Abstract:
An optical device for coupling the luminous output of a light-emitting diode (LED) to a predominantly spherical pattern comprises a transfer section that receives the LED&#39;s light within it and an ejector atop it that receives light from the transfer section and spreads it spherically. The base of the transfer section is optically coupled to the LED so that the LED&#39;s light goes inside the transfer section, which comprises a compound elliptic concentrator operating via total internal reflection. The ejector section can have a variety of shapes, and can have diffusive features on its surface as well. The device is circularly symmetric, with a height preferably only a few times its diameter. An important application is for use at the focus of the parabolic reflectors in flashlights. This version of the invention further comprises a base-can identical in outer shape and electric contacts to those of flashlight bulbs. Within the can is the current-control circuitry the interfaces the LED with battery voltage. Another version uses half a transfer section at each end of a long cylindrical ejector, illuminated by an LED at both ends. Ejection is via internal scattering by controlled sub-wavelength roughness. In one cylindrical embodiment, diffuse reflectivity varies from one third at the ends to two thirds at center.

Description:
[0001]    This application claims priority under 35 U.S.C. 119( e ) to U.S. Provisional Patent Application No. 60/470,691 filed May 13, 2003, of Juan C. Minano et al., for OPTICAL DEVICE FOR LED-BASED LIGHT-BULB SUBSTITUTE, which United States provisional patent application is hereby fully incorporated herein by reference. 
     
    
     
       BACKGROUND OF THE INVENTION  
         [0002]    The present invention relates to light-emitting diodes (LEDs), particularly optical means for substituting white LEDs for incandescent and fluorescent light bulbs.  
           [0003]    Conventional incandescent lamps of less than 100 lumens output can be matched by the latest white LEDs, albeit at a higher price. At this low end of the lumen range, the majority of incandescent applications are battery-powered. It is desirable to have an LED suitable for direct installation in the place of, for example, a burnt-out flashlight bulb.  
           [0004]    LED&#39;s can offer superior luminous efficacy over the conventional incandescent lamps used in, for example, battery-operated flashlights. Moreover, LEDs are far more tolerant of shock, vibration, and crush-stress. Although they currently cost more to produce than the incandescents, their lifetimes are ten thousand times longer. For the sake of efficacy, flashlight bulbs are run hot so they typically last only a few hours until filament failure. Also, the prices of LEDs continue to fall, along with those of the control-electronics to handle variations in battery voltage.  
           [0005]    Indeed, LED flashlights are commercially available already, but their optics have to be adapted to the geometry of light-emitting diodes, which only emit into a hemisphere. Conventional LED lamps are unsuitable for direct installation into conventional flashlights, both electrically and optically. LED lamps are electrically unsuitable because they are current-driven devices, whereas batteries are voltage sources. Typical variations in the voltage of fresh batteries are enough to exceed an LED&#39;s tolerable operating-voltage range. This causes such high currents that the Ohmic heating within the die of the LED lamp exceeds the ability of thermal conduction to remove it, causing a runaway temperature-rise that destroys the die. Therefore, a current-control device must accompany the LED lamp.  
           [0006]    Conventional LED lamps are optically unsuitable for direct installation into the parabolic reflectors of flashlights. This is because their bullet-lens configuration forms a narrow beam that would completely miss a nearby parabolic reflector typical of flashlights. Using instead a hemispherically emitting non-directional dome, centered on the luminous die, gives the maximum spread commercially available, a Lambertian pattern, with a sin 2 θ dependence of encircled flux on angle θ from the LED lamp center axis. Since θ for a typical parabolic flashlight reflector extends from 45° to 135°, an LED lamp with a hemispheric pattern is mismatched because it&#39;s emission falls to zero at only θ=90°. This results in a beam that is brightest on the outside and completely dark halfway in. Worse yet, even this inferior beam pattern from a hemispheric LED would require that it be held up at the parabola&#39;s focal point, several millimeters above the socket wherein a conventional incandescent bulb is installed.  
           [0007]    Another type of battery-powered lamp utilizes cylindrical fluorescent lamps. Although LEDs do not improve on their luminous efficacy, fluorescent lamps are relatively fragile and require high voltages.  
           [0008]    There is thus a need in the art for an effective, low voltage and optically suitable LED lamp with which current incandescent bulb flashlights can be retrofitted by direct installation of the LED lamp into the parabolic reflectors of current flashlights.  
         SUMMARY OF THE INVENTION  
         [0009]    The present invention advantageously addresses the needs above as well as other needs by providing an optical device for an LED-based light-bulb substitute, with preferred embodiments substituting for both incandescent and fluorescent lamps.  
           [0010]    In one embodiment, the invention can be characterized as an optical device for distributing the radiant emission of a light emitter comprising a lower transfer section and an upper ejector section situated upon the lower transfer section. The lower transfer section is operable for placement upon the light emitter and operable to transfer the radiant emission to said upper ejector section. The upper ejector section shaped such that the emission is redistributed externally into a substantial solid angle.  
           [0011]    In another embodiment, the invention can be characterized as an optical device for distributing the radiant emission of a light emitter, comprising multiple off-axis ellipsoids made of substantially transparent material. The ellipsoids are truncated at a focal point of each ellipsoid and they are coupled longitudinally to each other to provide a totally internally reflecting channel.  
           [0012]    In another embodiment, the invention can be characterized as an optical device for distributing radiant emission of a light emitter having an expander section and a cylindrical ejector section. The expander section is made of substantially transparent material and is operable for receiving said radiant emission and narrowing said radiant emission&#39;s angular range to that of light guiding via total internal reflection. The cylindrical ejector section is coupled to the expander section and is made of substantially transparent material having graded sub-wavelength roughness on a surface of the cylindrical ejector section and is operable for receiving and ejecting the angularly narrowed radiation.  
           [0013]    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 an illustrative embodiment in which the principles of the invention are utilized. 
