LEDs) generate light in zones so small (a few mm across) that it is a perennial challenge to spread their flux uniformly over a large target zone, especially one that is much wider than its distance from the LED. So-called short-throw lighting, of close targets, is the polar opposite of spot lighting, which aims at distant targets. Just as LEDs by themselves cannot produce a spotlight beam, and so need collimating lenses, they are equally unsuitable for wide-angle illumination as well, and so need illumination lenses to do the job. 
     A prime example of short throw lighting is the optical lens for the back light unit (BLU) for a direct-view liquid crystal display (LCD) TVs. Here the overall thickness of the BLU is usually 26 mm or less and the inter-distance between LEDs is about 200 mm. Prior art for LCD backlighting consisted of fluorescent tubes arrayed around the edge of a transparent waveguide, that inject their light into the waveguide, which performs the actual backlighting by uniform ejection. While fluorescent tubes are necessarily on the backlight perimeter due to their thickness, light-emitting diodes are so much smaller that they can be placed directly behind the LCD display, (so called “direct-view backlight”), but their punctuate nature makes uniformity more difficult, prompting a wide range of prior art over the last twenty years. Not all of this art, however, was suitable for ultra-thin displays. 
     Another striking application with nearly as restrictive an aspect ratio is that of reach-in refrigerator cabinets. Commercial refrigerator cabinets for retail trade commonly have glass doors with lighting means installed behind the door hinging posts, which in the trade are called mullions. Until recent times, tubular fluorescent lamps have been the only means of shelf lighting, in spite of how cold conditions negatively affect their luminosity and lifetime. Also, fluorescent lamps produce a very non-uniform lighting pattern on the cabinet shelves. Light-emitting diodes, however, are favored by cold conditions and are much smaller than fluorescent tubes, which allow for illumination lenses to be employed to provide a much more uniform pattern than fluorescent tubes ever could. Because fluorescent tubes radiate in all directions instead of just upon the shelves, much of their light is wasted. With the proper illumination lenses, however, LEDs can be much more efficient, allowing lower power levels than fluorescent tubes, in spite of the latter&#39;s good efficacy. 
     The prior art of LED illumination lenses can be classified into three groups, according to how many LEDs are used:
     (1) Extruded linear lenses with a line of small closely spaced LEDs, particularly U.S. Pat. Nos. 7,273,299 and 7,731,395, both by these Inventors, as well as References therein.   (2) A line of a dozen or more circularly symmetric illumination lenses, such as those commercially available from the Efficient Light Corporation.   (3) A line of a half-dozen (or fewer) free-form illumination lenses with rectangular patterns, such as U.S. Pat. No. 7,674,019 by these Inventors.   

