Abstract:
An illumination lens for hemispherically emitting light emitting diodes is disclosed that produces a square illumination pattern too narrow for a refractive lens to produce by itself. The lens is freeform in that it departs from circular symmetry in order to produce a square pattern. It is catadioptric in that it comprises a central refractive lens with a square output of desired angular width and a surrounding TIR prism that produces the same square output, overlapping the first for better uniformity of the sum. The central lens and circumambient TIR prism are joined in a monolithic configuration suitable for injection molding. Vector equations are disclosed for generating the shapes of the five optically active surfaces of the invention, two internal surfaces forming a central cavity surrounding the LED and three external surfaces, all five departing from circular symmetry.

Description:
RELATED APPLICATION DATA 
     This application claims the priority date of provisional application No. 61/478,458 filed on Apr. 22, 2011, entitled “Free-Form Catadioptric Illumination Lens”. 
    
    
     BACKGROUND 
     The latest high-power light-emitting diodes emit hundreds of lumens from a very small emission zone, only a few square millimeters. Although incandescent filament emits more lumens per square millimeter, their glass envelopes and spherical emission make them unsuitable for use with lenses. LEDs only have small encapsulating domes over them, and their hemispheric emission is much more amenable to being totally gathered by a lens. A characteristic common to nearly all illumination lenses of the prior art is rotational symmetry, due to the ease of mold fabrication by rotating machinery. 
     Round illumination patterns have been ubiquitous since the advent of the flashlight, not only because of fabrication restrictions but even more because of the large size of the source. The larger the light source the larger the luminaire must be to produce an illumination pattern with a sharply defined border. A square pattern necessarily requires a minimum degree of border sharpness, in that a corner is 41.4% farther out than an edge. Thus the initial circular pattern would need its edge to be no fuzzier than ±10% of its radius or it can&#39;t be turned into a square pattern. The laws of optics require than the luminaire aperture be much wider than the light source for any such sharpness to be attained. Thus only large specialty luminaire are commercially available to make square patterns from conventional light bulbs. 
     With the advent of the LED, its compactness provides a previously lacking ability to escape circular symmetry. This motivated the development of optical-theoretical means of delivering square or rectangular patterns from a circularly symmetric source. The prior art emphasizes wide-angle patterns of rectangular illumination, as in U.S. Pat. Nos. 5,924,788 and 7,674,019, both by Parkyn. In fact, the only way that preferred embodiments of the latter patent produce square beams narrower than ±45° is by discarding lateral rays beyond ±75°. This is because trying to bend light more than 30° with refraction alone results in disadvantageously thick lenses with excessive Fresnel reflectance and distortion. 
     When maximizing efficiency is a primary goal, a lens must completely surround the LED in order to collect all its light. Such a lens must collect the rays from the source that are nearly horizontal, and give them high deflection angles in order to send them toward the edge of a narrow-angle target. Refraction alone cannot deliver such high deflections, so that Total Internal Reflection (TIR) must be used.  FIG. 1  shows some relevant prior art: U.S. Pat. No. 1,977,689 by Muller (1929) discloses the basic concept of an annular TIR prism surrounding a central refractive lens. U.S. Pat. No. 2,215,900 by Bitner (1940) discloses more complex profiles on the various surfaces. U.S. Pat. No. 2,254,961 by Harris (1941) discloses a conical top surface on the TIR prism. U.S. Pat. No. 7,083,297 by Matthews et al. (2006) discloses a conical TIR surface with a straight-line profile rather than the customary curved one. It was designed for a hemispherically emitting LED rather than a spherically emitting incandescent light bulb. Note that the LED is off-center, to form an automotive low beam, so that the profile of the central lens is an off-axis ellipse. 
     Other, less similar prior art is listed in the References, and like these four all are collimating, and all are circularly symmetric. What this prior art lacks is any capability of uniformly illuminating rectangular planar targets. 
     SUMMARY 
     The presently preferred embodiment alleviates this lack in the current art by providing compact optical embodiments, and methods of designing them, that when installed over LEDs produce square illumination patterns. These useful novelties are applicable to reading lamps and downlights, for which today&#39;s markets only have round output. 
     Preferred embodiments comprise a dual free-form lens that produces two completely overlapping square illumination patterns. A central free-form refractive lens intercepts about half the light from the central LED, typically out to an off-axis angle between 45° and 50°, depending upon the particular model of LED. Rays with wider angles out to 90° are intercepted by a free-form annular TIR prism that surrounds the central lens. Both lens and prism are given excess thickness so they can be joined into a single monolithic optical element, an illumination lens that has five optically active surfaces: the top surface of the central lens, the top surface of the TIR prism, the prism&#39;s lateral TIR surface, the prism&#39;s interior surface, laterally surrounding the LED, and the bottom surface of the central lens, covering the LED. 