       
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0014]    The above and other aspects, features and advantages of the present invention will be more apparent from the following more particular description thereof, presented in conjunction with the following drawings wherein:  
         [0015]    [0015]FIG. 1 is a top perspective view of an optical device according to an embodiment of the present invention;  
         [0016]    [0016]FIG. 2 is an exploded top perspective view of an optical device according to an embodiment of the present invention incorporating the device of FIG. 1;  
         [0017]    [0017]FIG. 3 is a top perspective view of an optical device according to an embodiment of the present invention shown in cutaway and incorporating the device of FIG. 2;  
         [0018]    [0018]FIG. 4 is a top perspective view of an optical device according to an embodiment of the present invention incorporating the device of FIG. 3;  
         [0019]    [0019]FIG. 5 is a diagram of an off-axis, totally internally reflecting elliptical profile and shows the focal properties of an ellipse;  
         [0020]    [0020]FIG. 6 is a perspective view of two optical devices similar to that of FIG. 1 according to two embodiments of the present invention, the smaller of which shows the reduction in profile achieved by relaxing the TIR condition;  
         [0021]    [0021]FIG. 7 a  is a perspective view of an optical device according to an alternative embodiment of the present invention;  
         [0022]    [0022]FIG. 7 b  is a side view of a variant of the device of FIG. 7 a  according to an embodiment of the present invention;  
         [0023]    [0023]FIGS. 8 a  and  8   b  are side views of cone-topped optical devices according to alternative embodiments of the present invention;  
         [0024]    [0024]FIGS. 9 a  to  9   d  are side views of other optical devices according to alternative embodiments of the present invention;  
         [0025]    [0025]FIG. 9 e  is a side cross-sectional view of an LED package optical device according to an alternative embodiment of the present invention;  
         [0026]    [0026]FIG. 10 is a side view of an optical device with multiple transfer sections and a depiction of light propagating through the device according to an embodiment of the present invention that substitutes for tubular fluorescent lamps;  
         [0027]    [0027]FIG. 11 is a spot diagram depicting ray density at exit plane  14  of the transfer section of FIG. 2;  
         [0028]    [0028]FIG. 12 is a side view of a multi-section optical device according to an embodiment of the present invention with an LED at each end, acting as a substitute for fluorescent lamps;  
         [0029]    [0029]FIG. 13 is FIG. 12 with a depiction of rays coming from one source;  
         [0030]    [0030]FIG. 14 is FIG. 12 with a depiction of rays scattered from one ray being totally internally reflected;  
         [0031]    [0031]FIG. 15 is a side view of an optical device according to an embodiment of the present invention with an LED at each end, acting as a substitute for fluorescent lamps;  
         [0032]    [0032]FIG. 16 is a side view the optical device of FIG. 15 with an exemplary ray shown propagating to the right;  
         [0033]    [0033]FIG. 17 is a side view of the optical device of FIG. 15 with sub-wavelength surface roughness and exemplary rays shown propagating through the device;  
         [0034]    [0034]FIG. 18 is a front cross-sectional view of the cylindrical ejector section of FIG. 17 depicting the sphere-projection method of calculating view factors;  
         [0035]    [0035]FIG. 19 is a partial perspective cut-away view of the cylindrical ejector section of FIG. 17 depicting rays emitted at the critical angle with the surface normal of the interior of the cylinder and intersecting the cylinder at the same critical angle;  
         [0036]    [0036]FIG. 20 a  is a side perspective view of the device of FIG. 15 showing the coordinate system for calculating optimal roughness distribution;  
         [0037]    [0037]FIG. 20 b  is a top perspective close up view of an expander section according to an embodiment the present invention;  
         [0038]    [0038]FIG. 20 c  is a side perspective view of the expander section of FIG. 20 b  showing edge rays of the expander section;  
         [0039]    [0039]FIG. 20 d  is a graph showing the angular variation of spatially averaged luminance of expander section output;  
         [0040]    [0040]FIG. 20 e  is a graph showing the same luminance of FIG. 20 d  with sine squared;  
         [0041]    [0041]FIG. 21 a  is a diagram of a unit sphere of directions depicting a method of calculating radiant reception by the method of the unit sphere of direction;  
         [0042]    [0042]FIG. 21 b  is a diagram showing an equatorial plane of the same unit sphere of FIG. 21 a , with circles of sin θ from 10 to 90° from the local surface normal inside a cylinder;  
         [0043]    [0043]FIG. 22 is a partial side perspective cut-away view of the cylindrical ejector of FIG. 20 a  showing how the unit sphere of directions is placed inside the cylindrical ejector;  
         [0044]    [0044]FIG. 23 a  is a diagram showing the equatorial plane of projected directions, as an interior view of a cylindrical ejector according to an embodiment of the present invention;  
         [0045]    [0045]FIG. 23 b  is a diagram showing an equal-flux subdivision of the angular space of light rays trapped within a cylindrical ejector by total internal reflection according to an embodiment of the present invention;  
         [0046]    [0046]FIG. 23 c  is a close-up view of the upper left quadrant of FIG. 23 a , labeled according to number of reflections, from 0 to 500;  
         [0047]    [0047]FIG. 24 is a graph of one-lamp specular illuminance for different values of diffuse reflectance according to an embodiment of the present invention;  
         [0048]    [0048]FIG. 25 is a graph of brightness for different values of diffuse reflectance from two-lamp specular illuminance according to an embodiment of the present invention; and  
         [0049]    [0049]FIG. 26 is a graph showing a linear distribution of diffuse reflectance that gives uniform brightness as calculated by the view-factor method disclosed herein according to an embodiment of the present invention. 
     
    
       [0050]    Corresponding reference characters indicate corresponding components throughout the several views of the drawings.  
       DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS  
       [0051]    The following description of the presently contemplated best mode of practicing the invention 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 scope of the invention should be determined with reference to the claims.  
         [0052]    Referring to FIG. 1, shown is top perspective view of an optical device according to an embodiment of the present invention. Lens  10  comprises lower transfer section  11  and upper ejector section  12 .  
         [0053]    The lens  10  is a transparent solid in the general shape of a prolate ellipsoid and is single piece of a transparent material such as acrylic or polycarbonate. It has a rotationally symmetric shape, taller than its diameter. Upper ejector section  12  is cylindrical, with a conical indentation  13  on top of it. Exit plane  14  is the boundary between transfer section  11  and ejector section  12 .  
         [0054]    The lower transfer section  11  uses internal reflection to relocate an LED&#39;s emission upward to a parabola&#39;s focal point.  
         [0055]    The upper ejector section is shaped such that the light within it is redistributed out an external surface of the upper ejector section into a solid angle substantially greater than a hemisphere, and approximating that of an incandescent flashlight bulb. The ejector section must be positioned at the same height as a glowing filament of a light bulb it replaces. It is easier to optically move this emission point, using the transfer section, than to put an LED itself at such a height, which would make heat transfer difficult, among other problems that the present invention advantageously addresses.  
         [0056]    The upper ejector section sends the transferred light out to a parabolic reflector  40  (as described below and shown in FIG. 4), sideways and downward at angles to the axis extending all the way to at least 135°, or a little more, depending upon the reflector. At least half the ejected light should be at angles over 45°, in order to illuminate the reflector and form a sufficiently intense collimated beam.  
         [0057]    In order to avoid an external reflective coating on the surface of the transfer section  11 , the geometry of the transfer section  11  must promote total internal reflection. This is why polycarbonate, with its higher refractive index (1.5855), is preferable to acrylic (1.492). Its correspondingly smaller critical angle, θ c =sin −1  (1/n), of 39.°103 vs. 42.°1, reduces the height of the transfer section from 23.5 mm to 11.6 mm.  
         [0058]    Referring next to FIG. 2, shown is an exploded top perspective view of an optical device according to an embodiment of the present invention incorporating the device of FIG. 1. Shown are circularly-symmetric lens  10 , and tricolor LED  20 .  