     The first two approaches necessarily require many LEDs in order to achieve reasonable uniformity, but recent trends in LEDs have produced such high luminosity that fewer LEDs are needed, allowing significant power savings. This is the advantage of the last approach, but free-form lenses generating rectangular patterns have proved difficult to produce, via injection molding, with sufficient figural accuracy for their overlaps to be caustic-free. (Caustics are conspicuous small regions of elevated illuminance.) 
     What is needed instead is a circularly symmetric illumination lens that can be used in small numbers (such as five or six per mullion) and still attain uniformity, because the individual patterns are such that those few will add up to caustic-free uniformity. The objective of this Invention is to provide a lens with a circular illumination pattern that multiples of which will add up to uniformity across a rectangle. It is a further objective to attain a smaller lens size than the above mentioned approaches, leading to device compactness that results in lower manufacturing cost. The smaller lens size can be achieved by a specific tailoring of its individual illumination pattern. This pattern is an optimal annulus with a specific fall-off that enables the twelve patterns to add up to uniformity between the two illuminating mullions upon which each row of six illuminators are mounted. This fall-off at the most oblique directions is important, because this is what determines overall lens size. The alternative approaches are: (1) Each mullion illuminates 100% to mid-shelf and zero beyond, which leads to the aforementioned caustics; (2) Each mullion contributes 50% at the mid-point, falling off beyond it. The latter is the approach of this Invention, and has proven highly successful. 
     The prior art is even more challenged, moreover, when fewer LEDs are needed due to ongoing year-over-year improvements in LED flux output. After all, backlight thickness is actually relative to the inter-LED spacing, not to the overall width of the entire backlight. For example, in a 1″ thick LCD backlight with 4″ spacing between LEDs, the lens task is proportionally similar to the abovementioned refrigerator cabinet. Because of the smaller size of an LCD as compared to a 2.5 by 5 foot refrigerator door, lower-power LEDs with smaller emission area will be used, typically a Top-LED configuration with no dome-like silicone lens. 
     Regarding the prior art patents which have taught non-specific design methods for addressing this problem are: US 2006/0138437, U.S. Pat. No. 7,348,723, U.S. Pat. No. 7,445,370, U.S. Pat. Nos. 7,621,657 and 7,798,679 by Kokubo et al. shows the same cross-sectional lens profile as in FIG. 15A of U.S. Pat. No. 7,618,162 by Parkyn and Pelka, while failing to reference it. U.S. Pat. No. 7,798,679 furthermore contains only generically vague descriptions of that lens profile, and worse yet has no specific method of distinguishing the vast number of significantly different shapes fitting its vague verbiage, its many repetitively generic paragraphs notwithstanding. Experience has shown that illumination lenses are unforgiving of small shape errors, such as result from unskilled injection molding or subtle design flaws. Very small changes in local slope of a lens can result in highly visible illumination artifacts sufficient to ruin an attempt at a product. Therefore such generic descriptions are insufficient for practical use, because even the most erroneous and ill-performing lens fulfills them just as well as an accurate, high-performing lens. Thus U.S. Pat. No. 7,798,679 does not pertain to the preferred embodiments disclosed herein, because it never provides the specific, distinguishing shape-specifications whose precise details are so necessary for modern optical manufacturing. 
     SUMMARY 
     Commercial refrigerator display cabinets for retail sales have a range of distances from mullion to the front of the shelves, commonly from 3″ to 8″, with the smaller spacings becoming more prevalent as store owners seek to cram more and more product into their reach-in refrigerator cabinets. Fluorescent tubes have great difficulty with these tighter spacings, leading to an acceleration in the acceptance of LED lighting technology. Even though fluorescent tubes have efficacy comparable to current LEDs, their large size and omnidirectional emission hamper their efficiency, making it difficult to adequately illuminate the mid-shelf region. Early reach-in refrigerator LED illuminators utilized a large number of low-flux LEDs, but continuing advances in luminosity enable far fewer LEDs to be used to produce the same illuminance. This places a premium on having illumination lenses that when arrayed will sum up to uniformity while also having the smallest possible size relative to the size of the LED. 
     Disclosed herein are preferred embodiments that generate wide-angle illumination patterns suitable for short-throw lighting. Also disclosed is a general design method for generating their surface profiles, one based on nonimaging optics, specifically a new branch thereof, photometric nonimaging optics. This field applies the foundational nonimaging-optics idea of etendue in a new way, to analyze illumination patterns and classify them according to the difficulty of generating them, with difficulty defined as the minimum size lens required for a given size of the light source, in this case, the LED. 
     OBJECTIVES 
     It is the first objective to disclose numerically-specific lens configurations that in arrays will provide uniform illumination for a close planar target, especially in retail refrigeration displays and in thinnest-possible direct-view LCD backlights. 
     It is the second objective to provide compensation for the illumination-pattern distortions caused by volume scattering and scattering due to Fresnel reflections, which together act as an additional, undiscriminating secondary light source. 
     On fulfillment of the inventor&#39;s duty to go beyond superficial description, it is the third objective to disclose fully the design methods that generated the preferred embodiments disclosed herein, such that those skilled in the art of illumination optics could design further preferred embodiments for other illumination applications, in furtherance of the ultimate objective of the patent system that being to expand public knowledge. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The above and other aspects, features and advantages will be apparent from the following more particular description thereof, presented in conjunction with the following drawings wherein: 
         FIG. 1  shows how a rectangular door is illuminated by circular patterns. 
         FIG. 2  shows a graph of an individual illumination pattern. 
         FIG. 3  shows an end view of the door of  FIG. 1 , with slant angles. 
         FIG. 4  shows a graph of required source magnification. 
         FIG. 5  shows a cross-section of an illumination lens and LED. 
         FIG. 6A-6F  show source-image rays from across the target. 
         FIG. 7  shows how a rectangular door is illuminated by only 4 LEDs. 
         FIG. 8  shows a cross-section of a further illumination lens and LED. 
         FIG. 9  illustrates a mathematical description of volume scattering. 
         FIG. 10  is a graph of illumination patterns. 
         FIG. 11  sets up a 2D source-image method of profile generation. 
         FIG. 12  shows said method of profile generation. 
         FIGS. 13A and 13B  show the 3D source-image method of profile generation. 
         FIG. 14  shows a plano-convex lens-center, with defining rays. 
         FIG. 15  shows a concave-concave lens-center, with defining rays. 
         FIG. 16  shows a concave-plano lens-center, with defining rays. 
         FIG. 17  shows the complete lens made from the lens-center of  FIG. 14 . 
         FIG. 18  shows the complete lens made from the lens-center of  FIG. 15 . 
         FIG. 19  shows the complete lens made from the lens-center of  FIG. 16 . 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     A better understanding of the features and advantages of the present invention will be obtained by reference to the following detailed description of the invention and accompanying drawings, which set forth illustrative embodiments in which principles of the invention are utilized. 
       FIG. 1  shows rectangular outline  10  representing a typical refrigerator door that is 30″ wide and 60″ high, with other doors, not shown, to either side. Dashed rectangles  11  denote the mullions behind which the shelf lighting is mounted, typically at 3-6″ from the front of the illuminated shelves. This is much closer than the distance to the shelf center, denoted by centerline  12 . There are twelve illuminators  31  (six on either side), four of which are denoted by small circles  1 . Each illuminator  31  produces an illuminated circle with its peak on a ring denoted by solid circles  2  and its edge on dotted circles  3 . Here the circles  2  have radius of a quarter of the shelf width, or halfway to centerline  12 . The circles  3 , where illuminance has fallen to zero, are sized to meet the circles  2  from the opposite mullion. 
       FIG. 2  shows graph  20  with abscissa  21  that is horizontally scaled the same as  FIG. 1  above it. Ordinate  22  is scaled from 0 to 1, denoting the ideal illuminance I(x), as graphed by curve  23 , generated on the shelves by an illuminator  31  under the mullion. This illumination function is relative to the maximum on circle  2 , which has radius x M . It falls off to zero at radius x E . This gradually falling illuminance is paired with the gradually ascending one of the illuminator  31  on the opposite side of the door, so the two patterns add up to constant illuminance along the line  4  of  FIG. 1 . 
     An actual injection-molded plastic lens will exhibit volume scattering within its material, making the lens itself an emitter rather than a transmitter. This volume scattered light will be strongest just over the lens. The central dip in the pattern  23 , shown in  FIG. 2  to be at the ¾ level, compensates for this extra volume-scattered light, so that the total pattern (direct plus scattered) is flat within circle  2 . This effect becomes more pronounced with the larger lenses discussed below. 
     Another advantage of this type of gradually falling-off pattern is that any point on centerline  12  is lit by several illuminators  31  on each mullion, assuring good uniformity. The dotted curve  24  shows the illumination pattern of an LED alone. It is obviously incapable of adding up to satisfactory illumination, let alone uniform, hence the need for an illumination lens  51  to spread this light out properly. 
       FIG. 3  shows an end view of shelf-front rectangle  30  identical to that of  FIG. 1 . Illuminators  31  are located as shown by small rectangles  31 . This addresses the difficulty of lighting from so close to the shelf, in this case at a distance of z T =4″.  FIG. 3  shows the distances x m =15″ and x E =22.5″, respectively, to centerline  33  and edge-line  34 , at which the pattern of illuminator  31  has reached zero illuminance. These distances correspond to off-axis angles from the normal given by
 
