Patent Publication Number: US-2022228723-A1

Title: Asymmetrical Optics for Linear Lighting

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims priority to U.S. Provisional Patent Application No. 63/139,534, filed Jan. 20, 2021, the contents of which are incorporated by reference herein in their entirety. 
    
    
     TECHNICAL FIELD 
     The invention relates to optics for linear lighting, and more specifically, to asymmetrical optics. 
     BACKGROUND 
     Linear lighting is a class of solid-state lighting in which an elongate, narrow printed circuit board (PCB) is populated with a number of light-emitting diode (LED) light engines, spaced along the PCB at a regular pitch or spacing. In finished linear lighting luminaires, the PCB with the LED light engines is often installed in a channel, such as a metal or plastic extrusion, and covered with a cover. The cover serves a variety of purposes, for example, protecting the interior of the channel and preventing ingress of foreign material. 
     Some channel covers may also serve as lenses or other types of optical elements that modify the light emissions from the LED light engines, e.g., to constrain the emitted light beam to some smaller beam width than would otherwise be the case. As one example, U.S. Pat. No. 10,788,170, which is incorporated by reference in its entirety, discloses two-element optical systems suitable for installation in channels. The two elements may be, e.g., an inner lens and an outer lens, or an inner diffuser and an outer lens. While the lens systems taught by this patent are effective at constraining the beam width, and also address color issues specific to LED light engines, these systems emit light symmetrically in the same fundamental direction as it was originally emitted by the LED light engines. 
     There are many circumstances in which it is desirable for a linear luminaire to emit light in a specific direction different than the direction in which it would typically emit light. The usual solution in these circumstances is to use a custom channel profile that tilts or angles the PCB and its LED light engines to the desired emission angle. Alternatively, angled mounting brackets may be used with a conventional channel. However, these types of solutions are not appropriate for all installations, because they may consume more space than is available or have special mounting requirements that the installation cannot support. 
     BRIEF SUMMARY 
     One aspect of the invention relates to a cover lens for a linear luminaire. The cover lens has a body with a refractive portion and cover-engaging structure. The body has an inner surface with a plurality of facets, and an outer surface that is either continuously curved or splined. Each of the plurality of faces has a facet angle and a facet length. The plurality of facets are physically asymmetrical so as to cause or allow an asymmetrical refraction of light that is emitted toward the inner surface. The body of the cover lens has a constant cross section over its length. 
     Another aspect of the invention relates to luminaires that include the kind of cover lens described above. 
     Other aspects, features, and advantages of the invention will be set forth in the description that follows. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWING FIGURES 
       The invention will be described with respect to the following drawing figures, in which like numerals represent like features throughout the description, and in which: 
         FIG. 1  is a perspective view of a channel and cover lens according to one embodiment of the invention; 
         FIG. 2  is a cross-sectional view of the channel and cover lens of  FIG. 1 ; 
         FIG. 3  is a cross-sectional view and ray-trace diagram similar to the view of  FIG. 2 ; 
         FIG. 4  is an optical diagram illustrating the path of a single ray of light through two optical interfaces, used to illustrate a portion of the method of designing an asymmetrical lens like that of  FIGS. 1-3 . 
         FIG. 5  is an optical diagram illustrating the path of three principal rays through an asymmetrical lens; 
         FIG. 6  is a luminous intensity plot, shown in polar coordinates, for the asymmetrical lens of Example 1; 
         FIG. 7  is a luminous intensity plot, shown in polar coordinates, for the asymmetrical lens of Example 2; and 
         FIG. 8  is a luminous intensity plot, shown in polar coordinates, for the asymmetrical lens of Example 3. 
     