     Another innovative aspect is a class of preferred embodiments wherein the two top surfaces are circularly symmetric top surfaces and the lateral TIR surface is developable (i.e., quasi-conical). These features reduce the cost of mold fabrication by allowing the top half of the mold to be simply made with a lathe, and the lateral mold surface to be made with a straight edge. The interior prism surface and the bottom surface of the central refractive lens are fully free-form, acting to make their illumination patterns square. These two interior surfaces join to form a central cavity surrounding the LED light source. The particular shape of each preferred embodiment varies according to the subtended angle of the square target to be illuminated. The narrower this angle, the larger the optical element must be if the square pattern is to be well defined, with sharp enough borders to be distinguishable from a round pattern. In practical terms the primary operating range of this preferred embodiment is square targets subtending ±20° to ±60°. Smaller angles may lead to objectionable overall size, while wider angles can be done with refraction only. Also, refraction-only lenses can produce ±50° if some source light is abandoned 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The above and other aspects, features and advantages of the preferred embodiments will be apparent from the following more particular description thereof, presented in conjunction with the following drawings wherein: 
         FIG. 1  shows some prior art. 
         FIG. 2  graphs Lambertian intensity vs. that required for planar illumination. 
         FIG. 3  shows the radial expansion involved in illuminating a square. 
         FIG. 4  shows the azimuthal redistribution of flux towards a diagonal. 
         FIG. 5  is a graph of the required azimuthal deflections. 
         FIG. 6  compares small and large hemispheric encapsulants of LED chips. 
         FIG. 6A  shows how small domes over-magnify laterally. 
         FIG. 7  is a graph comparing Lambertian to supra-Lambertian intensity. 
         FIG. 7A  shows the partition of a supra-Lambertian distribution. 
         FIG. 8  shows an illumination lens of the present preferred embodiment. 
         FIG. 8A  shows the axial vs. diagonal profiles of same. 
         FIG. 8B  shows the characteristic out-of-round profiles of same. 
         FIG. 9  is a diagram of an iterative step in the vector-based profile-generation. 
         FIG. 10  is a perspective view from above of the illumination lens. 
         FIG. 10A  is a perspective cutaway view of the interior surface of the lens. 
         FIG. 11  shows a smaller, wider-angle version of the lens. 
         FIG. 12  shows how much thicker is the equivalent all-refraction lens. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     A better understanding of the features and advantages of the presently preferred embodiments will be obtained by reference to the following detailed description and accompanying drawings, which set forth illustrative embodiments in which the principles of the preferred embodiment are utilized. 
     Preferred embodiments comprise a central refraction-only lens and an annular TIR prism surrounding it. All those depicted herein are of the above-mentioned partially free-form class of lenses. Their design procedure begins by considering what is involved with producing rectangular illumination patterns. Uniform illumination of a plane requires that the luminaire intensity pattern follow a cos −3  profile.  FIG. 2  shows graph  20 , with abscissa  21  of off-axis angle to 70°, ordinate  22  of intensity relative to maximum. Dotted curve  23  is the cosine-dependent intensity of a Lambertian source, typical of an LED chip. Solid curve  24  is the cos −3  function relative to its value at 70°. This is the intensity output a luminaire must have to uniformly illuminate a plane subtending ±70°. It is quite different from the LED&#39;s raw output, as the two curves show. 
     The preferred embodiments disclosed herein produce a square illumination pattern through their free-form shape. If we suppose that we already have a lens that makes a round pattern of the desired width, then we can imagine modifying that lens to change the round pattern into a square one of the same width. Mild departures of the lens shape from circular symmetry suffice to expand a round distribution of LED light into a square output on the target. 
     Producing a square illumination pattern from a uniform circular one has two components, radial and azimuthal. Firstly, uniform illumination of a planar square means that light going to the corners must be deflected further off-axis (41% more) than rays going to the middle of an edge.  FIG. 3  is a quadrant diagram showing the radial expansion of circle  31  on the target plane into circumscribing square  32 , with greatest spreading towards the corner (for clarity, only one quadrant is shown). Radial dashed lines  33  extend from center  34  to circle  31 , each with its own uniformly spaced value of azimuth θ c . This uniform angular spacing is a consequence of the circle being uniformly illuminated. Radial arrows  35  extend outward from circle  31  to square  32 . They indicate how much the dashed radii  33  must be stretched to fill square  32 . Dotted circles  36  are 10% larger and smaller than circle  31 . They indicate the effect of the finite source-size relative to the size of the luminaire. Such blurring greatly diminishes the ‘squareness’ of  32 , so that an illumination lens has a minimum size requirement to keep blurring modest. 