γ m =tan −1 ( x   m   /z   T )=tan −1 (15/4)=75° γ E =tan −1 ( x   E   /z   T )=tan −1 (22.5/4)=80°
 
These large slant-angles drive the lens design, requiring considerable lateral magnification of the source by the lens. At low slant-angles, in contrast, the lens must demagnify.
 
     This concept of magnification and demagnification can be made more explicit via etendue considerations. The source-etendue is that of a chip of area A S =2.1 mm 2 , immersed in a dome of refractive index n=1.45:
 
 E   S   =πn   2    A   S  sin 2 θ=14 mm 2  
 
Here θ is 90° for a Lambertian source of which an LED is a very good approximation°.
 
     An illumination lens  51  basically redistributes this etendue over the target, which is much larger than the chip. In the case of the illumination pattern in  FIG. 2 , the target etendue relates to the area A T  of the 45″ illumination circle of  FIG. 1 , as weighted by the relative illumination function  23  of  FIG. 2 . This simple model of an actual illumination pattern has a central dip to illuminance I 0 , a rise to unity at x=x M , and a linear falloff to zero at x=x E . This is mathematically expressed as
 
 I ( x )= I   0   +x (1 −I   0 )/ x   M    x≦x   M  
 
 I ( x )=( x   E   −x )/( x   E   −x   M )  x   M   ≦x≦x   E  
 
Then the target etendue is given by an easily solved integral:
 
               E   T     =       π   ⁢           ⁢     sin   2     ⁢     θ   T     ⁢       ∫   0     x   ⁢           ⁢   e       ⁢     2   ⁢   π   ⁢           ⁢   x   ⁢           ⁢     I   ⁡     (   x   )       ⁢     ⅆ   x           =     2   ⁢     π   2     ⁢     sin   2     ⁢       θ   T     ⁡     (         [           I   0     ⁢     x   2       2     +         (     1   -     I   0       )     ⁢     x   3         3   ⁢           ⁢     x   M           ]     0     x   M       +       [           x   E       2   ⁢     (       x   E     -     x   M       )         ⁢     x   2       -       x   3       3   ⁢     (       x   E     -     x   M       )           ]       x   M       x   E         )                               ⁢       E   T     =       sin   2     ⁢     θ   T     ⁢   1.47   ⁢           ⁢   square   ⁢           ⁢   meters             
Here θ T  is the half angle of a narrow-angle collimated beam with the same etendue as the source, so that
 
sin 2  θ T ˜1 E− 5 θ T =±0.18°
 
At the center of the lens this is reduced by ¾, to ±0.13°. This can be contrasted with the angular subtense of the source alone, as seen from directly above it on the shelf, at distance z T  as shown in  FIG. 3 :
 
tan 2  θ S   =n   2    A   C /4 z   T   2  θ S =±0.61°
 
Thus the central demagnification of the lens needs to be 1:4.5, dictating that the central part of the lens be concave, in order to act as an expander with negative focal length. This can be attained on a continuum of concavity bounded by a flat-topped outer surface with a highly curved inside surface or a flat-topped inner surface with the outer surface highly curved. That of  FIG. 5  lies between these extremes.
 
     As shown in  FIG. 3 , a high slant angle γ means that to achieve uniform illumination the source image made by the lens must be correspondingly larger than for normal incidence, by a factor of 1/cos γ. The source itself will be foreshortened by a slant factor of cos γ, as well as looking smaller and smaller by being viewed from farther away, by a further factor of cos 2 γ. Thus the required lens magnification is 
               M   ⁡     (   γ   )       =     1     4.5   ⁢           ⁢     cos   4     ⁢   γ             
Note that magnification rises from ¼ on-axis to unity at an off-axis angle given by
 
     
       
         
           
             
               
                 
                   
                     γ 
                     ⁡ 
                     
                       ( 
                       M 
                       ) 
                     
                   
                   = 
                   
                     
                       cos 
                       
                         - 
                         1 
                       
                     
                     ⁢ 
                     
                       
                         1 
                         
                           4.5 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           M 
                         
                       
                       4 
                     
                   
                 
               
               
                 
                   
                     γ 
                     ⁡ 
                     
                       ( 
                       1 
                       ) 
                     
                   
                   = 
                   
                     47 
                     ⁢ 
                     ° 
                   
                 
               
             
           
         
       