         [0059]    An LED package  20  comprises a diffusely white reflector cup  21 , filled with transparent epoxy; light-emitting diode chips  22 ,  23 , and  24 , colored red, green, and blue, and balanced to mix into a white hue of selectable color temperature. This selection of color temperature is accomplished via the relative luminosities of the three chips  22 ,  23 , and  24 . The bottom diameter of the lens  10  matches the top diameter of reflector cup  21 , with lines  25  showing a path of joining the lens  10  and the reflector cup  21 . All the light from the LED  20  is injected into the lens  10 , and is totally internally reflected therein to cross exit plane  14  and enter the ejector section  12 . The LED package  20  completely fills the bottom of the lens  10 . In other embodiments, this optical filling might be incomplete, in order for the lens  10  to be joined with a pre-existing commercial LED package. As long as the focus of the elliptic profile is on the luminous perimeter of the LED package, the lens  10  intercepts all light from the LED package.  
         [0060]    Referring next to FIG. 3, shown is a top perspective view of an optical device, according to an embodiment of the present invention, shown in cutaway and incorporating the device of FIG. 2. FIG. 3 shows the device of FIG. 2 installed in a light-bulb substitute. Shown are the lens  10 , the LED  20 , the base can  30 , LED control circuitry on a board  33  with a power coupler  32 , a collar  34 , and a transparent glass bulb  35 .  
         [0061]    The lens  10  and the tricolor LED  20  are mounted on the base can  30 , shown in cutaway to reveal interior components. These include the board  33 , having LED control circuitry, with the power coupler  32  connecting to an electrically isolated bottom nub  31 , which is customarily the positive polarity point on identically configured incandescent flashlight bulbs, while the negative proceeds through collar  34 , on which the tricolor LED  20  is mounted. Protectively enclosing the lens  10  and the LED  20  is a transparent glass bulb  35 , shown in cutaway. Unlike an incandescent bulb, the glass bulb  35  of the present embodiment needs no vacuum seal, so that there are none of the elevated temperatures of bulb-forming and sealing with molten glass, as there are with incandescent bulb formation. The bulb  35  preferably has a front lens  36  for beam-forming of forward rays from the lens  10 .  
         [0062]    Not only does the bulb  35  protect the lens  10  from being knocked loose from its bond with the tricolor LED  20  during handling and from getting dirty in operation, but it also makes the entire device completely similar to the incandescent bulbs it is replacing. In this case, however, the glass in  35  will not be hot. Instead, the collar  34  will provide a heat path for the tricolor LED  20 .  
         [0063]    Yet another function of the bulb  35  is color mixing. During luminous operation of the LED  20 , close scrutiny of the lens body  10  will reveal numerous small colored sparkles that are tiny images of spots on each of the three colored chips (i.e. in FIG. 2, chips  22 ,  23 , and  24 ). Although such sparkles may prove to be aesthetically attractive, if suppressing them is desired then a diffuser on the surface of the bulb  35  could provide just enough of a haze effect as to blur out such sparkles into uniform whiteness. Such diffusers can be formed directly by the mold that manufactures the bulb  35 , through transfer of a diffuser pattern (which can be holographically generated) onto the metal of the mold. The holographic diffusers operate diffractively upon transmitted light and have high forward-scattering efficiency (i.e., very little light is scattered backwards).  
         [0064]    The base can  30  is mechanically and electrically compatible with the sockets for incandescent lamps and contains built-in current-control electronics, and heat-removal means  33  (such as a copper rod) to protect the light emitting diode chips as well as enable higher currents to be used for greater die luminosity.  
         [0065]    Referring next to FIG. 4, shown is a top perspective view of an optical device according to an embodiment of the present invention incorporating the device of FIG. 3. FIG. 4 depicts the device of FIG. 3 installed in parabolic reflector  40 , such that the upper part of the lens  10  is at a focus of the parabolic reflector  40 . The base can  30  is shaped and electrically configured to be substantially similar to the incandescent light bulb it replaces.  
         [0066]    At least half of the luminous emission from the device is directed into the side angles filled by the parabolic reflector  40 , typically 45° to 135° from the axis of the parabolic reflector  40 . The remainder of the luminous emission is sent forward and not returned to the die. Above the base can  30 , the bulb  35  and the lens  10  are approximately the same size as the incandescent lamp the device replaces, with its main emission center near the focal point of the parabolic reflector  40 . This is typically near the top of the incandescent lamp, in lowest-cost designs.  
         [0067]    Referring next to FIG. 5, shown is a diagram of an off-axis, totally internally reflecting elliptical profile, showing the focal properties of an ellipse.  
         [0068]    Shown is an ellipse segment  50  having a major axis  51 , of semi-length a, and a minor axis  52 , of semi-length b. For the sake of illustrating the full length of the ellipse, reflective segment  50  mathematically continues upwards with short upper segment  50   u  and short lower segment  50   b . The ellipse segment  50  is revolved around a central axis  53  to generate a lens shape for the transfer section  11  of FIG. 1. A lower focus  54  and an upper focus  56  are at an equal distance c from a minor axis  52 . The lower focus  54  radiates a ray fan  55  that is equivalent to the string method of generating an ellipse. This ray fan reflects off of the ellipse  50  to go to the upper focus  56 . An uppermost incidence angle  57  has the largest value of all the rays making up the ray fan. The off-axis ellipsoidal surface formed by rotation of the ellipse segment  50  around the axis  53  will totally internally reflect along its entire length if an incidence angle  57 , having a value hereinafter designated θ 0 , is not less than critical angle θ c =sin −1  (1/n).  
         [0069]    A radius  58  of the lens is designated as r w  and the radius  59  is designated as r s , the semi-width of a light source, shown as  54   s  in FIG. 5. The sum of these distances, R=(r s +r w ), and the edge incidence angle θ 0  determine the shape and size of the ellipse  50 . The distance between the two foci is 2c, and R is the base of the triangle formed by either the first or last of rays  55 , known as edge rays. From the triangle it can be seen that  
         2c=R tan 2θ 0    
         [0070]    The edge slope of the ellipse  50  is tan θ 0 , so that differentiation of the ellipse equation yields  
         b 2 =tan θ 0  R a 2 /c  
         [0071]    Using the standard identity for ellipses, a 2 =b 2 +c 2 , yields:  
         a=c/{square root}(1-2 tan θ 0 /tan 2θ 0 )  
         [0072]    or a=R tan 2θ 0 /{square root}(1-2 tan θ 0 /tan 2θ 0 )  
         [0073]    For polycarbonate this edge-total internal reflectance (TIR) condition gives a relatively tall profile, with aspect ratio a/R=5.89 at θ 0 =39.1° for polycarbonate. This creates a problem for a candidate LED with R=2.5 mm, the diameter of reflector cup  21  in FIG. 2. The resultant 14 mm height is twice the focal height of a typical 2″ parabolic reflector above its base, worse yet because an ejector section will add even more height.  
         [0074]    Referring next to FIG. 6, shown is a top perspective view of two optical devices similar to that of FIG. 1 according to two embodiments, the smaller of which shows the reduction in profile achieved by relaxing the TIR condition.  
