Patent Publication Number: US-11655957-B2

Title: Microstructures for transforming light having Lambertian distribution into batwing distributions

Description:
RELATED APPLICATION SECTION 
     The present application is a continuation of U.S. patent application Ser. No. 16/962,155, entitled “Microstructures for Transforming Light Having Lambertian Distribution Into Batwing Distributions”, filed on Jul. 14, 2020, which is a National Stage Entry into the United States Patent and Trademark Office from International Patent Application No. PCT/US2019/015600, filed on Jan. 29, 2019, which relies for priority on U.S. Provisional Patent Application Ser. No. 62/623,894, entitled “Microstructures for Transforming Light Having Lambertian Distribution into Batwing Distributions,” filed Jan. 30, 2018, the entire content contents all of these applications are hereby incorporated by reference. 
    
    
     FIELD 
     The present invention is related to micro light transmitting optics and microstructures for transforming light having a Lambertian distribution into batwing distributions for large area uniform illumination. 
     BACKGROUND 
     Light emitting diodes (LEDs) have quickly become the primary light generating device for current applications. Intrinsically, an LED emits the light in a Lambertian distribution, characterized by the strongest intensity at the emitting direction (zero degrees or “nadir”). Light intensity decreases following the cosine function of the angles deviated from the zero-degree (nadir) emitting direction and reduces to zero as the angle reaches 90 degrees from nadir, as illustrated in  FIG.  1   . When an LED is used to illuminate a flat surface target, the light traveling path length varies for different target locations. Typically, the path length is the shortest at the zero-degree direction where the LED emits the highest light intensity, which forces designers to increase the light source density to achieve a good illumination uniformity. 
     For applications that require uniform or even illumination over a desired area of a flat plane with low light source density, such as the back light units for displays or lighting projects for a large area, the light source should deliver light energy in the reverse fashion of a Lambertian distribution, i.e. reduced intensity at zero degrees (nadir) and high intensity at angles away from nadir, as shown in  FIG.  2   , for example. Such a distribution profile (illustrated in  FIG.  2   ) is often referred as a “batwing” distribution and is more desirable for achieving uniform illumination. 
     Transforming a Lambertian distribution emitted by, for example, an LED light source into a batwing distribution may be achieved efficiently for some applications, such as some lighting applications, by using bulk optical lenses with specifically designed shapes. Such structures may not be feasible for many applications in which LEDs are used, such as in displays of cell phones, smart phones, tablets, laptop computers, etc., due to the structure bulkiness of implanting such solutions. It is desirable to transform a Lambertian distribution into a batwing distribution with structures that are more compact than current optical lenses. 
     SUMMARY 
     It has been found that micro optical transmissive structures that are fabricated on a light transmissible substrate may be used to perform the desired transformation functions to transform a Lambertian distribution into a desired batwing distribution so that a substantially uniform illumination may be provided to a large area relative to the size of an LED light source. Embodiments of the present invention are described below. 
     According to an aspect of the invention, there is provided a light transmissive substrate for transforming a Lambertian light distribution into a batwing light distribution. The light transmissive substrate includes a first surface comprising a plurality of microstructures, and a second surface on a side of the substrate opposite the first surface. The substrate is configured to receive light in a Lambertian distribution from a light source at the first surface and transform the light into a batwing distribution exiting the second surface. The batwing distribution has a peak intensity at about ±30° to about ±60° from X and Y axes and a minimum intensity at nadir. 
     In an embodiment, each of the plurality of microstructures has a shape of a pyramid extending in a direction away from the second surface. In an embodiment, at least the microstructures are made from material having a refractive index of about 1.5, and the pyramid has a roof angle of between about 70° and about 95°. 
     In an embodiment, each pyramid has a base portion and a top portion connected to the base portion. The top portion includes a tip of the pyramid and has sides disposed at different angles than sides of the base portion. 
     In an embodiment, at least the microstructures are made from material having a refractive index of about 1.5, the sides of the base portion are disposed at angles of about 55° relative to a plane substantially parallel to the second surface, and the top portion has a roof angle of between about 85° and about 90°. 