     In all a free-form lens did was spread the light radially, as indicated by arrows  35  of  FIG. 3 , the illuminance along each radius would be accordingly reduced as the flux is radially spread out. As a result there would be four dark ‘spokes’ on the diagonals, with illuminance reduced thereupon to √½=71%. To prevent this, the lenses of the present invention also spread light azimuthally, towards the diagonals and away from what hereinafter will be termed the coordinate axes, or simply axes (and axial).  FIG. 4  is a further quadrant diagram, with circle  41  and circumscribed square  42 . Dashed lines  43  radiate from centre  44 , each with its own value of azimuth θ s . Dotted arrows  45  extend from circle  41  to square  42 . Their angular spacing decreases toward the diagonal, denoting how flux must be redistributed to uniformly illuminate the square. The tips of arrows  45  can be seen to be uniformly spaced along square  42  at distances tan θ s . 
     Morphing a uniform circle into a uniform square requires the azimuthal deflections indicated by the difference between dashed radii  33  of  FIG. 3  and radii  43  of  FIG. 4 . Due to symmetry, this deflection is zero along the coordinate axes and along the diagonals. Let the variable x be in the range from 0 to 1 to cover the angular range 0° to 45°, representing the fraction of the flux in that range (i.e., ⅛ th  of the entire circle).  FIG. 5  shows graph  50  with abscissa  51  denoting azimuth angle θ c  ranging from 0 to 45 degrees and ordinate  52  denoting deflection δ=θ s −θ c  in degrees. Uniform angular spacing means that the flux from 0° to θ=45x is simply x, a consequence of the uniform illumination of circle  31  of  FIG. 3 . Let square  42  of  FIG. 4  have unit half-width, so that the uniformly spaced tips of arrows  45  lie at evenly spaced values of the coordinate x, which is given by x=tan θ s , where θ s  is the azimuth of unevenly spaced dashed radii  43 . 
     The required azimuthal deflection is the difference between the these two azimuths at any given flux fraction x: δ=tan −1 x−45x, with a maximum value of 4.07456859622276, but not at the symmetrical 45/2=22.5° rather at 23.522544°, as can be seen from close inspection of the slightly asymmetric top of curve  53  of  FIG. 5 . These azimuthal deflections of a few degrees are in great contrast to the much larger radial deflections involved in transforming the Lambertian distribution  23  of  FIG. 2  into the cos −3  pattern  24 , which is what is required to produce the uniformly illuminated circles  31  of  FIG. 3 and 41  of  FIG. 4 . 
     The free-form shapes of the surfaces of the preferred embodiments disclosed herein are such as to bring about both the radial and azimuthal deflections that are required to redistribute the circularly symmetric Lambertian intensity distribution of an LED chip into one that uniformly illuminates a nearby planar square. 
     The Lambertian emission of an LED chip has a significant portion of its luminosity at high off-axis angles. In fact, a Lambertian source&#39;s cumulative off-axis emission is simply equal to sin 2 θ, as for example 50% at 45°, 75% at 60°, and 90% at 72°. Sometimes, however, LED chips are encapsulated in a hemisphere that for the sake of compactness is only a little bigger than the chip itself. When instead the hemisphere is at least n times bigger than the chip diagonal (n=refractive index of the encapsulant, typically around 1.45), the output distribution remains Lambertian and the chip simply appears magnified n times but otherwise undistorted. A smaller hemisphere will distort light from a corner, making it look peeled upward, so that horizontal emission is no longer zero but 10% of the on-axis maximum, thereby producing a ‘supra-Lambertian’ pattern. 
       FIG. 6  shows an LED  60  having the proportions of commercially available high-power LEDs such as the XP-G model sold by the Cree Corporation, truly a miniature device, with a base width of only 3.5 mm. Package base  61  supports 2.6 mm hemispheric encapsulation dome  62  and square chip  63 , shown as a diagonal cross section to show how it ‘crowds’ the dome  62 . Double-arrow  64  shows the chip&#39;s width across its square profile. Viewing-rays  65  are drawn at a typical low angle of 16° from the horizontal (74° off-axis), as they project from a viewer&#39;s eye (not shown). They are refracted downward by dome  62  to intercept chip  63 . Thus double arrow  66  shows the height of the source image as seen by the viewer. 
       FIG. 6A  depicts what that viewer sees, with tilted package  61  and dome  61 . Dotted rectangle  62  outlines the actual chip while the chip&#39;s magnified image  67  can be seen outlined by front edge  67 F, lateral edges  67 L, and rear edge  67 R. Double arrow  66  is again the height of trapezoidal source image  67 . Conventional LEDs have domes that are larger than the chip by a factor at least that of the encapsulant refractive index, typically about 1.46. Such a dome is shown in  FIG. 6  by dash-dot arc  62 L, only partially representing a hemisphere so as not to interfere with lines  65 . In the case of actual dome  62 , however, the relatively large chip ‘crowds’ the dome and produces more source magnification at large off-axis angles than the n 2  magnification of a large-enough dome. This advantageously produces a ‘supra-Lambertian’ emission, helping the preferred embodiments disclosed herein to better utilize lateral rays 