         [0075]    The lens  10  on the tricolor LED  20  is shown as in previous FIGS. 1 through 4, while a full-TIR version  10   a  is shown with a tricolor LED  20   a . Moreover, this leakage is usefully forward. Reducing the value of θ 0  enables the height of the lens  10  to be reduced, but edge rays leak out. The total internal reflection condition is not enforced over the entire surface of the transfer section  11 , leading to a major reduction in device height at a modest leakage cost of less than 10%, as determined by ray-tracing. If the tricolor LED  20  is Lambertian, not much energy is at the large angles from normal that strike near the bottom of the ellipse  50 . Monte-Carlo simulations with a Lambertian emitter that fills the bottom of the lens (r s =r w ) will suffer a surprisingly small loss of only 7% for a better than 2:1 height reduction, as shown in FIG. 6.  
         [0076]    Referring next to FIG. 7 a , shown is a top perspective view of an optical device according to an alternative embodiment of the present invention. Shown is lens  70  comprised of an off-axis ellipsoidal transfer section  71  and a spherical, diffusive ejector section  72 .  
         [0077]    The surface of the ejector section  72  has diffusive characteristics, so that each point on the ejector section  72  has a brightness proportional to the light received from the transfer section  71 . The advantage of this kind of ejector section is that the multiple wavelengths from a tricolor LED are mixed before they leave the ejector section  72 . In the non-diffusive ejector section  12  discussed above, the color mixing may be incomplete, leading to coloration of the output beam of a parabolic reflector, such as that in FIG. 4. The ejector section  72  is larger than the transfer section  71  (i.e., having a larger diameter than a middle diameter of the transfer section  71 ), so that it has some downward facing surfaces to send light towards the bottom of a parabolic reflector. Such a diffuser also acts to mix the colors of the red, green, and blue source chips within the LED light source, as previously discussed for FIG. 3. Other ejector sections can be used as well, for producing different output patterns.  
         [0078]    Referring next to FIG. 7 b , shown is a side view of a variant of the device of FIG. 7 a  according to an embodiment of the present invention.  
         [0079]    [0079]FIG. 7 b  shows a variant of the previous ball design. Lens  75  comprises an off-axis ellipsoidal lower section  76  and an upper spherical ejector section  77 . Due to the smaller ejector size the variant radiates less in the angles beyond 900° than the previous ball design in FIG. 7 a.    
         [0080]    Referring next to FIGS. 8 a  and  8   b , shown are side views of cone-topped optical devices according to alternative embodiments of the present invention.  
         [0081]    Shown are two embodiments of a lense  85  with conicoid ejectors. Conicoids include curvilinear as well as the straight-line cones. FIG. 8 a  shows a lens  80  comprising a lower transfer section  81  and a conicoid ejector  82  starting at a point on along the top of the transfer section  81  where the diameter of the base of the conicoid ejector  82  is smaller than that of the transfer section  81 , making an abrupt transition from the transfer section  81  to the ejector  82 . FIG. 8 b  shows lens  85  comprising lower transfer section  86  and a larger conical ejector  87  than that of FIG. 8A. Here the ejector  87  starts at a point on along the top of the transfer section  86  where the diameter of the base of the conicoid ejector  87  is the same as that that of the transfer section  86 , making a smooth transition from the transfer section  86  to the ejector  87 . Such conicoid ejectors can be used to send less light forward and more to the side, such as for the parabolic reflector of FIG. 4.  
         [0082]    Referring next to FIGS. 9 a  to  9   d , shown are side views of other optical devices according to alternative embodiments.  
         [0083]    [0083]FIG. 9 a  shows a lens  90  comprising a lower transfer section  90   t  and an ellipsoidal ejector  90   e , tending to produce a relatively narrow forward beam and thus needing no parabolic mirror. FIG. 9 b  shows lens  92  comprising lower transfer section  92   t  and indented ejector  92   e  with a spherical depression on top of the ejector  92   e . FIG. 9 c  shows lens  94  with transfer section  94   t  and cylindrical ejector  94   e . FIG. 9 d  depicts lens  96  comprising transfer section  96   t  and internal diffuse cylindrical ejector  96   e.    
         [0084]    Referring next to FIG. 9 e , shown is a side cross-sectional view of an LED package optical device according to an alternative embodiment of the present invention.  
         [0085]    Regarding the use of the present invention with LEDs, an alternative embodiment is illustrated in FIG. 9 e . Shown is LED package  190  having source chip  191 , transparent dielectric  192 , lead frame  193 , ejector section  194 , conical indentation  195 , upper transfer section  196 , optically inactive base  197 , reflector cup  198  and lead wire  199 .  
         [0086]    In order to facilitate manufacturing integration, this preferred embodiment can be produced integrally as a monolithic LED package comprising device  10  of FIG. 3. LED package  190  has source chip  191  immersed in transparent dielectric  192  and mounted on lead frame  193 . The lead wire  199  from the source chip  191  is operably connected to the lead frame  193 . Ejector section  194  has conical indentation  195  and is atop upper transfer section  196 . Optically inactive base  197 , like all other near-vertical surfaces in FIG. 9 e , has a positive mold release. Reflector cup  198  acts as a lower transfer section. Upper transfer  196  operates via total internal reflection. Both transfer sections, upper  196  and lower  198 , have common foci on chip  191  and on the outer edge of ejector section  194 .  
         [0087]    Turning to embodiments that substitute for tubular fluorescent lamps, there are two defining novelties. First is a serial joining of a multiplicity of successive transfer sections with intercalated ejector sections. Second is the substitution of ejector sections by making the surfaces of these sections diffusely emissive themselves.  
         [0088]    When multiple transfer sections are joined into a single lens body, light injected at one end will propagate all the way to the other end.  
         [0089]    Referring next to FIG. 10, shown is a side view of an optical device with multiple transfer sections and a depiction of light propagating through the device according to an embodiment of the present invention that substitutes for tubular fluorescent lamps.  
         [0090]    Shown is a cross-section of a dielectric body  800 , comprising three identical off-axis ellipsoids  810 ,  820 , and  830 . A light source  805  emits rays  850  that proceed rightward to an end-plane  801 . The above mentioned TIR condition is fulfilled by these off-axis elliptical profiles, so that they have the proportions of the taller section  10   a  of FIG. 6. Thus, all rays  850  arrive via numerous internal reflections to the end-plane  801 . The only reason for having multiple such sections would be a limit on overall diameter  802 . For example, duplicating the proportions of a small, battery-powered fluorescent lamp could involve more than three such ellipsoidal sections. There is preferably an LED at both ends of such a lens.  
         [0091]    There are two embodiments for duplicating the emission of a fluorescent lamp. The first is a succession of transfer sections with intercalated ejector sections. These ejectors will be functionally different from those of the flashlight-bulb substitutes previously discussed, for two reasons. First, light will be coming from both directions into the ejector section, because there is an LED light-source at each end. Second, the angular output range is only lateral, without the forward emission required of the ejector section of a flashlight-bulb substitute, as previously discussed.  
         [0092]    The exit plane of a transfer section can be considered as a source in itself, with a particular spatio-angular distribution of outward-going light. Each of the three colors from the source has its separate distribution, originally emanating from one of the LED chips  22 ,  23 , or  24  of FIG. 2, as well as its diffuse reflection from conical sidewall  21 . The off-axis ellipsoidal shape of the transfer section acts as a non-imaging optical device, in that the original spatio-angular distribution at its entrance is strongly scrambled at its exit.  