     In an embodiment, each of the plurality of microstructures has a shape of a frustum of a pyramid and a recess in a shape of a reverse pyramid. In an embodiment, at least the microstructures are made from material having a refractive index of about 1.5, sides of the frustum are disposed at angles of about 55° relative to a plane substantially parallel to the second surface, and the reverse pyramid has a roof angle of between about 85° and about 90°. 
     In an embodiment, each of the plurality of microstructures has a shape of a corner cube. 
     In an embodiment, the second surface is substantially planar. 
     In an embodiment, the second surface comprises a texture. 
     These and other aspects, features, and characteristics of the present invention, as well as the methods of operation and functions of the related elements of structure and the combination of parts and economies of manufacture, will become more apparent upon consideration of the following description and the appended claims with reference to the accompanying drawings, all of which form a part of this specification. It is to be expressly understood, however, that the drawings are for the purpose of illustration and description only and are not intended as a definition of the limits of the invention. As used in the specification and in the claims, the singular form of “a”, “an”, and “the” include plural referents unless the context clearly dictates otherwise. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The components of the following figures are illustrated to emphasize the general principles of the present disclosure and are not necessarily drawn to scale, although at least one of the figures may be drawn to scale. Reference characters designating corresponding components are repeated as necessary throughout the figures for the sake of consistency and clarity. 
         FIG.  1    is a two-dimensional polar chart of a Lambertian intensity distribution; 
         FIG.  2    is a two-dimensional polar chart of a batwing-type intensity distribution; 
         FIG.  3    is a schematic side view of a light transmissive substrate in accordance with embodiments of the invention; 
         FIG.  4 A  is an isometric schematic view of an LED light source and a pair of light transmissive substrates with microstructures; 
         FIG.  4 B  is an isometric schematic view of a single microstructure of the substrates of  FIG.  4 A ; 
         FIG.  5 A  is an isometric view of a transferred batwing intensity distribution three-dimensional polar chart for the embodiment of  FIG.  4 A  having microstructures with roof angles of 90 degrees; 
         FIG.  5 B  is a top view of the three-dimensional polar chart of  FIG.  5 A ; 
         FIG.  5 C  is a two-dimensional polar chart of the transferred batwing intensity distribution for the embodiment of  FIG.  4 A  having microstructures with roof angles of 90 degrees; 
         FIG.  5 D  is an isometric view of a transferred batwing intensity distribution three-dimensional polar chart for the embodiment of  FIG.  4 A  having microstructures with roof angles of 85 degrees; 
         FIG.  5 E  is a two-dimensional polar chart of the transferred batwing intensity distribution for the embodiment of  FIG.  4 A  with the microstructures having a refractive index of 1.5 and roof angles of 85 degrees; 
         FIG.  5 F  is a two-dimensional polar chart of the transferred batwing intensity distribution for the embodiment of  FIG.  4 A  with the microstructures having a refractive index of 1.6 and roof angles of 85 degrees; 
         FIG.  6 A  is an isometric schematic view of an LED light source and a single light transmissive substrate with microstructures in accordance with an embodiment of the invention; 
         FIG.  6 B  is an isometric schematic view of a single microstructure of the substrate of  FIG.  6 A ; 
         FIG.  6 C  is a top schematic view of the single microstructure of  FIG.  6 B ; 
         FIG.  7 A  illustrates an isometric view of a transferred batwing intensity distribution three-dimensional polar chart for the embodiment of  FIG.  6 A  having microstructures with roof angles of 90 degrees; 
         FIG.  7 B  illustrates a top view of the three-dimensional polar chart of  FIG.  7 A ; 
         FIG.  7 C  illustrates a two-dimensional polar chart of the transferred batwing intensity distribution for the embodiment of  FIG.  6 A  with the microstructures having a refractive index of 1.5 and roof angles of 90 degrees; 
         FIG.  7 D  illustrates a two-dimensional polar chart of the transferred batwing intensity distribution for the embodiment of  FIG.  6 A  with the microstructures a refractive index of 1.6 and roof angles of 90 degrees; 