     
     These angles dilute the illuminance by a cosine-cubed factor, so that the farther out light must be thrown, the more intense must be the lens output. Considering that the LED source has a cosine fall-off of its own, the total source magnification required is the well-known cos −4  factor, amounting to 223 at 75° 1100 at 80° respectively. Here lies the advantage of the fall-off in the illumination pattern of  FIG. 2 , since these deleterious factors are reduced accordingly. 
       FIG. 4  shows graph  40  with abscissa  41  running from 0 to 80° in off-axis angle γ and ordinate  42  showing the source magnification M(γ) required for uniform illumination. Unit magnification is defined as a source image the same size as if there were no lens. What this magnification means is that the illumination lens  51  of the present invention must produce an image of the glowing source, as seen from the shelf, that is much bigger than the Lambertian LED source without any lens. For uniformly illuminating a 4″ shelf-distance, curve  43  shows that the required magnification peaks at 77.5°, while lower curve  44  is for the much easier case of a 6″ shelf distance, peaking at 71°. This required image-size distribution is the rationale for the configuration of the present invention. 
       FIG. 5  is a cross-section of illuminator  31 , comprising illumination lens  51 , bounded by an upper surface  46  comprising a central spherical dimple with arc  52  as its profile and a surrounding toroid with elliptical arc  53  as its profile, and also bounded by a lower surface  48  comprising a central cavity  54  with bell-shaped profile  54  and surrounding it an optically inactive cone  55  joining the upper surface  46 , with straight-line profile and pegs  56  going into circuit board  57 . Illuminator  31  further comprises LED package  58  with emissive chip  58 C immersed in transparent hemispheric dome  58 D. The term ‘toroid’ distinguishes from the conventional term ‘torus’, which solely covers the case of zero tilt angle. The highly oblique lighting setup of refrigerator-cabinet shelf-fronts involves tilting the torus so that the lensing effect of the elliptical arc  53  points toward the center of the shelf. 
     Arc  52  of  FIG. 5  extends to tilt angle τ, which in this case is 17°, its importance being that it is the tilt angle of major axis  53 A of elliptical arc  53 . Its minor axis  53 B lines up with the radius at the edge of arc  52 , ensuring profile-alignment with equal surface tangency. There are three free parameters which define a particular outer surface of illumination lens  51 , as intended for different shelf distances. The first is the radius of arc  52 , which controls the amount of de-magnification by the central portion of illumination lens  51 . The second is the tilt angle τ, which defines the orientation of elliptical arc  53 , namely towards the shelf center of  FIG. 1 . The third free parameter of the upper surface  46  is the ratio of the radius to the elliptical arc  53  at major axis  53 A to the radius to the elliptical arc  53  at minor axis  53 B, in this case 1.3:1, defining the above-discussed source magnification. Ray-fan  59  comprises central rays (i.e., originating from the center of chip  58 C) at 2° intervals of off-axis angle. The central ten rays designated by dotted arc  59 C, outbound from the centerline or central axis  59 , illustrate the diverging character of the center of lens  51 , which provide the central demagnification required for uniform illumination. The remaining rays are all sent at steep angles to the horizontal, providing the lateral source magnification of  FIG. 4 . 
     The central cavity  54  surrounding LED  58  has bell-shaped profile  54  defined by the standard aspheric formula for a parabola (i.e., conic constant of −1) with vertex at z v , vertex radius of curvature r c , 4 th -order coefficient d and 6 th -order coefficient e:
 
 z ( x )= z   v   +x/r   c   +dx   4   +ex   6  
 
In order for profile  54  to arc downward rather than upward, the radius of curvature r c  is negative. The aspheric coefficients provide an upward curl  49  at the bottom of the bell, to help with cutting off the illumination pattern. The particular preferred embodiment of  FIG. 5 , with a cavity entrance-diameter set at 6.45 mm, is defined by:
 
 z   v =6 mm  r   c =−1.69 mm  d=− 0.05215  e= 0.003034
 
This profile only needs minor modification to be suitable for preferred embodiments illuminating other shelf distances.
 