         [0093]    Referring next to FIG. 11, shown is a spot diagram depicting ray density at exit plane  14  of the transfer section of FIG. 2. These rays are leaving transfer-section  11  after being emitted by one of the colored chips and being totally internally reflected therein. Shown is distribution  900  of 20000 rays traced from one chip, with cluster  910  its blurry image, and annulus  920  corresponding to sidewall  21 , but most of the spots are distributed at random, due to multiple internal reflections. These more randomized, multiply reflected rays have a Lambertian angular distribution. The spatio-angular distribution of this light is important to ejector design, because it determines how much light intercepts the ejector.  
         [0094]    Referring next to FIG. 12, shown is a side view of a multi-section optical device according to an embodiment of the present invention with an LED at each end, acting as a substitute for fluorescent lamps.  
         [0095]    [0095]FIG. 12 illustrates a preferred embodiment exemplified by luminaire  100  with LED packages  101  and  102  optically joined to each end of a plastic body comprising off-axis ellipsoidal transfer sections  111  through  116  and intercalated diffuse ejector sections  121  through  125 . Their relative length determines how much light they intercept. Their diffuse transmittance is produced by wavelength-scale roughness, the amplitude of which determines how much of this light is scattered.  
         [0096]    Referring next to FIG. 13, shown is the device of FIG. 12 with a depiction of rays coming from one source.  
         [0097]    Shown is a luminaire  100  with an active source  101  with single-chip emission represented by rays  130 , seen to be propagating rightwards via total internal reflection. At each of successive cylindrical ejector sections  121  through  125 , about a third of the light entering is intercepted, so that the small number of remnant rays  131  represent only a few percent of all the light from the active source  101 . It is not shown what happens to the light intercepted by the ejector sections.  
         [0098]    Referring next to FIG. 14, shown is the device of FIG. 12 with a depiction of rays scattered from one ray being totally internally reflected.  
         [0099]    Shown is a luminaire  100  with an end source  101  and rays  130  emitted therefrom. The rays  132  have refracted into outside air  133 . Rays  134  are shown being internally reflected by an ejector section  124 . Such rays remain within the body of luminaire  100 , so that there are many more rays  131  remaining at the opposite end of luminaire  100  from the source  101 . A rough surface on the ejector sections  121 ,  122 ,  123 ,  124 ,  125  will cause the ejector sections to become diffuse scatterers of light impinging on them. A proper distribution of roughness (determined as described infra) will see to it that there are only the small amount of remnant rays  131  shown in FIG. 14.  
         [0100]    Exactly duplicating the luminous surface-emission of a fluorescent lamp requires firstly that the entire surface of the device seem to be glowing, that is, it all is an ejector. Secondly, there are is an alternative to the multiple-ellipsoid approach of FIG. 13 and FIG. 14, namely the cylindrical shape of fluorescent lamps themselves.  
         [0101]    Referring next to FIG. 15, shown is a side view of an optical device according to an embodiment with an LED at each end, acting as a substitute for fluorescent lamps.  
         [0102]    Shown is luminaire  200  with LED packages  201  and  202  at each end. The expander sections  211  and  212  channel light from sources  201  and  202  into cylindrical section  213 , which will act as the ejector section via a phenomenon that ordinarily is a source of optical losses in devices utilizing total internal reflection; namely, scattering by sub-wavelength roughness. Plastic fiber optic devices are limited in length by such scattering to much shorter runs than with glass. The embodiment depicted in FIG. 15 incorporates this phenomenon in a deliberately calibrated manner so as to produce a desired surface emission. This emission, however, is inward from the surface section  213 , not outwards as in a fluorescent lamp.  
         [0103]    Referring next to FIG. 16, shown is the optical device of FIG. 15 with an exemplary ray shown propagating to the right.  
         [0104]    The same lens shape as in FIG. 15 is shown as luminaire  200  with LED package  201  and tubular ejector section  213 . Exemplary ray  220  is shown propagating to the right, with total internal reflection at points  230 .  
         [0105]    Reference is hereby made to ’Loss mechanisms in optical light pipes’ by Remillard, Everson, &amp; Weber,  Applied Optics,  Vol. 31, #34, pp 7232-7241, December 1992, the entirety of which is incorporated herein. Specifically therein, equation (8) shows how internal reflectance R 0  (TIR means R 0 =1) is reduced to reflectance R according to rms surface roughness σ, when the roughness is described by Gaussian statistics:  
         R spec =R 0 exp[−(2k⊥ σ) 2 ] 
         [0106]    where k⊥=2πncosθ/wavelength is the wave number normal to the surface, for incidence angle θ&gt;θ c , within a medium of refractive index n. For 0.5 micrometer wavelength (500 nm) and n=1.58 (θ c =sin −1 (1/n)=39.3°, so cos θ&lt;0&lt;0.77 and k⊥=15,288/mm=15.3 μm  
                       TABLE I                       σ(x), nm   R spec  (x)   R diff  (x)                    1   0.999   0.001        2   0.996   0.004        5   0.976   0.024       10   0.911   0.089       15   0.810   0.190       20   0.688   0.312       25   0.558   0.442       30   0.431   0.569       35   0.318   0.682       40   0.224   0.776       45   0.151   0.849       50   0.097   0.903       75   0.005   0.995                  
 
         [0107]    The luminance B(x) of scattered radiation is the product of diffuse reflectance R diff (x)=1−R spec (x) and the illuminance I(x) of impinging light, and thus it grows rapidly with roughness σ. By varying the roughness at different distances x along the ejector section, graded amounts of scattering can be produced, so as to attain approximately uniform brightness B(x). Note that the higher scattering levels are more wavelength-sensitive and would be less preferred in applications requiring color mixing.  
         [0108]    The angular pattern of emission by this controlled sub-wavelength roughness depends upon the structure of its sub-wavelength spatial correlation function C(x, Δx), which measures the extent to which the particular random roughness profile at x resembles the profile at small distance Δx away. Completely uncorrelated roughness results in the even glow of Lambertian emission, while spatial correlations at the same level of roughness will lead to angularly periodic directional variation in luminance, visible as undesirable non-uniformities in brightness.  
         [0109]    The lowest (1-5 nm) amounts of roughness listed above, however, are more typical of residual roughness from the best diamond turning anyway. The intermediate levels, in the range 0.8&gt;R diff &gt;0.2, are preferred in this invention, although complete scattering (R diff ˜1) might be called for near the very center of an ejector-tube&#39;s length.  
         [0110]    The significance of sub-wavelength roughness for total internal reflection is that at the scale of this roughness all light is coherent and thus subject to diffraction, which in turn leads to non-specular reflection via phase disturbances of the reflected light. Non-specular reflection is another name for scattering, but it is important to note than none of this scattering is out of the dielectric body, so none passes through the surface into the surrounding air. Roughness only does this when it is near the wavelength scale, as in the case of the above-mentioned holographic diffuser. When roughness is at 1% of the wavelength, as in the case of this preferred embodiment, all scattering is reflective. Furthermore, this scattering is intended to be inflicted on totally internally reflected light guided within a cylinder. Thus it is not akin to holographic diffusers, which operate on transmitted light. Instead, sub-wavelength roughness scatters a fraction of reflected light, whether it is a Fresnel reflection at a refractive interface or totally internally reflected light.  