         FIG.  7 E  illustrates an isometric view of a transferred batwing intensity distribution three-dimensional polar chart for the embodiment of  FIG.  6 A  having microstructures with roof angles of 80 degrees; 
         FIG.  7 F  illustrates a two-dimensional polar chart of the transferred batwing intensity distribution for the embodiment of  FIG.  6 A  having microstructures with roof angles of 80 degrees; 
         FIG.  7 G  illustrates an isometric view of a transferred batwing intensity distribution three-dimensional polar chart for the embodiment of  FIG.  6 A  having microstructures with roof angles of 70 degrees; 
         FIG.  7 H  illustrates a two-dimensional polar chart of the transferred batwing intensity distribution for the embodiment of  FIG.  6 A  having microstructures with roof angles of 70 degrees; 
         FIG.  7 I  illustrates an isometric view of a transferred batwing intensity distribution three-dimensional polar chart for the embodiment of  FIG.  6 A  having microstructures with roof angles of 60 degrees; 
         FIG.  7 J  illustrates a two-dimensional polar chart of the transferred batwing intensity distribution for the embodiment of  FIG.  6 A  having microstructures with roof angles of 60 degrees; 
         FIG.  7 K  illustrates an isometric view of a transferred batwing intensity distribution three-dimensional polar chart for the embodiment of  FIG.  6 A  having microstructures with roof angles of 100 degrees; 
         FIG.  7 L  illustrates a two-dimensional polar chart of the transferred batwing intensity distribution for the embodiment of  FIG.  6 A  having microstructures with roof angles of 100 degrees; 
         FIG.  8 A  is an isometric schematic view of an LED light source and a single light transmissive substrate with microstructures in accordance with an embodiment of the invention; 
         FIG.  8 B  is an isometric schematic view of a single microstructure of the substrate of  FIG.  8 A ; 
         FIG.  8 C  is a top schematic view of the single microstructure of  FIG.  8 B ; 
         FIG.  9 A  is an isometric view of a transferred batwing intensity distribution three-dimensional polar chart for the embodiment of  FIG.  8 A ; 
         FIG.  9 B  is a top view of the three-dimensional polar chart of  FIG.  9 A ; 
         FIG.  9 C  is a two-dimensional polar chart of the transferred batwing intensity distribution for the embodiment of  FIG.  8 A  with the microstructures having a refractive index of 1.5; 
         FIG.  9 D  illustrates a two-dimensional polar chart of the transferred batwing intensity distribution for the embodiment of  FIG.  8 A  with the microstructures having a refractive index of 1.6; 
         FIG.  10    is an isometric schematic view of a microstructure in accordance with an embodiment of the invention; 
         FIG.  11    is an isometric schematic view of a microstructure in accordance with an embodiment of the invention; 
         FIG.  12 A  illustrates an LED light source and a single light transmissive substrate with microstructures in accordance with an embodiment of the invention; 
         FIG.  12 B  illustrates a single microstructure of  FIG.  12 A ; 
         FIG.  13 A  illustrates an isometric view of a transferred batwing intensity distribution three-dimensional polar chart for the embodiment of  FIG.  12 A ; 
         FIG.  13 B  illustrates a top view of the three-dimensional polar chart of  FIG.  13 A ; 
         FIG.  13 C  illustrates a two-dimensional polar chart of the transferred batwing intensity distribution for the embodiment of  FIG.  12 A , 
         FIG.  14 A  illustrates an isometric view of a light transmissive substrate with microstructures in accordance with an embodiment of the invention; 
         FIG.  14 B  illustrates an isometric view of a transferred batwing intensity distribution three-dimensional polar chart for the embodiment of  FIG.  14 A ; 
         FIG.  14 C  illustrates a two-dimensional polar chart of the transferred batwing intensity distribution for the embodiment of  FIG.  14 A , 
         FIG.  15 A  illustrates an isometric view of a light transmissive substrate with microstructures in accordance with an embodiment of the invention; 
         FIG.  15 B  illustrates an isometric view of a transferred batwing intensity distribution three-dimensional polar chart for the embodiment of  FIG.  15 A ; 
         FIG.  15 C  illustrates a two-dimensional polar chart of the transferred batwing intensity distribution for the embodiment of  FIG.  15 A  with the microstructures having a refractive index of 1.5; and 
         FIG.  15 D  illustrates a two-dimensional polar chart of the transferred batwing intensity distribution for the embodiment of  FIG.  15 A  with the microstructures having a refractive index of 1.6. 