    
    
     DETAILED DESCRIPTION 
       FIG. 1  is a perspective view of a linear luminaire, generally indicated at  10 . The linear luminaire  10  includes a channel  12 . The channel  12  in the illustrated embodiment includes an upper compartment  14  and a lower compartment  16 . A strip of linear lighting including a narrow, elongate printed circuit board (PCB)  18  with a number of LED light engines  20  mounted on it and spaced at a regular interval or pitch is mounted at the bottom of the upper compartment  14 . An LED light engine, as the term is used here, refers to one or more LEDs in a package. The package allows the light engine to be mounted on a PCB by a common technique, such as surface mounting. A cover lens  22  is mounted overtop the upper compartment  14 . Typically, the ends of the channel  12  would be covered with endcaps  24 , one of which is shown in the view of  FIG. 1 , and the other of which has been removed in order to illustrate the interior arrangement of the luminaire  10 . 
     The channel  12  and the cover lens  22  are each assumed to have a constant cross-sectional shape over their respective lengths. Both elements  12 ,  22  are typically manufactured by extrusion, although they may also be injection molded, machined, or manufactured by other methods. There is no theoretical limit to the length of a channel  12  or its cover lens  22 , although as a practical matter, these components may be limited to 2.5-3 meters in length in order to facilitate packaging and transportation. 
     The illustrated channel  12  is the channel described in U.S. patent application Ser. No. 17/130,935, filed Dec. 22, 2020, which is incorporated by reference in its entirety. The engagement between the channel  12  and the cover lens  22  is as described in that application. However, the cover lens  22 , which will be described below in more detail, can be adapted for use with any type of channel. Typically, the channel has some sort of cooperating engaging structure in its sidewalls that allows it to engage with the cover lens  22 . In this case, as described in the &#39;935 application, the cover lens  22  has a pair of depending legs  26  that engage with complementary structure  28  on the upper, inner sidewalls  30  of the channel  12 . In other embodiments, any such cooperating engaging structures that keep the cover lens on the channel may be used. This includes situations in which a cover lens without special mechanical engaging structure may be adhered or sealed to the channel  10  with an adhesive or encapsulant, rather than mechanically seated on it, in which case, the adherent or the surface(s) to which it is applied should be considered to be cooperating engaging structure or channel-engaging structure. 
       FIG. 2  is a cross-sectional view of the luminaire  10 , illustrating, among other things, the shape of the cover lens  22 .  FIG. 3  is a ray-trace diagram, using a view similar to that of  FIG. 2  to illustrate the paths of light rays emitted by the LED light engines  20  as they are refracted by the cover lens  22  out into the environment. In the following description, unless otherwise noted, it is assumed that the LED light engines  20  emit light symmetrically, centered about an axis indicated at  31  in  FIGS. 2 and 3 . It is also assumed that, as shown, the PCB  18  is centered on the bottom of the upper compartment  18  of the channel  12 , and the LED light engines  20  are in a line aligned with the center of the PCB  18 . The ray-trace diagram of  FIG. 3  also assumes that the rays of light are emitted into air, a point that will be addressed in more detail below. 
     The cover lens  22  is designed to refract light asymmetrically, and to produce a beam width that is narrower than an unmodified beam width of the LED light engines. Here, the terms “asymmetric” and “asymmetrical,” when applied to light emission, refer to light emission that is more to one side than the other of an axis aligned with the usual centers of emission of the LED light engines  20 . In this case, with no lens installed, light would typically be emitted along the axis  31  and symmetrically to both sides of it. With the cover lens  22  installed, instead of the peak luminous flux being emitted along a plane or axis  31  aligned with the centers of the LED light engines  20 , the peak luminous flux is centered around a plane or axis  33  that lies at an angle α away from the axis  31 , as shown in  FIG. 3 . 
     As a point of reference, a typical LED light engine  20  used in a luminaire like the luminaire  10  may have a beam width of approximately 120°. The cover lens  22  may produce a beam width of any lesser width, directed toward any angle α. In the illustrated embodiment, the angle α is 35°, and the beam width is 60°, half that of the typical unmodified beam width. In this description, beam widths are given as full width, half maximum (FWHM), unless otherwise noted. In this case, 60° FWHM means that the beam is 60° edge-to-edge, and at the edges, the luminous flux is half the luminous flux at the center of the beam. The asymmetrical refraction of the cover lens  22  and the more restricted beam width it offers can be appreciated from the ray-trace diagram of  FIG. 3 . 