         [0111]    From the lens  10  surface, roughness-scattered light is reflected back into the body of the ejector section  213 , but in all directions, not just along the direction of specular reflection. Some of this scattered light will subsequently have incidence angles from the local surface normal that are less than the critical angle, and thus will exit the device when next encountering the side opposite its scattering point. The internal origin of this scattered light will be visible upon scrutiny of the lens, when inspection discloses that the light comes from inside. Sub-wavelength roughness does not affect the specular transmittance of light, so that light exits the lens governed only by Snell&#39;s law, unlike the previously discussed holographic diffuser, which imparts additional deflection to that of refraction alone. Unlike the wavelength-scale roughness of holographic diffusers, the sub-wavelength roughness utilized by the present invention has no effect upon light refracted through it—only its Fresnel reflection is scattered.  
         [0112]    Referring next to FIG. 17, shown is the optical device of FIG. 15 with sub-wavelength surface roughness and exemplary rays shown propagating through the device.  
         [0113]    Shown is the same luminaire  200  with tubular ejector section  213 , now with sub-wavelength surface roughness. Thus ray  220  from LED package  201  of FIG. 16 gives rise to hemispherical ray-fan  240  emanating from TIR point  230 . Rays  241  are transmitted out of surface  213  into the air. Rays  242  stay inside lens  213  because they are trapped by total internal reflection. Each time such a ray is internally reflected, some of its flux is scattered in turn.  
         [0114]    The cylindrical shape of the ejector  213  acts as a lens to magnify the interior surface opposite the viewer. Approximately 4-power for such plastics as acrylic or polycarbonate, this magnification tends to give salience to scratches and other surface flaws, including any non-uniformities in the sub-wavelength surface roughness. Such non-uniformities in surface roughness are made visible by the non-uniformities they cause in perceived brightness. This cylindrical magnification means that only a small portion, ¼/π=8%, of the glowing perimeter is actually visible from any particular vantage point.  
         [0115]    Referring next to FIG. 18, shown is a front cross-sectional view of the cylindrical ejector section  213  of FIG. 17 depicting the sphere-projection method of calculating view factors.  
         [0116]    Meridional ray fan  180  represents light originating by scattering from the inside surface of the ejector  213 . It is refracted out the opposite side of the ejector  213  and propagates to meet at a viewpoint to the left, not shown. From that viewpoint, zone  182  seems magnified to fill the entire image of the ejector section  213 . Edge ray  181  appears to originate from the edge of the ejector section  213 , although it actually comes from the center of zone  182 . The large incidence angle of this edge ray reduces its transmittance, giving a slight edge-dimming to the glowing appearance of the ejector section  213 . The proper distribution of roughness σ(x), for location x along the tube, will give fairly uniform brightness to the ejector section.  
         [0117]    Calculating the proper roughness distribution σ(x) falls in the category of inverse problems. The cylindrical geometry of the ejector section  213  lends itself to a mathematical analysis based on thin rings of interior surface of the tube. The aspect ratio of tube length to diameter, equal to 13:1 in FIG. 15, will play a role as well. The guided light from the expander section comes down the tube and, at each length coordinate x, some of it is scattered by roughness σ(x), while the remainder is specularly reflected, with reflectivity R given in the table above. Only 40% of this roughness-scattered light refracts out the other side of the tube, while the remainder adds to the guided light, helping to make brightness more uniform.  
         [0118]    Once emitted into the body of the ejector section, roughness-scattered light is subject to the additional effects of total internal reflection.  
         [0119]    Referring next to FIG. 19, shown is a partial perspective cut-away view of the cylindrical ejector section  213  of FIG. 17 depicting rays emitted at the critical angle with the surface normal of the interior of the cylinder and intersecting the cylinder at the same critical angle.  
         [0120]    [0120]FIG. 19 shows scattering point  190  on the interior surface of cylinder  191 , only half of which is shown, for clarity. Point  190  acts as a source for ray-cone  192 , emitting at angle θ c  from the local surface normal, with three quadrants of it shown, and one quadrant being refracted into external air. Exemplary ray  193  is shown being refracted to nearly 90° incidence angle as it passes into the external air. Uncorrelated surface roughness causes scattered light to be Lambertian, so that the fraction of ejected light is that emitted at less than the angle θ c  from the normal, which will be sin 2 θ c . Thus the escape fraction of the scattered light for n=1.59 will be  
         [0121]    sin 2 θ c =1/n 2 =39.6%  
         [0122]    Referring next to FIG. 20 a , shown is a side perspective view of the device of FIG. 15 showing the coordinate system for calculating optimal roughness distribution.  
         [0123]    [0123]FIG. 20 a  shows a perspective version of FIG. 15, with device  200  comprising light sources  201  and  202 , the expander sections  211  and  212 , and the cylindrical ejector section  213 . Deploying the origin x=0 at the ejector endpoint, and assuming unit radius r=1, gives problem domain 0&lt;x&lt;A, for aspect ratio A=L/r, given ejector length L.  
         [0124]    Referring next to FIG. 20 b , shown is a top perspective close up view of the expander section  211  according to an embodiment the present invention.  
         [0125]    [0125]FIG. 20 b  is a view inside the expander section  211 , atop LED package  201 . Within package  201  are differently colored LED chips  204 ,  205 , and  206 . Common electrode  203  is connected to the chips by wires  207 . White cup-shaped diffuse reflector  208  is filled with a protective transparent epoxy (not shown).  
         [0126]    Referring next to FIG. 20 c , shown is a side perspective view of the expander section  211  of FIG. 20 b  showing edge rays of the expander section  211 .  
         [0127]    [0127]FIG. 20 c  depicts the expander section  211  and LED package  201 . Circular edge  210  delimits output rays to the angle of ray-fan  209 . The area enclosed by circle  210  acts as a spatially uniform source in a simplified analysis of ejector performance. Section  211  is specifically shaped to restrict its output angle, θ&lt;θ g . Edge rays  209  shine from the perimeter  210  of section  211  and mark the limit of the light pattern to less than angle θ g .  
         [0128]    The view-factor approach is the basis for calculating interior illuminance inside the ejector  213  of FIG. 20 a . The illumination I(x) at coordinate x along the interior surface of the ejector section is a result of luminance adding up over a hemispherical field of view.  
         [0129]    Referring next to FIG. 20 d , shown is a graph depicting the angular variation of spatially averaged luminance of the expander section  211  output.  
         [0130]    [0130]FIG. 20 d  depicts the spatially averaged output pattern of rays  209  of FIG. 20 c . Output rays  209  issue from exit-plane  210  of the expander  211 . It is much easier to use this averaged luminance function for all rays than to account for inhomogeneities due to LED chips  204 ,  205 , and  206  of FIG. 20 b , as well as to diffuse-white cup-reflector  208 .  