       FIG. 6A through 6F  shows illumination lens  51  and LED chip  61 . In  FIG. 6A , rays  62  come from points on the shelf at the indicated x coordinates of 0, 2″, and 4″ laterally from the lens. Each bundle is just wide enough that its rays end at the edges of chip  61 , which is the definition of a source image. Each bundle is narrower than chip  61  would appear by itself, in accordance with the previously discussed demagnification. The central portion of lens  60  that is traversed by rays  62  can be seen to be a concave, diverging lens, as previously mentioned. 
       FIG. 6B  shows ray bundle  63  proceeding from the distance x M  to the maximum of the illumination pattern in  FIG. 2 . It is twice the width of those in  FIG. 6A . 
       FIG. 6C  shows ray bundle  64  proceeding from the distance x m  to the middle of the shelf, as shown in  FIG. 2 . 
       FIG. 6D  shows ray bundle  65  proceeding from beyond mid shelf, at 18″. 
       FIG. 6E  shows ray bundle  66  proceeding from beyond mid shelf, at 20″, nearly filling the lens. This is the maximum source magnification this sized lens can handle. 
       FIG. 6F  shows ray bundle  67  proceeding from the edge of the illumination pattern, at x E =22″. Note that these rays miss chip  61 , indicating that there will be no light falling there, which is required by the pattern cutoff. 
     The progression of  FIG. 6A through 6F  is the basis for the numerical generation of the upper and lower surface profiles of the lens, starting at the center and working outwards, as will be disclosed below. The results of this method can sometimes be closely approximated by the geometry of  FIG. 5 . 
     The illumination lens  51  of  FIG. 5  has elliptical and aspheric-parabolic surfaces with shapes that are exactly replicable by anyone skilled in the art. In the illumination pattern of  FIG. 2 , the central depression to ¾ the maximum value was empirically found to work with the lens array of  FIG. 1 , with six lenses on each side. This lens is the first commercially available design enabling only six LEDs to be used, rather than the dozen or more of the prior art. More recently, however, even higher-power LEDs have become available that only require two per door, as  FIG. 7  illustrates. 
       FIG. 7  shows rectangular outline  70  representing a typical refrigerator door that is 30″ wide and 60″ high, with other doors, not shown, to either side. Dashed rectangles  71  denote the mullions behind which the shelf lighting is mounted, typically at 3-6″ from the front of the illuminated shelves. This is much closer than the distance to the shelf center, denoted by centerline  72 . There are four illuminators  31  (two on either side), denoted by small circles  73 . Each illuminator  31  produces an illuminated circle with its peak on a ring denoted by solid circles  74  and its edge on dotted circles  75 . Here the circles  74  have radius of about a fifth of the shelf width, or a third the way to centerline  72 . The circles  75 , where illuminance has fallen to zero, are sized to reach nearly all the way across the shelf. As in  FIG. 1 , each pattern has the value ½ at centerline  72 , so two lenses add to unity. Also, at shelf center-point  76  the four patterns overlap, so at this distance each pattern must have the value ¼, and thus add to unity. This same configuration is applicable for LCD backlights comprising square-arrayed LEDs, merely on a smaller scale. This arrangement of precisely configured illumination lenses  51  is capable of generating uniformity satisfactory for LCD backlights. 
     The LEDs used in the arrangement of  FIG. 7  must be three times as powerful as those used for  FIG. 1 . This greater flux has unwanted consequences of triply enhanced scattered light, strengthened even more by the greater size of the lenses used for  FIG. 7  versus the smaller ones which would suffice for  FIG. 1 . The illumination pattern of  FIG. 2  has a central dip in order to compensate for the close spacing of the lenses. When scattering is significant, however, the scattered light can be strong enough to provide all the illumination near the lens. The upshot is that the illumination pattern shown in  FIG. 2  would have nearly zero intensity on-axis. The resultant lens has a previously unseen feature: either or both surfaces have a central cusp  82  that leaves no direct light on the axis, resulting in a dark center for the pattern, in order to compensate for the scattered light. 
       FIG. 8  is a cross-section of illuminator  31 , comprising circularly symmetric illumination lens  51 , bounded by an upper surface comprising a central cusp  82  formed by a surrounding toroid with tailored arc  83  as its profile. Lens  81  is also bounded by a lower surface comprising a central cavity  54  with tailored profile  84  preferably peaking at its tip, and surrounding it an optically inactive cone  55  joining the upper surface, with straight-line profile  85  and pegs  56  going into circuit board  87 . Illuminator  31  further comprises centrally located LED package  88  with emissive chip  88 C immersed in transparent hemispheric dome  88 D. 
     The optically active profiles  83  and  84  of  FIG. 8  are said to be tailored due to the specific numerical method of generating it from an illumination pattern analogous to that of  FIG. 2 , but with little or no on-axis output. The reason for this is, as aforementioned, to compensate for real-world scattering from the lens. The profiles  83  and  84  only control light propagating directly from chip  83 C, through dome  83 D, and thence refracted to a final direction that ensures attainment of the required illumination pattern. This direct pattern will be added to the scattering pattern of indirect light, which thus needs to be determined first. 
       FIG. 9  shows illumination lens  51 , identical to lens  81  of  FIG. 8 , with other items thereof omitted for clarity. From LED chip  98 C issues ray bundle  92 , comprising a left ray (dash-dot line), a central ray (solid line), and a right ray (dashed line), issuing respectively from the left edge, center, and right edge of LED chip  98 C. Anywhere within lens  91 , these rays define the apparent size of chip  98 C and thus how much light is passing through a particular point. Any light scattered from such a point will be a fixed fraction of that propagating light. The closer to the LED the more light is present at any point, and the greater the amount scattered. This scattering gives the lens its own glow, separate from the brightness of the LED itself when directly viewed. 
     Strictly speaking, scattering does not take place at a point but within a small test volume, shown as infinitesimal cube  93  in  FIG. 9 , magnified for clarity. It is oriented along the propagation direction of ray bundle  92 . It has cross-section  93 A of area dA and propagation length dl, such that its volume is simply dV=dl dA. Within cube  93  can be seen the left, central, and right rays of bundle  92 , now switched sides. The right and left rays define solid angle Ω, indicating the apparent angular size of LED chip  98 C as seen from cube  93  within lens  91 . The greater this solid angle the more light will be going through cube  93 . LED chip  98 C has luminance L, specified in millions of candela per square meter. This is reduced when ray bundle  92  goes into lens  91 , due to less-than-unity transmittance τ caused by Fresnel reflections. Going into cube  93  the ray bundle  92  has intensity I given simply by I=τ L dA. The total flux F passing through cube  93  is then given, simply again, by F=IΩ. 
     Volume scattering removes a fixed fraction of this intensity I per unit length of propagation, similar to absorption. Both are described by Beer&#39;s law:
 
 I ( l )= I (0) e   −κl  
 
Here I(0) is the original intensity and I(l) is what remains after propagation by a distance l, while scattering coefficient κ has the dimension of inverse length. It can easily be determined by measuring the loss in chip luminance as seen through the lens along the path l of  FIG. 9 .
 