       FIG. 7  shows graph  70 , with abscissa  71  in degrees off-axis and ordinate  72  scaled from 0 to 1. Dashed curve  73  is the cosine function of a Lambertian source and dashed curve  74  is its cumulative function, sin 2 θ, which reaches unity at 90°. Solid curve  75  lies above curve  73 , hence the ‘supra-Lambertian’ term. Solid curve  76  is its cumulative distribution, lying to the right of curve  74 , because of the greater lateral fraction of its emission. Such supra-Lambertian LEDs as the XP series by Cree will thus pose an efficiency challenge to any lenses that do not totally surround them, because of their strong lateral emission. 
       FIG. 7A  redraws graph  70  and supra-Lambertian cumulative distribution  76 , along with cumulative distributions  77  and  78  derived therefrom. Their utilization will be discussed below. 
     Full collection of the LED&#39;s flux, however, limits how narrow the output beam can be, because of the high deflections required for lateral light to become forward-going light. The above-mentioned free-form lenses of the prior art only produce wide-angle patterns (±60° or more), since they are refraction-only. It is the objective of the preferred embodiment to provide uniformly illuminated square patterns for narrower patterns than this. 
     Preferred embodiments are lens elements comprising four surfaces: a top light-emitting surface, a lateral TIR surface, an interior cavity, and a flat base suitable for mounting means. The interior cavity is defined by its upper and lateral interior surfaces, both of which markedly depart from circular symmetry in order to bend more light toward the target&#39;s diagonals. The top light-emitting surface, however, is circularly symmetric, which is advantageous for mold fabrication. The lateral surface is a cone that departs only a few degrees from circular symmetry, leaning a little more horizontally on the diagonals. (This also alleviates mold costs since it is still a developable surface.) This nearly indiscernible departure from circularity of the TIR surface suffices for the different off-axis angles going to the outside edge or the corner, on the target square to be illuminated. The lateral surface must tilt far up enough from horizontal else TIR will fail within it, but otherwise the different preferred embodiments can differ considerably in the proportions of these surfaces. 
     A relatively small light source enables the lens to produce a square with a well-defined border, so the corners are well-defined. Increasing the source size will fuzz the pattern, eventually making it circular. This idea is illustrated by dotted circles  36  of  FIG. 3 , denoting a ±10% smearing of the circle  31 . Such a smearing would make the formation of square  32  difficult. The sharpness of the edge of an illumination pattern can only be attained by a luminaire with a minimum size relative to its source. While a sharp edge is uncommon with the round outputs of commercial luminaires, a square can&#39;t much be fuzzy and still be a square. 
     Key to attaining high efficiency of an illumination lens is the collection of the LED&#39;s side-going light, such as rays more than 60° off-axis, because of their significant fraction of the LED&#39;s luminosity. The difficulty arises whenever the rectangular output of moderately wide angles, such as 20-55°. The farther away is a square target and the smaller the angle it subtends at the illumination lens, the greater the deflection which lateral rays must receive from the lens. A single refractive surface can in practice deflect a ray 20-25°, requiring incidence angles of 52-60°, beyond which reflections and distortion are excessive. Thus the two surfaces of a refraction-only lens can in practice bend edge rays up to 40-45°. 
     A target at a distance of only 1.5 times its width will subtend a relatively narrow ±18° across and ±25° diagonally, much smaller angles than emitted by an LED, so that large deflections are necessary, requiring high incidence angles. In turn, high incidence angles require a thick lens. A two-surface refractive illumination lens that is thick enough for 60° incidence angles will bend the edge rays a total of 40°. Thus it can only utilize source light out to 68° off-axis, beyond which is another 16% of the light of a Lambertian distribution. For the supra-Lambertian distribution  76  of  FIG. 7 , this is 24%, one that the preferred embodiment can fully utilize. 
     Rather than such a thick refraction-only lens that must let unusable lateral light escape beneath it, preferred embodiments disclosed herein that produce the same desired square will instead comprise an annular TIR prism surrounding a smaller and thinner central refraction-only lens.  FIG. 8  shows free-form illumination lens  80 , bounded by upper spherical surface  81 , upper conical surface  82 , lateral quasi-conic TIR surface  83 , circumferential entry surface  84 , and free-form lower lens-surface  85 . Lens  80  comprises central refraction-only lens  80 L, bounded by upper surface  81  and lower surface  84 , and annular TIR prism  80 P, bounded by surfaces  82 ,  83 , and  84 . Gap  80 T joins central lens  80 L and annular TIR prism  80 P into a single monolithic illumination lens. Gap  80 T has a minimum size, typically a few millimeters, dictated by mold-flow and to provide a shadow for mounting base  80 B. Free-form surfaces  84  and  85  enclose cavity  80 C, which completely surrounds the central LED light source  86 , so that all its light is collected. 
     In  FIG. 8 , central refraction-only lens  80 L generates a complete output pattern, shown by dotted arc  87  of output angle γ ranging from 0 to edge-ray angle γ A , the angular half-width of the target. For target width W and height H above LED chip  86 , target angular half-width is given by
 