     As can be seen in  FIG. 2 , the cover lens  22  includes a number of features that make possible the asymmetric light emission and narrowed beam width. First, while the cover lens  22  itself is made of a plastic material with an index of refraction higher than that of air, such as acrylic, polycarbonate, or PVC, only those portions from which light is to be emitted are transparent. In the illustrated embodiment, this means that only the center-left portion  32  of the cover lens  22  is transparent; the legs  26  and a far-right portion  34  of the cover lens  22  are colored with an opaque colorant. Here, the terms “left” and “right” are used with respect to the coordinate system of  FIGS. 2 and 3 . The opacity of some sections of the cover lens  22  prevents or retards the transmission of light through those sections. 
     The transparent portion of the cover lens  22  has an inner surface with a number of facets that face the LED light engines  20 , and an outer surface  36  that is either continuously convexly curved or convexly splined. In  FIGS. 2 and 3 , there are six facets, labeled A through F, on the inner surface. 
     Given this arrangement, refraction occurs at the facets A, B, C, D, E, F and at the outer surface  36 . That is, the angles and lengths of the facets A, B, C, D, E, F, as well as the characteristics of the outer surface  36 , define where light goes and what the beam width is. There may be any number of facets in a cover lens  22 , more or fewer than the six facets A, B, C, D, E, F of the illustrated embodiment. The facets A, B, C, D, E, F may be of equal angle and facet length, or they may differ in one or both of angle or facet length. As was noted briefly above, the outer surface  36  may form a continuous curve, or it may be a spline (i.e., a discontinuous set of curves) that provides a different curvature, and thus, a different refractive behavior, corresponding to each of the facets A, B, C, D, E, F. 
     The design of a lens like the cover lens  22  may initially begin with certain assumptions. For example, for design purposes, it may be assumed initially that the facets A, B, C, D, E, F and the outer surface  36  will each perform half of the refraction necessary to refract the light toward the angle α. A design may also initially begin with the assumption that the facets A, B, C, D, E, F will be of equal size, and that the outer surface  36  will be in the form of a spline with a segment corresponding to each of the facets. The angles of the facets A, B, C, D, E, F can be derived, under these assumptions, from an iterative process using Snell&#39;s Law, given the desired angle α and the refractive index of the material of which the cover lens  22  is to be made. The lengths and angles of the facets A, B, C, D, E, F can then be adjusted, if needed, to create a desired beam angle. If the splines that comprise the outer surface  36  approximate a single continuous curve closely enough, that single curve may replace the splines. 
     With respect to facet angles and lengths, the present inventor has found that if one calculates an ideal solution (i.e., number of facets, facet angles, facet lengths) for refracting light toward the angle α, the result will likely be cover that produces a light beam that is indeed centered at the angle α, but with a narrow beam width on the order of 10-15°. If a wider beam width is desired, adjusting the lengths and angles of the facets somewhat can help to create that wider beam width. 
     A cover lens according to embodiments of the invention may contain any number of facets, although considerations like manufacturability and the fineness of the features may influence the number of facets. In designing a cover lens and determining the number of facets, it may be helpful to begin by examining the emitted light at some regular angular interval from the axis of emission  31  of the LED light engines  20 . (The axis of emission  31  may also be referred to as the normal to the center of the emitting surface of the LED light engine  20 .). For example, tracing the path of a light ray at 10° intervals from the axis  31  may be a suitable way to determine appropriate properties for the facets without incurring an overwhelming computational burden. 
     In the illustrated embodiment, facet A has an angle of 45° with respect to the axis  31  and a facet length of 2.00 mm; facet B has an angle of 45° and a facet length of 2.50 mm; facet C has an angle of 45° and a facet length of 3.00 mm; facet D has an angle of 45° and a facet length of 3.00 mm; facet E has an angle of 45° and a facet length of 3.00 mm; and facet F has an angle of 50° and a facet length of 2.77 mm. With these dimensions, the term “facet length” refers to the length of the facet as measured along its length (i.e., its angled length); it does not refer to the vertical height of the facet as measured from its base or root. In most cases, radii of curvature may be added at the roots and tips of the facets in order to avoid sharp angles, aid in manufacturability, and prevent stress concentrators that may cause mechanical failure in use. The lengths and distances specified here are given as distances before the addition of any radii. 
     Because many of the facets A, B, C, D, E have the same facet angles but different lengths, they give the visual impression of a ragged or uneven set of teeth. The unlabeled return surfaces opposite the facets A, B, C, D, E are not critical to the overall refractive properties of the cover lens  22  and may be specified as needed. That said, it may be advantageous to choose angles for the return surfaces such that the return surfaces are substantially aligned with the light rays coming from the LED light engines. Choosing the angles of the return surfaces in this way ensures that the return surfaces have minimal interaction with the incoming light rays and, as much as possible, do not block the light rays from reaching the refractive facets A, B, C, D, E, F. In this case, the return surface for facet A has an angle of 11.96°, the return surface for facet B has an angle of 6.76°, the return surface for facet C has an angle of 6.29°, the return surface for facet D has an angle of 12.72°, and the return surface for facet E has an angle of 35.71°. 