     Returning to cube  93  of  FIG. 9 , the ingoing intensity I is reduced by the small amount dI=e −κdl . This results in a flux decrement dF=dIΩ that is subtracted from F. Then the emission per unit volume is dF/dV. Integrating this over the entire optically active volume of lens  91  gives the total scattered light.  FIG. 9  further shows observer  94  gazing along line of sight  95 , along which direct rays  97  give rise to scattering points  96 , summing into a lens glow that acts as a secondary light source surrounding the LED. 
     These scattering phenomena are usually looked upon as disadvantageously parasitic, acting only to detract from optical performance. There is a new aspect to this, however, where some volume scattering would be beneficial. It arises in the subtle failings of current high-brightness LEDs, namely that of not delivering the same color in all directions. More specifically, many commercially available LEDs with multi-hundreds of lumens output, look much yellower when seen laterally than face-on. This is because of the longer path through the phosphor taken by light from the blue chip. 
     Thick phosphors have uniform whiteness, or color temperature, in all directions, but they reduce luminance due to the white light being emitted from a much bigger area than that of the blue chip. Conformal coatings, however, are thin precisely in order to avoid enlarging the emitter, but they will therefore scatter light much less than a thick phosphor and therefore do much less color mixing. As a result, lateral light is much yellower (2000 degrees color temp) and the face-on light much bluer (7000 degrees) than the mean of all directions. As a result of this unfortunate side-effect of higher lumen output, the lenses disclosed herein will exhibit distinct yellowing of the lateral illumination, and a distinct bluing of the vertical illumination. 
     The remedy for this inherent color defect is to use a small quantity of blue dye in the lens material. Since the yellow light goes through the thickest part of the lens, the dye will automatically have its strongest action precisely for the yellowest of the LEDs rays, those with larger slant angles. The dye embedded in the injection-molding material should have an absorption spectrum that only absorbs wavelengths longer than about 500 nm, the typical spectral crossover between the blue LED and the yellow phosphor. The exact concentration will be inversely proportional to lens size as well as to the absorption strength of the specific dye utilized. 
     A further form of scattering arises from Fresnel reflections, aforementioned as reducing the luminance of rays as they are being refracted.  FIG. 9  further shows first Fresnel-reflected ray  92 F 1  coming off the inside surface of lens  91 , then proceeding into the lens to be doubly reflected out of the lens  92 F 1  and onto the printed circuit board. This ray has strength of (1−τ) relative to the original ray  92 , where tau is the coefficient of transmission at the particular point where the ray impinges upon the exit face. Of similar strength is the other Fresnel-reflected ray,  92 F 2 , which proceeds from the outer surface to the bottom of the lens. These two rays are illustrative of the general problem of stray light going where it isn&#39;t wanted. Unlike the volume scattering at points  96 , these Fresnel-reflected rays can travel afar to produce very displeasing artifacts and greatly destroy the uniformity of the optical system. It has been well-known for many decades of optical engineering that the easiest way to deal with this is to institute surface scattering or absorption of these stray rays. Since the flat conical bottom surface  91 C of lens  91  intercepts most of these stray Fresnel reflections, the tried-and-true traditional solution is simply to roughen the corresponding mold surface so that the Fresnel light is dissipated to become part of the above-described volume scattering. At the termination of ray  92 F 2  can be seen the scattered rays, some of which illuminate the top of substrate or printed circuit board (PCB)  99 , which of course could also be scattered by using a white diffuse white paint, say on the PCB. Another method that leads to some loss in overall optical efficiency of the system is to simply paint the bottom of the lens or the PCB with a highly absorbing black paint. This method has been found to produce excellent uniformity by these inventors for the illumination on LCD screens or for the reach-in refrigeration application, where really good uniformity on the illuminated packages has been produced. 
       FIG. 10  shows graph  100  with abscissa  101  denoting distance in millimeters from the center of the lens of  FIG. 9  and ordinate  102  denoting illuminance relative to the pattern maximum (in order to generalize to any illumination level). Dashed curve  103  is the ideal illumination pattern desired for the configuration of  FIG. 7 , given an inter-lens spacing of 125 mm and a target distance of 23 mm. These dimensions represent a backlight application, where the LEDs are arrayed within a white-painted box, and the target is a diffuser screen, with a liquid-crystal display (LCD) just above it. Increased LED luminosity mandates fewer LEDs, to save on cost, while aesthetics push for a thinner backlight. These two factors comprise a design-pressure towards very short-throw lighting. 
     The ‘conical pattern’ of curve  103  and its converse (not shown) from an illuminator  31  at 125 mm, will add to unity, which assures uniform illumination. Dash-dot curve  104  depicts the combined parasitic illuminance on that target plane caused by the above-discussed volume and surface scattering from a lens at x=0. This curve is basically the cosine 4  of the off-axis angle to a point on the target. Solid curve  105  is the normalized difference between the other two curves, representing the pattern that when scaled will add to curve  104  to get a total illuminance following curve  103 . In this case the scattered light of curve  104  is strong enough to deliver 100% of the required illuminance just above the lens. In such a case the central cusp  82  of  FIG. 8  will ensure that the central illuminance is zero when only counting direct light that is delivered through the lens. 
     The illumination pattern represented by curve  105  of  FIG. 10  can be used to numerically generate the inner and outer profiles of the lens  81  of  FIG. 8 , utilizing rays from the right and left edges of the source. Dotted curve  106  of  FIG. 10  graphs the relative size of the source image height (as shown in  FIG. 6A-F ) required by the illuminance pattern of curve  105 . This height function is directly used to generate the lens profiles. 
       FIG. 11  shows LED  110  and illumination lens  51 , of 20 mm diameter, sending right ray  112  and left ray  113  to point  114 , which has coordinate x on planar target  115 , located 23 mm above LED  110 . Right ray  112  hits point  114  at slant angle γ, and left ray  113  at slant angle γ+Δγ. In the two-dimensional analysis of  FIG. 11 , the illuminance I(x) at point x is proportional to the difference between the sines of the left and right rays&#39; slant angles:
 