γ A =tan −1 ( W/√ 2 H )
 
Here the subscript A stands for axial. A second, completely overlapping square output pattern is produced by prism  80 P, shown as dotted arc  88  spanning the same edge-ray angle γ A . Both square patterns have diagonal angular half-width γ D , given by
 
γ D =tan −1 (√2 W/ 2 H )=tan −1 ( W/√ 2 H ).
 
     For various relative target heights, Table 1 lists their values of target angular semi-width, on-axis and diagonal.  FIG. 8  is for H/W=1.16. 
                                               TABLE 1                   Angular half-widths of rectangular targets of width W at distance H            H/W   2   1.5   1   0.75   0.5               γ A     14.0°   18.4°   26.6°   33.7°   45°       γ D     19.5°   25.3°   35.3°   43.3°   54.7°       (γ D /γ A )   (1.387)   (1.369)   (1.327)   (1.286)   (1.216)                    
Note the declining ratio of the two angles, while their tangents keep the ratio √2. Their absolute difference actually levels out, and it is γ D −γ A  against which the above-mentioned ‘fuzziness’ must be specified, say as not to exceed 20%.
 
     Central lens  80 L collects input light designated by arc  89 L, comprising all directions of emission within an off-axis angle α L . Annular TIR prism  80 P collects the remaining input light, designated by arc  89 P, comprising all light beyond α L . This edge-ray angle α L  is a freely chosen design parameter, in this case being at 50°, the 50% cumulative flux angle of supra-Lambertian LED  86 , as shown by curve  76  of  FIG. 7 . Other values will be illustrated below. It is of course also possible for the two output patterns to be otherwise than coincident, but the result is likely to be less aesthetic, since the overlap helps each pattern hide the flaws of the other. It is also possible for the output pattern to be annular, with a dark center, if desired, simply by deflecting central rays  87 A and  88 A off-axis by minor alterations of the lens profile. This will be obvious to anyone skilled in optical design who is capable of utilizing the design procedures disclosed herein. 
     An important advantage is the free-form character of the surfaces, even though the mold-fabrication costs are higher than for rotationally symmetric surfaces. In particular, preferred embodiments are disclosed wherein only the interior surface is fully free-form. This approach is illustrated by illumination lens  80  of  FIG. 8A . Its upper surface, wherefrom light exits, is completely circularly symmetric, being the combination of a spherical refraction-only central lens  81  and a cone  82  sweeping upward from circular perimeter  81 P of the central lens  81 . This refractive cone  82  terminates along a somewhat non-circular perimeter  82 P, where the lateral surface  83  intercepts it. 
     As seen in  FIG. 8 , from non-circular tip-perimeter  82 P the lateral TIR surface  83  sweeps down and inward, acting as the TIR surface of illumination lens  80 . TIR surface  83  is a quasi-cone with its apex angle varying a few degrees going around it, specifically leaning down a few degrees more on the diagonal than on the axis. The lateral TIR surface, like a true cone, is developable (i.e. made by wrapping a planar cutout), in that its profile is an uncurved straight line of slightly varying tilt angle ρ T , which is a few degrees less along a diagonal than along an axis. The variations in cone angle act to radially spread the light from circle to corners. 
     The interior surface  84  of the TIR prism  80 P is fully free-form, acting to both radially and azimuthally redistribute the light of the central LED  86 . It is mirror-symmetrical about its axes and its diagonals, so that mathematically it is only required to numerically generate lens coordinates from 0 to 45 degrees. With rectangular targets a full 90 degrees would have to be generated. The interior surface&#39;s local tilt angle from vertical, angle ρ I , is adjusted point-to-point so that exemplary ray  88 R is properly refracted as it enters TIR prism  80 P. Thereafter its course is fixed, because surfaces  82  and  83  have straight-line profiles, so that all rays encounter them at the same slope. Thus only interior surface  84  remains adjustable in order to properly distribute the flux from LED source  86  onto a rectangular target. As the closest surface to LED source  86 , cavity  80 C scales the overall size of illumination lens  80  relative to the LED&#39;s width. Cavity  80 C must be large enough to assure that the light it receives has relatively narrow angular extent ±σ, where narrow usually means under ±10°. 
       FIG. 8A  magnifies the right side of  FIG. 8  in order to show both axial and diagonal lens profiles. Upper surfaces  81  and  82  are circularly symmetric and thus have identical profiles, except that cone  82  extends to point  82 P on-axis but to somewhat farther out point  82 D on the diagonals. Lateral TIR surface  83  has a straight-line profile slightly varying in tilt from horizontal. On-axis profile  83 A is tilted from horizontal at the angle ρ A , extending from circular base  83 B to tip  82 P. Only slightly different, diagonal profile  83 D is tilted from horizontal at the angle ρ D , extending from circular base  83 B to tip  82 D. 
     In contrast, interior surface  84  differs oppositely in axial vs. diagonal profiles, as shown in  FIG. 8A , in that the on-axis profile  84 A is slightly outside the diagonal profile  84 D. On-axis profile  84 A extends vertically downward from upper corner  85 P while slightly inward diagonal profile  84 D extends downward from upper corner  85 D. These corners,  85 P and  85 D, are the ends of the axial and diagonal profiles of free-form surface  85 , the lower surface of the central refraction-only lens  80 L. Both corners  85 P &amp;  85 D lie at off-axis angle α L  from the center of LED  86 . Perimeter  85 P is non-circular, as shown in  FIG. 8C , because of this radial change going from on-axis to diagonal. At both points, interior surface  84  has vertical slope to minimize the angle μ L , but only for a fraction of a millimeter, so that mold release is not compromised. 
       FIG. 8B  is a view from above of the non-round edges of illumination lens  80 . Perimeter  82 P on upper cone  82  can be compared with circumscribed dashed reference circle RC 1  to see that it has modest diagonal swelling. Base perimeter  84 B of interior surface  84  can be compared with circumscribed dashed reference circle RC 2  to see its shrunken diagonals. Perimeter  85 P is the least out-of-round of the three, with only minor departures from inscribed dashed reference circle RC 3 . This is because it is constrained to lie along the cone defined by central angle α L  subtended at the center of source LED  86  of  FIG. 8 . 
     Perimeter  85 P is defined by rays arriving at the angle α L  from the center of the LED. (Rays from other parts of the chip will be discussed below) As the defining edge ray is incident upon that vertical slope on the top edge of surface  84 , it will be deflected downward by refraction to become ray  84 R, at angle μ L  from horizontal, given by
 
μ L =sin −1 (cos α L   /n )=24°
 
This particular value is for n=1.58 of polycarbonate and α L =50°.
 