     As those of skill in the art may note, the cover lens  22  is not a Fresnel lens, at least because the facets A, B, C, D, E, F are neither identical nor concentric about a center. In fact, in addition to providing asymmetrical light emission, the facets A, B, C, D, E, F are physically asymmetrical, in that there is no axis of symmetry along the inner face of the cover lens  22  about which the facets A, B, C, D, E, F are concentric or reflected. However, the facets A, B, C, D, E, F share some conceptual heritage with the facets of a Fresnel lens, in that, in both cases, it is the angle of the facet, and not its thickness, that determines its refractive effect. Along those lines, while the thickness of the facets A, B, C, D, E, F may vary from embodiment to embodiment, and they may be thicker in some cases to satisfy mechanical strength requirements or other concerns, they should generally be as thin as possible. In understanding the meaning of the terms “faceted lens” and “faceted surface,” it may be helpful to consider that while a Fresnel lens is a type of faceted lens, not all faceted lenses are Fresnel lenses. 
     In this embodiment, the outer surface  36  has the form of a convex lens of constant curvature. It has a radius of curvature of 50 mm centered at a point 5.00 mm to the right of the central axis  31 , given the coordinate system of  FIG. 2 . The radius of curvature of the outer surface  36  intersects with an apex line  38  that is a distance X from the bottoms of the legs  26 , as shown in  FIG. 2 . In embodiment of  FIG. 2 , the distance X is 6.27 mm, plus or minus 0.05 mm. 
     As can also be appreciated from  FIG. 2 , the roots of the facets A, B, C, D, E, F lie closer to the bottoms of the legs  26  than the apex line  38 , at positions dependent on their facet lengths and angles. In this case, the base or root of facet A lies along a line 5.73 mm from the bottoms of the legs  26 ; the base or root of facet B lines along a line 5.90 mm from the bottoms of the legs  26 ; the base or root of facet C lies along a line 6.26 mm from the bottoms of the legs  26 ; the bases or roots of facets D and E lie along a line 6.08 mm from the bottoms of the legs  26 ; and the base or root of facet F lies along a line 5.91 mm from the bottoms of the legs  26 . All of these dimensions may have a specified tolerance of, e.g., plus or minus 0.200 mm. 
     In the design and construction of a cover lens  22 , the material into which light is to be emitted is taken into account during the design process, as its refractive index is used in Snell&#39;s Law calculations. That material should also be taken into account in determining the environments where the luminaire  10  can and should be installed. For example, the shapes and dimensions illustrated in  FIGS. 1-3  assume that the luminaire  10  will be installed and emit into air. It is also assumed that the material of which the cover lens  22  is made will have a refractive index in the range of about 1.4-1.6, which covers most plastics. For example, an acrylic plastic such as Evonik Acrylite 8N (Evonik Industries, Essen, Germany), which is a particularly suitable material for the cover lens  22 , has a refractive index of 1.492. Particularly for certain special applications, a cover lens according to an embodiment of the invention could be made of a material with a higher refractive index, such as sapphire. The facet lengths and angles would be different with different materials. 
     Luminaires that have water resistance and that can be operationally immersed in water and other fluids can be made, either by sealing or encapsulating portions of the channel  12 . If the luminaire is to emit into water, for a refractive effect similar to the effect of the luminaire  10  described above, the facet lengths and angles would be recalculated and a custom cover lens would be constructed for the environment. 
     The process of determining the angles and extents of the facets is the same regardless of the desired angle α at which light is to be directed.  FIG. 4  is an optical ray diagram illustrating the path of a single ray of light, indicated as R 1 , as it is emitted by an LED light engine  20  and passes from air into an optical medium  50 . The diagram of  FIG. 4  follows from Snell&#39;s Law, the basic law of refractive optics, and assumes that the LED light engine  20  emits the light ray R 1  into air, refractive index (n) of 1. The optical medium  50  has an inner surface  52 , which would be a facet in a faceted lens, and an outer surface  54 , which would typically be a curve or a spline in a faceted lens. In the diagram of  FIG. 4 , U 0  is the angle between the normal  31  to the surface of the LED light engine  20  and the angle of emission of the ray R 1 . The ray R 1  is emitted toward the inner surface  52  and makes an angle with the normal  56  to the inner surface  52  of θ 1 . Relative to the normal  56  to the inner surface  52 , the ray R 1  is bent at the interface between air and the inner surface  52  to an angle θ 2 . The bent ray R 1  makes an angle of θ 3  with the normal  58  to the outer surface  54  as it strikes the outer surface  54 , is bent at the interface between the outer surface  54  and the air and is emitted at an angle of θ 4  with respect to the normal  58  to the outer surface  54 . The overall relationship between the angles can be expressed as follows: 
       ∝− U   0 =θ 1 −θ 2 +θ 4 −θ 3    (1)
 