 I ( x )α sin(γ+Δγ)−sin(γ)
 
This angular requirement can be met by the proper height H of the source image, namely the perpendicular spacing between right ray  112  and left ray  113 , at the lens exit of  112 . Curve  106  of  FIG. 10  is a plot of this height H, relative to its maximum value. From this geometric requirement the lens profiles can be directly generated by an iterative procedure that adds new surface to the previously generated surface.
 
       FIG. 12  shows incomplete illumination lens  51 , positioned over LED  120 . It is incomplete in that it represents a typical iteration-stage of generating the entire lens of  FIG. 11 . The portion of Lens  111  of  FIG. 11  that is shown as a slightly thickened curve terminates at its intersection, shown as point  124 , with right ray  122 . In  FIG. 12 , a new left ray  123  is launched that is barely to the right of left ray  113  of  FIG. 11 . After going through terminal point  126  and then through previously generated upper surface  121 , it will intercept the target (not shown) at a new point x+dx, just to the right of point x of  FIG. 10 . This point will have an already calculated source-height requirement such as curve  105  of  FIG. 19 , fulfilled by launching a new right ray  122  from x+dx. Ray  122  will intercept the lens surface at point  125 , upon new surface that has been extended from point  124 . The new surface has a slope determined by the necessity to deflect ray  122  towards point  126  on the interior surface of lens  121 . The location of this point  127  is determined by right ray  122 -S coming from the right edge of LED chip  120 C. The off-axis angle of this ray is determined by the usual requirement that the angular-cumulative intensity of right ray  122 S equal the spatially cumulative illumination at point x+dx, which is known from the desired illumination pattern, such as that shown by curve  105  of  FIG. 10 . The slope of this new interior surface, from point  126  to new point  127 , is determined by the necessity of refracting ray  122 S so it joins ray  122  to produce the proper source-image height for the illumination of the target at point x+dx. In this fashion, the generation of lens  121  will be continued until all rays from chip  120 C are sent to their proper target coordinates, and its full shape is completed. 
     The profile-generation method just described is two-dimensional and thus does not account for skew rays (i.e., out-of-plane rays), which in the case of a relatively large source can give rise to noticeable secondary errors in the output pattern, due to lateral variations in the size of the source image. This effect necessitates a fully three-dimensional source-image analysis for generating the lens shape, as shown in  FIG. 13 . 
     The lens-generation method of  FIG. 12  traces left ray  123  through the previously generated inner and outer surfaces to a target point with lateral coordinate x+dx. The pertinent variable is the height H of the source image. In three dimensions, however, rays must be traced from the entire periphery of the LED&#39;s emission window out to the target point, where they limit the image of the source as seen through the lens from that point. An illumination lens  51  acts to alter the sources&#39; apparent size from what it would be by itself. The size of the source image is what determines how much illumination the lens will produce at any target point. 
       FIG. 13A  is a schematic view from above of circular illumination lens  51 , with dotted lines showing is incomplete, its design iteration having only extended so far to boundary  131 . Circular source  132  is shown at the center of lens  130 , and oval  133  represents the source image it projects to target point x+dx (not shown). This source image is established by reverse ray tracing from the target point back through the lens to the periphery of the source. The source image is the oval outline  133  on the upper surface where these rays intercept it. Thus the already completed part of the lens will partially illuminate the target point, and a small element of new surface must be synthesized for full illumination. 
       FIG. 13B  is a close-up view showing source ellipse  133  and boundary  131 , also showing curve  134 , representing a small element of new surface that will be added in order to complete source image  133  and achieve the desired illumination level at target point x+dx. Of course, when new upper surface is added there will have to be a corresponding element of new lower surface added as well. Just as enough there must be enough new upper surface to finish the source image, so too must there be enough lower surface to provide the source image to the upper surface. Since both the extent and slope of this new lower surface must be determined as free variables, the design method must be able to calculate both unknowns, but in general the slopes of the new elements of upper and lower surface will be smooth continuations of the previous curvatures of the surfaces. 
     Traditionally, non-imaging optics deals only with rays from the edge of the source, but the illumination lenses  51  disclosed herein go beyond this when assessing the source image at each target point. The incomplete source image of  FIG. 13B  gives rise to a less-than-required illuminance at the target point of interest, at lateral coordinate x. In order to calculate this illuminance, however, rays must be reverse traced back to the entire source, not just its periphery. This is especially true when the source has variations in luminance and chrominance. Then the flux from each small elemental area dA of  FIG. 13B  is separately calculated and integrated over the source image as seen through already completed surface. The deficit from the required illuminance will then be made up by the new surface  134  of  FIG. 13B . Its size is such that the additional source image area will just finish the deficit. When the illumination pattern only changes gradually, as with the linear ramps just discussed, the deficit is always modest because the previously generated surface has done a good job of getting close to the required illuminance. The new surface will not have to scrunch the new source image, due to a tiny deficit, nor expand it wildly for a large deficit, because the target pattern is ‘tame’ enough to prevent this. 
     This design method can be called ‘photometric non-imaging optics’, because of its utilization of photometric flux accounting in conjunction with reverse ray tracing to augment the edge-ray theorem of traditional non-imaging optics. 