     
    
    
     DETAILED DESCRIPTION 
     Embodiments of the present invention provide light transmissive substrates having microstructures that may provide the desired effect of transforming a Lambertian intensity distribution received from a light source, such as an LED, into a batwing intensity distribution that has maximum intensity away from nadir and at about ±30° to about ±60° from X and Y axes, and minimum intensity at nadir. 
       FIG.  3    is a schematic illustration of a light transmissive substrate  100  for transforming a Lambertian light distribution into a batwing light distribution in accordance with embodiments of the invention. The substrate  100  includes a first surface  110  that includes a plurality of microstructures  112 , and a second surface  120  on a side of the substrate  100  opposite the first surface  110 . 
     As discussed in further detail below, the substrate  100  is configured to receive light in a Lambertian distribution from a light source at the first surface  110  and transform the light into a batwing distribution exiting the second surface  120 . The resulting batwing distribution desirably has a peak intensity in a range of about ±30° to about ±60° from X and Y axes and a minimum intensity at nadir. In an embodiment, the light transmissive substrate  100  may provide a batwing distribution that has a peak intensity at about ±45 from X and Y axes and a minimum (near zero) intensity at nadir. In some embodiments of the invention, at least the light transmissive microstructures are made from a material having a refractive index of about 1.5, although materials having different refractive indices may also be used as long as the desired effect can be achieved. In some embodiments of the invention, the rest of the substrate is a film made of a material that also has a refractive index of about 1.5, or a refractive index that matches or substantially matches the refractive index of the microstructures. For light sources that emit infrared beams, infrared transmitting materials that may not be transparent in the visible range of light may be used. Various embodiments of the invention are described in further detail below. 
       FIG.  4 A  illustrates two light transmissive substrates  400 A,  400 B, each having a plurality of microstructures  412  on first surfaces  410 A,  410 B thereof, that are oriented orthogonally to each other and placed above a light source  430  that outputs light in a Lambertian distribution. The first surfaces  410 A,  410 B of the light transmissive substrates  400 A,  400 B are oriented towards the light source  430  and second surfaces  420 A,  420 B of the substrates  400 A,  400 B are oriented away from the light source  430 .  FIG.  4 B  illustrates a single microstructure  412  in further detail. As illustrated, the microstructure  412  is in the form of a ridge that has a so-called roof angle or vertex a (see  FIG.  3   ) of 90 degrees. 
     Light emitting from the light source  430  enters the first substrate  400 A closest to the light source  430  via its first surface  410 A, exits the first substrate  400 A at its second surface  420 A, enters the second substrate  400 B at its first surface  410 B, and exits the second substrate  400 B at its second surface  420 B. The different orientations of the microstructures  412  (i.e. being substantially perpendicular to each other) cause the light to bend and spread in two different directions and result in a net spread that is that is stronger and in a different direction relative to X and Y axes than if only one of the light transmissive substrates  400 A,  400 B is used. 
       FIGS.  5 A- 5 C  illustrate three dimensional and two-dimensional representations of the light distribution provided by the combination of the two light transmissive substrates  400  having a refractive index of 1.5 and arranged as illustrated in  FIG.  4 A . As illustrated, light energy is not only steered away from the 0 degree (nadir) emitting direction, but also pushed toward four directions approximately 45 degrees from the primary X and Y axes as shown in  FIGS.  5 A and  5 B . Along those directions, light typically travels the longest path length reaching the target area where stronger intensity is desired. Such a distribution may be desirable when there are multiple light sources arranged in a substantially square array, such as in back-lit displays or large area lighting applications (such as when lighting a warehouse).  FIG.  5 C  is a 2D polar plot of the light intensity distributions represented by  FIGS.  5 A and  5 B . 
     The prism angles α on both substrates  400  may be adjusted to optimize the output distribution. For example, in an embodiment, the roof angle α of the ridges  412  on the substrates  400  may be 85 degrees.  FIG.  5 D  illustrates a three dimensional representation of the light distribution provided by the combination of the two light transmissive substrates  400  having a refractive index of 1.5 with the ridges  412  having a roof angles of 85 degrees, and arranged as illustrated in  FIG.  4 A .  FIG.  5 E  is a 2D polar plot of the light intensity distribution represented by  FIG.  5 D .  FIG.  5 F  is a 2D polar plot of the light intensity distribution provided by the combination of the two light transmissive substrates  400  having a refractive index of 1.6 with the ridges  412  having a roof angles of 85 degrees, and arranged as illustrated in  FIG.  4 A . A comparison of  FIGS.  5 E and  5 F  shows the influence the refractive index has on the batwing spreading performance of the substrates. 