         [0131]    The particular proportions of the expander  211  of FIG. 20 c  is somewhat larger than the minimum possible, further restricting its output angle. Also, this emission is non-ideal in that the intensity is nonuniform as well.  
         [0132]    [0132]FIG. 20 d  graphs the average output luminance of the expander section  211 , with all three LEDs emitting. This graph is key to a preferred method of calculating proper patterns of sub-wavelength roughness. Although this graph was obtained by a computer ray-trace, goniophotometric measurements could supply such data as well. Curve  2001  gives the relative luminance as a function of off-axis angle, while curve  2002  gives the cumulative intensity with off-axis angle. This curve will be used to calculate illumination inside the ejector section.  
         [0133]    In order to evaluate the effects of this narrowed distribution for a particular roughness function a (x), the illuminance function I(x) is calculated from next to the expander section (x=0) all the way to the middle of the ejector section. The geometry of the cylinder ensures that incidence angle is preserved as view-factor rays are sent out from a viewpoint on the inside of the dielectric material, making it easy to calculate how many bounces each ray undergoes before it arrives at x=0. From its angle of arrival at exit plane  210  of FIG. 20 c.    
         [0134]    Referring next to FIG. 20 e , shown is a graph depicting the same luminance of FIG. 20 d  with sine squared.  
         [0135]    Referring next to FIG. 21 a , shown is a diagram of a unit sphere of directions  2100  depicting a method of calculating radiant reception by the method of the unit sphere of direction.  
         [0136]    [0136]FIG. 21 a  depicts the unit-hemisphere method of illumination calculation. Unit hemisphere  2100  is symmetrical about surface normal  2101 , which is normal to elemental surface patch dA 1 . Illumination from distance surface A 2 , including elemental patch dA 2 , is evaluated on unit circle  2102  via sphere projection A s  including elemental area dA s , with dA s =dA 2  cosθ 2 /S 2 . The projection of A 2  onto A s  thus accounts for distance S and inclination θ 2  of surface A 2 . To account for the effect of inclination θ 1  of rays from dA 2 , sphere projection A 2  projects to area Ab on unit circle  2102 , with elemental area d s  multiplied by cos θ 1 . This methodology is applied to the interior surface of the ejector  213 .  
         [0137]    Referring next to FIG. 21 b , shown is a diagram depicting an equatorial plane of the same unit sphere of FIG. 21 a , with circles of sin θ from 10 to 90° from the local surface normal inside a cylinder.  
         [0138]    [0138]FIG. 21 b  depicts the unit circle  2102  of FIG. 21 a . It represents the projected view from point  190  of FIG. 19. It also represents the inside surface of the ejector tube  213  of FIG. 20 a . Outer circle  2190  represents the local tangent plane, 90° from the local surface normal. Concentric circles  2180 ,  2170 ,  2160 , to  2110  represent angles θ n  of 80’0, 70’0, 60°, to 10° from the surface normal, respectively, with their radii equal to their sines. Instead of circle  2140 , however, circle  2139  is shown, representing the critical angle θ c  at n=1.59, the refractive index of transparent polycarbonate, a widely used injection-molding plastic for the present invention. Circle  2139  corresponds to ray-cone  192  of FIG. 19.  
         [0139]    Referring next to FIG. 22, shown is a partial side perspective cut-away view of the cylindrical ejector  213  of FIG. 20 a  showing how the unit sphere of directions is placed inside the cylindrical ejector  213  of FIG. 20 a.    
         [0140]    Shown is the LED  201 , the expander section  211  with periphery  210 , and the ejector cylinder  213 . Directional hemisphere  2200  is used for the calculation of illumination at distance  250  along the x-axis from periphery  210 . The z-axis is towards the center of the cylinder and the y-axis is lateral therefrom. Cylindrical mathematical grid  240  helps visualize the surface of the ejector cylinder  213 . Ray vector t1 intercepts the opposite side of the ejector cylinder  213  from the center of directional hemisphere  2200 .  
         [0141]    Referring next to FIG. 23 a , shown is a diagram depicting the equatorial plane of projected directions, as an interior view of the cylindrical ejector  213 .  
         [0142]    [0142]FIG. 23 a  recapitulates unit circle  2190  of FIG. 21 a , as well as critical-angle circle  2139 . Curvilinear grid  230  is the projection of cylindrical grid  240  of FIG. 22 upon hemisphere  2200 . Oval  231  is the projection of periphery  210  in FIG. 22 as seen from 1.5 diameters away. Opposite and much smaller oval  232  is the projection of the other expander section  212  of FIG. 20 a , much farther away. Exemplary circumferential line  233  and exemplary axial line  234  are elements of curvilinear grid  230 . Light shining out of oval  232  will contribute to local illumination in accordance with the area A v  of  232  weighted by its luminance.  
         [0143]    Referring next to FIG. 23 b , shown is a diagram depicting an equal-flux subdivision of the angular space of light rays trapped within the cylindrical ejector  213  by total internal reflection according to an embodiment of the present invention.  
         [0144]    [0144]FIG. 23 b  has unit circle  2190 , representing all possible directions of rays and identical to that of FIG. 23 a , but evenly subdivided by radial grid  235 , with radius sin θ and polar angle Φ. Grid  235  extends from ring m=1 at critical-angle circle  2139  to ring m=M at periphery  2190 . Circumferential index n extends from n=1 to oppositely situated n=N. Incidence angle θ(m) is given by sinθ={square root}(0.4+0.6 m/M and polar-angle is Φ=n/N. These M rings and N spokes divide half of this annular zone of direction-space into MN (here  540 ) evenly-spaced and nearly square cells  236 . A square shape makes for uniform sampling. Only a semi-annulus need be considered, due to the bilateral symmetry of the cylinder. A computer-implemented calculation using such a grid would use at least a hundred times the few drawn here for the sake of clarity, essentially by subdividing grid  235 , as shown in magnified close-up  237 . Each cell  236  accounts for a fixed portion (1/MN) of the 60% of the area of the unit circle  2190  that is outside circle  2139 .  
         [0145]    Because incidence angle is preserved for rays inside a cylinder, it is only from the directions represented by cells within grid  235  that guided light can come, and it is only this light that will be scattered by the inside surface of the ejector  213 , thereby generating perceived brightness. As a projected solid angle of area 0.3/MN, each of cells  236  multiplies the luminance L(m,n) of light coming from its direction, so that I(m,n)=0.3L(m,n)/MN is the illuminance contribution of that cell. The sum total ΣI(m,n), of the illuminance contributions from all MN cells, gives the local illuminance I(x) at that distance x from the expander section  211  of FIG. 22. When the specular reflectivity is R spec (x), as listed above being determined by sub-wavelength roughness σ(x), the brightness of the ejected light is B(x)=I(x)R diff (x).  