     Cone surface  82  is in this case given a cone angle ρ C  equal to μ L , so that it will not interact with any rays going toward lateral TIR surface  83 . Due to thickness  80 T of  FIG. 8 , the distance between cone  82  and ray  84 R allows the cone to take a lesser angle if desired, limited only by TIR failure. This option will be disclosed below. 
     As shown in  FIG. 8 , the angle α ranges from 0 to 90°, or even a little more in supra-Lambertian LEDs. It is the input variable for deriving the lens profile, while the off-axis output angle γ(α) varies from 0, towards the center of the target, then across target to the output angle γ A , towards the edge of the target. The exact value of γ(α) will depend upon the source&#39;s cumulative flux distribution, as shown in  FIG. 7A , where the abscissa is now identified with the input angle α and the ordinate with cumulative flux Cum, ranging from 0 on-axis to unity at 95 degrees. Curve  76  reads out the cumulative flux function Cum(α) of a supra-Lambertian source such as  86  of  FIG. 8 . Cumulative flux is also known as encircled flux, and its square root is the radius of the encirclement. When uniformly illuminating a square, for instance, a square that contains a fraction f of the total will be √f times smaller. In this case the scale factor √f will be applied to the tangent of γ A  or γ D . 
     Because illumination lens  80  generates two output patterns, there are two output angle assignments γ(α), using two partitions of the cumulative flux distribution, under and over the angle α L . The first is for central refraction-only lens  80 L, utilizing the flux from 0 to α L , so that its cumulative flux distribution is 100% at that angle: Cum A (α L )=1, giving
 
Cum A (α)=Cum(α)/Cum(α L ) α&lt;α L  
 
 FIG. 7A  shows this function as curve  77 .
 
     The value of output angle γ a  (α) assigned to each value of the input angle α is
 
Tan γ a (α)=√Cum A (α)Tan γ A .
 
 FIG. 7A  shows this square root as dashed curve  77 S. It is nearly a straight line.
 
     The second partition of the cumulative flux distribution Cum(α) is for TIR prism  80 P and runs in the opposite direction, due to the mirror nature of the TIR prism.
 
Cum D (α)=[1−Cum(α)]/[1−Cum(α L )]
 
 FIG. 7A  shows this function as curve  78 .
 
     Similarly, along the diagonal the output-angle assignment is
 
Tan γ d (α)=√Cum D (α)Tan γ D  
 
 FIG. 7A  shows this square root as dashed curve  78 S. It too is nearly a straight line.
 
     These two functions γ a (α) and γ d (α) will be used to generate slightly differing profiles of surfaces  84  and  85 , in the axial and diagonal directions. Any direction intermediate between axial and diagonal will have a profile lying between them that is a linear combination of the two. A conveniently smooth way to calculate such a linear combination is via the azimuthal angle θ, shown at the top of  FIG. 8A . 
     In  FIG. 8A  the tilt angles ρ A  and ρ D  of TIR surface  83  have the values necessary to internally reflect light at angle μ L  from horizontal into a ray that will be refracted by exit cone  82  into the external off-axis angle γ A  toward the edge of the target, or angle γ A  toward the corner. These values are given by
 
 I   A =ρ C +γ A    I   D =ρ C +γ D  
 
 R   A =sin −1 (sin  I   A   /n )  R   D =sin −1 (sin  I   D   /n )
 
β A   =R   A −ρ C  β D   =R   D −ρ C  
 
ρ A =½(90°+ρ C −β A ) ρ D =½(90°+ρ C −β D )
 
In  FIG. 8  the result is ρ A =55.1° and ρ D =53.3°.
 
     In  FIG. 8A , generic ray  88 R going from exit cone  82  toward the target at an output angle γ will leave the cone surface  82  at an incidence angle I, with the surface normal N, given by
 
 I=ρ   C +γ,
 
     Inside surface  82  the ray has an angle R with the normal N given by
 
 R =sin −1 (sin  I/n )
 
     Within TIR prism  80 P the ray has the angle β from vertical, given by
 
β= R−ρ   C  
 
     The incidence angle of the ray with the TIR surface  83  is given by
 
 I   T =ρ T +β
 
     Before reflection off TIR surface  83  the angle μ from horizontal given by
 
μ=180 °−I   T −ρ T  
 
     Recall that the cone angle ρ T  of lateral surface  83  varies between ρ A  and ρ D . 
     Table 1 above can be augmented with the characteristics of the various lenses for illuminating those targets. Table 2 further lists the edge-ray incidence angles at the tip of exit surface  82 . The rightmost column shows substantial flux loss to the diagonals. Compensation for this will be discussed below, but these losses and the proximity to the critical angle (bottom row) show that this is the nearest target, with the widest subtended angle, of this lens&#39;s illumination capability.  FIG. 11 , discussed below, shows an example of a preferred embodiment for this illumination angle. For closer targets a refraction-only lens is generally preferable, but the lens of  FIG. 11  is actually smaller than its all-refraction counterpart, shown in  FIG. 12  at the same scale. 
     