     And further: 
       θ 5 =α−θ 4    (2)
 
     As described above, it may be assumed in at least some lenses that about half of the refraction is done at the inner surface  52  and about half the refraction is done at the outer surface  54 . In the terms of  FIG. 4 , this means that θ 1 =θ 4 . By Snell&#39;s Law: 
     
       
         
           
             
               
                 
                   
                     
                       θ 
                       2 
                     
                     = 
                     
                       
                         sin 
                         
                           - 
                           1 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             sin 
                             ⁢ 
                             
                               θ 
                               1 
                             
                           
                           n 
                         
                         ) 
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
             
               
                 
                   
                     θ 
                     3 
                   
                   = 
                   
                     
                       sin 
                       
                         - 
                         1 
                       
                     
                     ⁡ 
                     
                       ( 
                       
                         
                           sin 
                           ⁢ 
                           
                             θ 
                             4 
                           
                         
                         n 
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     Where n is the index of refraction of the lens material. 
     When Equations (1)-(4) are manipulated algebraically, they yield: 
     
       
         
           
             
               
                 
                   
                     θ 
                     1 
                   
                   = 
                   
                     
                       
                         a 
                         - 
                         
                           U 
                           0 
                         
                       
                       2 
                     
                     + 
                     
                       
                         sin 
                         
                           - 
                           1 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               sin 
                               ⁢ 
                               θ 
                             
                             1 
                           
                           n 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
     As those of skill in the art might appreciate, Equation (5) is self-referential and thus not readily solved algebraically. It can be solved numerically by choosing values for θ 1  in the expression on the right side of the equation and solving iteratively until the equation is true. Once θ 1 , the angle between the ray R 1  and the normal  56  to the inner surface  52  is found, the angle of the inner surface  52  relative to the normal  31  to the surface of the LED light engine  20 , also called the facet angle, and referred to mathematically as θ 6  in this description, can be calculated as follows: 
       θ 6 =U 0 +θ 1    (6)
 
       FIG. 8  is a diagram that extends this concept. In  FIG. 8 , three light rays R 2 , R 3 , R 4  are modeled using the same basic computational technique described above. Light ray R 2  is emitted from the LED light engine  20  at an angle equal to the desired angle α. In this case, the first surface  60  and the second surface  62  are both set normal to ray R 2 , which means that ray R 2  exits the lens at the same angle at which it was emitted. 
     Light ray R 3  is emitted by the LED light engine  20  at an angle U 0  relative to the normal  31  to the surface of the LED light engine  20 . For this ray, θ 1  is calculated from Equation (5) above. Once again, the value of θ 1  can be found by iteratively selecting values for θ 1  on the right side of the equation until a value emerges that makes the equation true. This relationship holds for any angle U 0 . 
     In the case of light ray R 4 , U 0  is zero, since ray R 4  is aligned with the normal to the surface of the LED light engine  20 . Thus, for light ray R 4 , Equation (5) simplifies to: 
     