         [0146]    To calculate the specular luminance L(m,n) of the cells of FIG. 23 b , a reverse ray-trace is done for each cell. One ray per cell is sent out from the observation point on the inside surface of the ejector section  213 , at a distance x from the exit plane of the expander section  212  of FIG. 20 a . For coordinates y across the cylinder and z into it, the direction cosines of the ray are  
         [0147]    x1(m,n)=sin(θ)cos(Φ)=cos(nπ/N){square root}[1−(0.6 m/M)] 
         [0148]    y1(m,n)=sin(θ)sin(Φ)=sin(nπ/N){square root}[1−(0.6 m/M)] 
         [0149]    z1(m,n)=cos(θ)={square root}(0.6 m/M)  
         [0150]    The factor 0.6 describes light guided by a medium of refractive index n=1.581. This factor would be altered by a different refractive index n, to the value 1-(1/n 2 ).  
         [0151]    A ray sent in direction of cell (m,n) will travel distance t1 to interception with the interior wall, given by  
         t 1 (m,n)=2z1/(z1 2 +y1 2 )  
         [0152]    This is the distance to the ray&#39;s first bounce, at coordinate xB=x−x1(m,n)t1. If xB&lt;0 then there is no bounce and the ray is in oval  231  of FIG. 23 a . Thus the ray&#39;s luminance, L(m,n), is given by FIG. 20 d , at an off-axis angle cos −1  (x1(m,n)]. Reading off curve  2003  of FIG. 20 e  gives the value L(x1). The number nB of bounces of a ray at distance x from the expander section  211  is  
         nB=trunc(t1/x)  
         [0153]    Referring next to FIG. 23 c , shown is a close-up view of the upper left quadrant of FIG. 23 a , labeled according to number of reflections, from 0 to 500. At each bounce, the local diffuse reflectance R diff (xB) reduces ray luminance. Thus these nB bounces multiply their diffuse reflectance values to reduce luminance according to  
         L(m,n)=L(x1)Π nB R(xB)  
         [0154]    The particular luminance function of FIG. 20 d  tends to give little or no luminance for incidence angles over 30°. A cylinder of n=1.58, however, will trap all rays coming out of the expander section  211  to 50.8° from its axis. An ejector diameter only 30% larger than LED cup  208  would enable the expander section  211  to cover such a wide angle, in accordance with the conservation of etendue. Practical considerations, however, dictate a diameter several times that of the 2.4 mm of cup  208  of FIG. 20 b . This means a narrower angular distribution than the maximum±±41°. The narrower distribution of FIG. 20 d  means that the endmost parts of an ejector section mostly determine specular illumination. This advantageously reduces the mathematical sensitivity of perceived brightness to small changes in the form of the monotonic increase of sub-wavelength roughness with distance x from the ejector section  213 .  
         [0155]    In the situation of light guiding, internal reflection is not subject to deliberate sub-wavelength scattering so there would be no need for grid  235  of FIG. 23 b . Every cell  236  would have luminance equal to the undiminished expander-section luminance L(x1). With scattering by sub-wavelength roughness σ(x), however, ray luminance L(m,n) decreases with every internal reflection, as scattering takes it toll at each of nB bounce points. Illuminance I(x) is constant with x when there is no roughness, but with roughness σ(x), illuminance I(x) will decrease with distance x from the expander section  211 , as guided light is lost to scattering. Accordingly, roughness σ(x) must increase with x so that the fraction R diff (x) of scattered light will increase in compensation, keeping brightness B(x) approximately uniform along the ejector section  213 .  
         [0156]    The numerical implementation of this method is straightforward, especially with the abovementioned 30° angular restriction of FIG. 21 c . Using the 13:1 aspect ratio depicted in FIG. 20 b , the method disclosed herein gives a total illuminance at x=0 that is equivalent to 695 full cells, out of a total of 27,000 cells from Φ=0 to Φ=90°.  
         [0157]    Referring next to FIG. 24 shown is a graph of illuminance for different values of diffuse reflectance according to an embodiment of the present invention.  
         [0158]    Shown is graph  2400  of first-pass specular illuminance I(x) along the full length of the ejector section  213  of FIG. 22. Curve  2401  is for diffuse reflectance R diff (x)=0.1, while curves  2402 ,  2403 ,  2404 ,  2406 , and  2408  are respectively for values 0.2, 0.3, 0.4, 0.6, and 0.8. The values of I[26] at the right ends of these curves represent unextracted light that re-enters the expander section  211  from the other end. Because some of this light is absorbed by the LED chips  203 - 205  of FIG. 20A, there will be a return factor R exp . This returned light adds to the original light coming out of the expander section  211  of FIG. 20 c . This causes each curve of FIG. 24 to be multiplied by a factor  
       F   =       1   +       R   exp        I          {   26   ]     /     I        [   0   ]           +       {       R   exp        I          {   26   ]     /     I        [   0   ]           }     2     +   …          
                =     1   /     {     1   -       R   exp        I          {   26   ]     /     I        [   0   ]             }                               
 
         [0159]    Brightness is calculated using the multiplied illuminance from both expander sections:  
         [0160]    B(x)=F R diff (x) [I(x)+I(A−x)] 
         [0161]    Referring next to FIG. 25, shown is a graph of brightness for different values of diffuse reflectance according to an embodiment of the present invention.  
         [0162]    Shown are the results of the calculation above in graph  2500 , with brightness curves  2501  through  2508  corresponding respectively to diffuse reflectance values 0.1 through 0.8. Curve  2503  has the highest central value I[13]. This indicates that a good initial value for R diff (0) is about 0.3, with increasing values towards the center.  
         [0163]    Referring next to FIG. 26, shown is a graph depicting a linear distribution of diffuse reflectance that gives uniform brightness according to an embodiment of the present invention.  
         [0164]    Shown is a distribution of diffuse reflectance that gave a constant brightness of 285. This result would be a zero-order estimate of output brightness, with only specular light as its source of light to be scattered. But the 60% of the scattered light that is trapped within the ejector will add a diffuse illuminance I D (x) along its inside surface. According to FIG. 23 a , everything outside circle  2139  will have diffuse luminance, including oval  231  representing the expander section  211  of FIG. 15.  
         [0165]    Although the expander section  211  does not scatter light itself, most scattered light going into it will be retroreflected back into the ejector  213 , excepting only the chips. Thus diffuse illuminance will vary little with x, tending to make first-order illuminance smoother than the zero-order estimate. Several more orders will add diminishing amounts of illuminance, as the diffusely scattered light specularly recirculates and is rescattered. Thus the zero-order brightness BT(x) needn&#39;t be absolutely uniform in order for the final brightness to be satisfactorily uniform.  
         [0166]    Both uncorrelated and correlated roughness, with calibrated σ, are available via diamond-turning treatment of the ejector  213  shape in a metal master-part. This part could form a mold using conventional replicant mold-making. A preferred method of diamond-turning the master part in metal with calibrated sub-wavelength roughness is sonic vibration of the diamond tool to an amplitude-level equal to σ(x), using a wideband white-noise signal to provide uncorrelated roughness.  
         [0167]    While the invention herein disclosed has been described by means of specific embodiments and applications thereof, numerous modifications and variations could be made thereto by those skilled in the art without departing from the scope of the invention set forth in the claims.