       
         
               
             
               
               
               
               
               
               
             
           
               
                 TABLE 2 
               
             
             
               
                   
               
               
                 Lens parameters for Cone ρ C  = 24°, rectangular target width W at distance H 
               
             
          
           
               
                 H/W 
                 2 
                 1.5 
                 1 
                 0.75 
                 0.5 
               
               
                   
               
               
                 γ A   
                 14.0° 
                 18.4° 
                 26.6° 
                 33.7° 
                 45° 
               
               
                 γ D  (γ D /γ A ) 
                 19.5° (1.387) 
                 25.3° (1.369) 
                 35.3° (1.327) 
                 43.3° (1.286) 
                 54.7° (1.216) 
               
               
                 I = ρ C  + γ 
                 38°, 43.5° 
                 42.4°, 49.3° 
                 50.6°, 59.3° 
                 57.7°, 64.3° 
                 69°, 78.7° 
               
               
                 R 
                 22.9°, 25.8° 
                 25.3°, 28.7 
                 29.3°, 33.0° 
                 32.3°, 34.8° 
                 36.2°, 38.4° 
               
               
                 Fr refl. 
                 5.6%, 6.0% 
                 5.9%, 6.8% 
                 7.1%, 9.9% 
                 9.2%, 15.4% 
                 17.2%, 36% 
               
               
                 ρ A , ρ D   
                 57.5°, 56.1° 
                 56.4°, 54.7° 
                 54.4°, 52.5° 
                 52.8°, 51.1° 
                 50.9°, 49.8° 
               
               
                 I T   
                 46.8° 
                 45.4° 
                 43.3° 
                 41.9° 
                 40.6° 
               
               
                   
               
             
          
         
       
     
     In  FIG. 8 , ray  88 A (γ=0) will have angle
 
β(0)=sin −1 (sin ρ C   /n )−ρ C =−9.1°
 
     Consequently, incidence angle I T  is 46° on-axis and 44.2° diagonally. Both values are greater than the critical angle
 
θ C =sin −1 (1 /n )=39.26° for polycarbonate.
 
     The bottom row of Table 2 shows that the central-ray diagonal incidence angle is not a problem for polycarbonate, though it might be for acrylic (n=1.485). 
     The interior surface  84  has to have its local tangent at the angle ρ I  from vertical that will refract a ray at angle α into the angle μ, at the value required to be reflected and then refracted into exiting at the angle γ it must have to produce uniform illumination. Calculating the value of this angle ρ I  is most easily done with a vector approach, whereby rays have length equal to the refractive index, so that a unit vector T along the ray in air, upon entering the medium, is refracted into vector U of length n. Then the local normal vector N must have the same direction as the difference vector U-T. Using the a horizontal unit vector i and a vertical unit vector k, the ray vector in air is
 
 T=i  sin α+ k  cos α
 
Inside the lens the ray vector becomes
 
 U=n ( i  cos μ+ k  sin μ)
 
The surface normal vector is simply
 
 N =( U−T )/| U−T| 
 
This immediately yields the angle ρ I  according to
 
 N=i  cos ρ I   −k  sin ρ I  
 
This angle is used to locally generate the profiles shown in  FIG. 8A .
 
     For a particular target distance, a design is specified by the central height of surface  85 , namely z(0), as well as the parameter α L , which determines where the central lens  85  will terminate. Typically the angle α L  is selected to be near that of 50% cumulative encircled flux, so both parts of the lens handle about the same amount of flux. From α L , Snell&#39;s Law determines the cone angle
 
ρ C ≦sin −1 (cos α L   /n )
 
     In  FIG. 8A , surface  85  is shown with barely separated axial and diagonal profiles, which can be specified by the height z(r) of surface  85  at a distance r from the center. Thus there will be two profiles z A (r) and z D (r), generated with a point-by-point mathematical algorithm that calculates coordinates at α+dα from those a previously known coordinate for α.  FIG. 9  delineates the upper and lateral lens surfaces with grids that are spaced every degree (dα=1°), with ray vector T given above. Within the lens the ray vector is given by
 
 U=n ( i  sin β+ k  cos β)
 
In lieu of ray-tracing through the spherical exit surface  81 , the angle β within the lens can be approximated by
 
β=⅓(α+γ)
 
     From the vectors T and U, the surface normal vector N for surface  85  is derived in the same way as previously discussed for surface  84 , for all values of α up to α L , for both on-axis and diagonal profiles. Each value of α has a corresponding directional unit vector given by
 
 S (α)= i  sin α+ k  cos α
 
       FIG. 9  is a schematic diagram of the segment-by-segment derivation of profile  90 , similar to that of surface  85  of  FIG. 8 . The center of segment  91  is at an already known location (r,z) and the next segment  92  is to be calculated.  FIG. 9  further shows directional unit vectors i, k, and S(α), all moved off the origin for the sake of clarity. The unit normal vector N(α) is known for every α. Position vector R(α) is given by
 
 R (α)=| R (α)| S (α)
 
     Its length |R(α)| (scalar magnitude) is given by the previously know values of r and z
 
| R (α)|=√( r   2   +z   2 )
 
     Along the known direction S(α+½dα) will lie the control-point vector C(α+½dα), given by 
     
       
         
           
             
               C 
               ⁡ 
               
                 ( 
                 
                   α 
                   + 
                   
                     d 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       α 
                       / 
                       2 
                     
                   
                 
                 ) 
               
             
             = 
             
               
                 S 
                 ⁡ 
                 
                   ( 
                   
                     α 
                     + 
                     
                       d 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         α 
                         / 
                         2 
                       
                     
                   
                   ) 
                 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 
                   
                     N 
                     ⁡ 
                     
                       ( 
                       α 
                       ) 
                     
                   
                   · 
                   
                     R 
                     ⁡ 
                     
                       ( 
                       α 
                       ) 
                     
                   
                 
                 
                   