       
         
           
             
               
                 
                   
                     θ 
                     1 
                   
                   = 
                   
                     
                       a 
                       2 
                     
                     + 
                     
                       
                         sin 
                         
                           - 
                           1 
                         
                       
                       ⁡ 
                       
                         ( 
                         
                           
                             
                               sin 
                               ⁢ 
                               θ 
                             
                             1 
                           
                           n 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     This technique thus specifies the angle θ 1 . As those of skill in the art will note, all of the rays R 2 , R 3 , R 4  in  FIG. 8  are emitted on the same side of the LED light engine  20 . The same technique may be used to calculate facet angles for rays on the other side of the normal  31  to the surface of the LED light engine  20 , with U 0  as a negative angle. 
     By the diagram of  FIG. 8 , θ 1  is the angle that the ray R 2 , R 3 , R 4  makes with respect to the normal to the first or facet surface  60 ,  66 ,  68 . It should be noted that the value of θ 1  is different for each of the three rays R 2 , R 3 , R 4 . As was noted above, θ 6  is the angle that the facet surface  60 ,  66 ,  68  makes with respect to the normal  31  to the emitting surface of the LED light engine  20 , and is what this description refers to as the facet angle. The facet angle, θ 6 , is calculated using Equation (6), as described above. 
     EXAMPLES 
     Example 1: Narrow Beam Asymmetrical Lens 
     A five-facet asymmetrical lens in acrylic, n=1.492, was modeled assuming a desired angle α of 35° using five principal rays emitted at an angle U 0  relative to the normal of the LED light engine of 0°, 17.5°, 35°, −17.5° and −35° as shown below in Table 1: 
     
       
         
           
               
             
               
                 TABLE 1 
               
             
            
               
                   
               
               
                 Calculated angles for specified principal ray angles in Example 1. 
               
            
           
           
               
               
               
            
               
                 Principal 
                 Calculated θ 1   
                 Calculated θ 6   
               
               
                 Ray Angle 
                 (Equation (5)) 
                 (Facet Angle, Equation (6)) 
               
               
                   
               
            
           
           
               
               
               
            
               
                     0° 
                 46.7° 
                 46.7° 
               
               
                  17.5° 
                 25.6° 
                 43.1° 
               
               
                      35° 
                 0.0° 
                 35.0° 
               
               
                 −17.5° 
                 62.8° 
                 45.3° 
               
               
                     −35° 
                 75.5° 
                 40.5° 
               
               
                   
               
            
           
         
       
     
     The facets were assumed to have an equal facet length of 3 mm. It was also assumed that half the refraction would be done by the inner facet and half the refraction would be done by the outer surface of the lens. The resulting lens was modeled using ray-trace modeling software and a polar light emission plot in candela was created. This polar luminous intensity plot, generally indicated at  100  in  FIG. 6 , and in units of candela (Cd) showed a tight beam with an approximately 10° beam width having a center of emission at 35°. This result verifies the basic calculation techniques described here and demonstrates that an asymmetrical lens can be designed with these techniques to produce an extremely narrow beam centered at a desired angle. 
     Example 2: Initial 60° Beam-Width Lens 
     As an asymmetrical lens with a broader beam was desired, a five-facet asymmetrical lens similar to the lens of Example 1 was modeled with the same assumptions as to optical material and the same principal rays. The overall desired angle for the lens remained 35°. However, in contrast to Example 1, the individual facets were aimed differently. That is, instead of using the same desired angle α for each facet, each facet was given its own desired angle α 1 . The desired angles α 1  were in the range of 5°-65° in this example, spaced from one another at 15° intervals. This, it was hoped, would center the resulting light beam around the overall desired angle α of 35°, while the different desired angles α 1  for each facet would spread the beam more. The facet lengths in this example were equal. The calculations are shown below in Table 2. 
     
       
         
           
               
             
               
                 TABLE 2 
               
             
            
               
                   
               
               
                 Calculated angles for specified principal ray angles in Example 2. 
               