     The iterative process that numerically calculates the shape of a particular illumination lens  51  can begin, alternatively, at either the center or the periphery. If the lens diameter is constrained, the initial conditions would be the positions of the outer edges of the top and bottom surfaces, which then totally determines the lens shape, in particular its central thickness. If this thickness goes below a minimum value then the initial starting points must be altered. While this is conceptually feasible, in practical terms it leaves the problem underdetermined, whereas the reverse ray tracing of  FIG. 13A  utilizes the previously generated surface via reverse ray tracing. Thus it is easier to begin the design iteration at the center of the lens using some minimum thickness criterion, e.g., 0.75 mm. The height of the lens center above the source would be the primary parameter in determining the overall size of the lens. The other prime factor is how the central part of the lens is configured as a negative lens, that is, whether concave-plano, concave-concave, or plano-concave. Also, a concave surface can either be smooth or have the cusp-type center as shown in  FIG. 8 , in the case of strong parasitic losses. 
       FIG. 14  shows concave-plano lens-center  140 , to be used as a seed-nucleus for generating an entire illumination lens  51 . Its diameter is determined by the width of ray fan  141 , which propagates leftward from the target center (not shown) x=0 at a distance of 23 mm above (to the right of) LED chip  142 , the size of which has been exaggerated for clarity. Ray fan  141  has the width necessary to achieve the desired illumination level at the center of the target, and in short-throw lighting this is less than what the LED would do by itself. This means the central part of the illumination lens  51  must demagnify the source, which is why the lens-center is diverging, with negative focal length. In fact, the very function of lens-center  140  is to provide the proper size of source image (of which ray fan  141  is a cross-section) for the target center, x=0. 
       FIG. 14  also shows expanding ray fan  143 , originating at the left edge of chip  142 . The will mark the upper edge of a source image as seen from the x-positions at which these left rays intercept the target plane (not shown, but to the right). These rays exemplify how edge rays are sent through previously established surfaces. 
       FIG. 15  shows concave-concave lens-center  150 , central ray-fan  151 , chip  152 , and left-ray fan  153 . The lens surfaces have about half the curvature of the concave surface of  FIG. 14 . 
       FIG. 16  shows plano-concave lens-center  160 , central ray-fan  161 , chip  162 , and left-ray fan  163 . 
     In the progression from  FIGS. 14 to 16 , the lowest left ray (the one with an arrow) lies at a shrinking slant angle ψ, indicating different illumination behavior and setting a different course towards the final design. All three configurations produce the same illuminance at target x=0, that is to say the same size source image, as shown by the ray fans  141 ,  151 , &amp;  161  being of identical size as they arrive at each lens, which is equivalent to saying they produce the same target illuminance at center x=0. 
       FIG. 17  shows illumination lens  51 , numerically generated from a concave-plano center-lens, as in  FIG. 14 . Planar source  171  is the light source from which it was designed. 
       FIG. 18  shows illumination lens  51 , numerically generated from a concave-concave center-lens, as in  FIG. 15 . Planar source  181  is the light source from which it was designed. 
       FIG. 19  shows illumination lens  51 , numerically generated from a plano-concave center-lens, as in  FIG. 16 . Planar source  191  is the light source from which it was designed. 
     These three lenses were designed utilizing rays from the periphery of the light source, in this case circular. The size of the lens is a free parameter, but etendue considerations dictate that a price be paid for a lens that is too small. In the case of a collimator, the output beam will be inescapably wider than the goal if the lens is too small. In the case of the short-throw illumination lenses  51  disclosed herein, the result will be an inability to maintain an output illumination pattern that is the ideal linear ramp of curve  103  of  FIG. 10 , because it requires the source image of curve  106 . If the lens is smaller than the required source image size, then it cannot supply the required illumination. Thus the lens size will be a parameter fixed by the goal of a linear ramp. Lenses that are too small will have some rays trapped by total internal reflection instead of going to the edge of the pattern. If this is encountered in the design process then the iteration will have to re-start with a greater height of the lens-center above the LED. 
     In conclusion, the preferred embodiments disclosed herein fulfill a most challenging illumination task, the uniform illumination of close planar targets  115  by widely spaced lenses. Deviations from this lens shape that are not visible to casual inspection may nevertheless suffice to produce detractive visual artifacts in the output pattern. Experienced molders know that sometimes it is necessary to measure the shape of the lenses to a nearly microscopic degree, so as to adjust the mold-parameters until the proper shape is achieved. Experienced manufacturers also know that LED placement is critical to illumination success, with small tolerance for positional error. Thus a complete specification of a lens shape necessarily requires a high-resolution numerical listing of points mathematically generated by a fully disclosed algorithm. Qualitative shape descriptors mean nothing to computer-machined injection molds, nor to the light passing through the lens. Unlike the era of manual grinding of lenses, the exactitude of LED illumination lens  51  slope errors, means that without an iterative numerical method of producing these lens-profile coordinates, there can be no successful lenses produced. 
     The preceding description of the presently contemplated preferred embodiments is not to be taken in a limiting sense, but is made merely for the purpose of describing the general principles of the invention. The full scope of the invention should be determined with reference to the Claims.