                     N 
                     ⁡ 
                     
                       ( 
                       α 
                       ) 
                     
                   
                   · 
                   
                     S 
                     ⁡ 
                     
                       ( 
                       
                         α 
                         + 
                         
                           d 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             α 
                             / 
                             2 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
     This defines the outer edge of segment  91  of  FIG. 9 , from which will ‘hinge’ segment  92  so it adopts the angle given by the surface normal vector N(α+½dα). The next coordinate point R(α+dα) is where segment  92  is intercepted by an extension of directional unit vector S(α+dα), which for the sake of clarity is not shown. 
               R   ⁡     (     α   +     d   ⁢           ⁢   α       )       =       S   ⁡     (     α   +     d   ⁢           ⁢   α       )       ⁢           ⁢         N   ⁡     (     α   +     d   ⁢           ⁢   α       )       ·     C   ⁡     (     α   +     d   ⁢           ⁢     α   /   2         )             N   ⁡     (     α   +     d   ⁢           ⁢   α       )       ·     S   ⁡     (     α   +     d   ⁢           ⁢   α       )                   
This iteration continues until the input variable α reaches its limit α L . The segments  91  and  92  uniquely define a tangent parabola  93 , shown dotted and slightly offset for clarity. This curve is known as a second-order Bezier curve, from which the term ‘control point’ originates. Moving a control point changes the shape of its Bezier parabola while keeping it tangent to the segments at its two endpoints. For the sake of symmetry the control points are put halfway between the desired coordinate points. The vector P to any point on parabola  93  between its points of tangency with segments  91  and  92  is indexed by a parameter t varying from 0 to 1:
 
 P   α ( t )=(1 −t ) 2   R (α)+2 t (1− t ) C (α+½ dα )+ t   2   R (α+ d α) 0&lt; t&lt; 1.
 
The well-known NURBS methods in computer-aided graphics will use such a series of parabolic segments  93  to construct a smooth higher-order polynomial formula for profile  90 .
 
     This vector method is applied to generate the on-axis and diagonal profiles of surface  85  of  FIG. 8 . Thereafter the on-axis and diagonal profiles of interior surface  84  can be similarly generated downward, starting at the perimeter of surface  85  and ending slightly at base  84 B, below the level of LED source  86 , in order to catch all its light, even the slightly down-going rays at the fringe of a supra-Lambertian source. 
     Once the on-axis and diagonal profiles z A (r) and z A (r) of surface  85  and then r A (z) and r A (z) of surface  84  have been determined, they can be combined for to generate the entirety of surfaces  84  and  85 . Using azimuth θ as shown in  FIG. 8A , anywhere on the surface that is in between the on-axis and diagonal profiles will has a profile given by a simple trigonometric weighting:
 
 z ( r ,θ)= z   A ( r )cos 2θ+ z   D ( r )sin 2θ0&lt;θ&lt;45° for surface 85
 
 r ( z ,θ)= r   A ( z )cos 2θ+ r   D ( z )sin 2θ0&lt;θ&lt;45° for surface 84
 
     This gives the surfaces an eight-fold mirror symmetry, about the four axial and four diagonal directions, a symmetry that literally means these surfaces are not 100% freeform, though not circularly symmetric either. 
       FIG. 10  is a perspective view from above of illumination lens  100 , showing central refraction-only lens  101 , exit cone  102  with non-circular perimeter  102 P, lateral TIR surface  103 , and base  100 B, from which protrude mounting legs  100 L. 
       FIG. 10A  is a perspective cutaway view of the interior surface of illumination lens  100 , showing freeform lower surface  105  with perimeter  105 P and freeform interior surface  104  with base perimeter  104 B. Central LED source  106  illuminates the two surfaces. Axis direction A and diagonal direction D are the defining directions in that the surface profiles along them, as shown in  FIG. 8A , will determine the entire freeform surface of both. 
     The chosen parameter α L  marks the divide between the central refraction-only lens and the surrounding TIR prism. It also fixes the refracted ray angle μ L , and the cone angle ρ C , which as previously mentioned may differ somewhat from μ L . Its value of 50° was chosen because it divides into two halves the flux of a super-Lambertian source, with each half forming its own square pattern. If the division was not even, one pattern would be weaker than the other, but for some situations this may be desirable. 
     When the square target is much closer, as in the rightmost column of Table 2, the target angle γ begins to exceed the above-mentioned 50°.  FIG. 11  shows illumination lens  110 , comprising central refractive lens  111  and surrounding TIR prism  112 , both generated by the same rules disclosed above that generated the lens of  FIG. 8 . Lens  111  is basically a quasi-spherical shell that deforms the flux into a square with very little net deflection. 
     As a comparison,  FIG. 12  shows a refraction-only lens  120  of the prior art, configured to make the same illumination pattern with the same LED  121  as in  FIG. 11 . This lens is taller and thicker than that of  FIG. 11 , but not as wide. It is less advantageous in that it collects light only out to 72° from the LED axis, with the remainder escaping uselessly beneath the lens. In the case of a Lambertian LED this loss is an acceptable 10% but in super-Lambertian LEDs it would be a less palatable 20%. 
     The preceding description of the presently contemplated best mode of practicing the preferred embodiment is not to be taken in a limiting sense, but is made merely for the purpose of describing the general principles of the preferred embodiment. The full scope of the invention should be determined with reference to the Claims.