            
           
           
               
               
               
               
            
               
                   
                   
                   
                 Calculated θ 6   
               
               
                 Principal 
                 Facet Desired 
                 Calculated θ 1   
                 (Facet Angle, 
               
               
                 Ray Angle 
                 Aim Angle α 1   
                 (Equation (5)) 
                 Equation (6)) 
               
               
                   
               
            
           
           
               
               
               
               
            
               
                     0° 
                 35° 
                 46.7° 
                 46.7° 
               
               
                  17.5° 
                 50° 
                 44.0° 
                 61.5° 
               
               
                      35° 
                 65° 
                 41.2° 
                 76.2° 
               
               
                 −17.5° 
                 20° 
                 49.2° 
                 31.7° 
               
               
                     −35° 
                  5° 
                 51.8° 
                 16.8° 
               
               
                   
               
            
           
         
       
     
     Examples 1 and 2, as well as the description above, outline a general method for constructing asymmetric lenses of this type. One begins by choosing a defined angle α at which the light is to be aimed, as well as the optical material of which the lens is to be made. Based on the defined angle α, the size of the lens, and manufacturing considerations, one can choose the number of facets and select the angle of a principal ray (U 0 ) for each facet. In Examples 1 and 2, these principal rays were chosen as α, α/2, 0°, −α/2, and −α. If a narrow beam is required, each facet may be aimed at the defined angle α. If a wider beam is required, the facets can each be aimed separately at different angles in order to spread the beam. The effects of the facet angles can be tested and checked using ray-trace modeling. 
     Once the basic beam angle and beam width are set, small changes in facet angle and facet length can be used to improve the uniformity of the emitted beam, or to accentuate non-uniformity, if such is desired. 
     Example 3: Use of Luminous Intensity Plot to Finalize Facet Characteristics 
     A six-facet asymmetrical lens in acrylic, n=1.492, was modeled assuming a desired angle α of 35°. The facet angles were set as described above with respect to facets A-F of  FIGS. 1-3 : 45°, 45°, 45°, 45°, 45°, and 50° with a facet length in each case of 3.00 mm. A luminous intensity plot was created for this modeled lens using ray trace software. That luminous intensity plot is generally indicated at  150  in  FIG. 7 . 
     The luminous intensity plot  150  of  FIG. 7  shows a pronounced dip  152  in luminous intensity at 10° and a falloff  154  in luminous intensity beyond 40°. The dip  152  is interpreted as adjacent facets being too far apart, facets A and B are shortened in length to 2.00 mm and 2.50 mm, respectively, to address the dip  152 ; and facet F is shortened slightly to address a dip. Remaining facets C, D, and E have unchanged facet lengths at 3.00 mm. 
     The adjusted lens is modeled using ray-trace software and a new luminous intensity plot is created. This luminous intensity plot, generally indicated at  200  in  FIG. 8 , shows a lens that provides a 60° beam width centered at about 35° with about a 10% variation in luminous intensity across its width. 
     Although Examples 1-3 used ray-tracing technology to model the behavior of a lens, and particularly its luminous intensity over a range of angles, that need not be the case in all embodiments. In some embodiments, it may be simpler to determine a basic set of facet angles and lengths, construct an asymmetrical lens with those facet angles and lengths, and measure the luminous intensity of that actual, manufactured lens with an instrument such as a goniophotometer. For example, additive manufacturing techniques may be used to rapidly prototype asymmetrical lenses in some embodiments. 
     It should also be apparent that while luminous intensity plots are used in certain cases to determine the beam width and any variations in beam intensity, the plots shown in the drawing figures are but one tool that may be used for that purpose. Luminous intensity may be reported in any convenient manner, and other measures of the uniformity of a beam of light may be used in other embodiments. 
     As used in this description, the term “about” refers to the fact that the quoted number or range can change without changing the described effect or outcome. If it cannot be determined what number or range would cause the described effect or outcome to change, the term “about” should be construed to refer to the quoted number or range plus or minus 5%. 
     While the invention has been described with respect to certain embodiments, the description is intended to be exemplary, rather than limiting. Modifications and changes may be made within the scope of the invention, which is defined by the appended claims.