     Textures may be added to the second surface  420  of either or both substrates to fine tune the distribution profile and to enhance the optical transmission efficiencies. 
       FIG.  6 A  illustrates an embodiment of a light transmissive substrate  600  that has a plurality of microstructures  612  on a first surface  610  thereof. In this embodiment, the microstructures  612  are in the form of an array of micro-pyramids, each having four faces, that are placed above an LED light source  630  that outputs light in a Lambertian distribution. As depicted in  FIG.  6 A , the light enters the substrate  600  via the first surface  610  having the array of micro-pyramids  612  and exits a second surface  620  on an opposite side of the substrate  600  as the first surface  610 . Each of the microstructures  612  has a roof angle α (see  FIG.  3   ) of 90 degrees, and is shown in further detail in  FIGS.  6 B  (perspective view) and  6 C (top view). Pyramid roof angles may be adjusted to optimize the output distribution, and textures may be added to the second surface  620  of the substrate  600  to fine tune the distribution profile and to enhance the optical transmission efficiencies. 
     A representation of the three dimensional transformation of the light distribution provided by the substrate  600  having a refractive index of 1.5 is shown in  FIGS.  7 A and  7 B . In this embodiment, light energy is not only steered away from the 0 degree emitting direction, but also pushed toward four directions approximately 45 degrees from the primary X and Y axes as shown. Along those directions, light typically travels the longest path length reaching the target area where stronger intensity is desired.  FIG.  7 C  is a 2D polar plot of the light intensity distributions represented by  FIGS.  7 A and  7 B . 
       FIG.  7 D  is a 2D polar plot of a representation of the light intensity distribution provided by the substrate  600  having a refractive index of 1.6 with the micro-pyramids having a roof angle α of 90 degrees. A comparison of  FIGS.  7 C and  7 D  shows the influence the refractive index has on the batwing spreading performance of the substrate  600 . 
     The roof angle α of the micro-pyramids  612  affects the light distribution provided by the substrate  600  having a refractive index of 1.5, as illustrated by  FIGS.  7 A- 7 C and  7 E- 7 L . As illustrated, roof angles α of 80 degrees (represented by  FIGS.  7 E and  7 F ) and 70 degrees (represented by  FIGS.  7 G and  7 H ) as compared to 90 degrees (represented by  FIGS.  7 A- 7 C ) provide different zero degree light intensities as well as shapes of the batwing distribution. Roof angles α of 60 degrees (represented by  FIGS.  71  and  7 J ) and 100 degrees (represented by  FIGS.  7 K and  7 L ) provide different zero degree light intensities, but do not provide batwing distributions. According to embodiments of the invention, the roof angle α of the micro-pyramids  612  is in the range of 70 degrees to 95 degrees for substrates having a refractive index of 1.5. 
       FIG.  8 A  illustrates a light transmissive substrate  800  that has an array of microstructures  812  on a first surface  810  thereof. In this embodiment, the microstructures  812  are in the form of an array of hybrid micro-pyramids that are placed above an LED light source  830  that outputs light in a Lambertian distribution. As illustrated in  FIG.  8 A , the light enters the substrate  800  via the first surface  810  having the array of hybrid micro-pyramids  812  and exits a second surface  820  on an opposite side of the substrate  800  as the first surface  810 .  FIGS.  8 B and  8 C  illustrate the hybrid micro-pyramid  812  in further detail. As illustrated, a top portion  814  of the hybrid micro-pyramid  812  may have a roof angle α (see  FIG.  3   ) of 85 degrees, and a bottom portion (frustum)  816  of the hybrid micro-pyramid  812  may have a roof angle α of 70 degrees. In embodiments in which the roof angle α of the bottom portion  816  is 70 degrees, sides  818  of the bottom portion  816  are each disposed at an angle β (see  FIG.  3   ) of 55 degrees. In an embodiment, the top portion  814  may have a roof angle α of between 85 degrees and 90 degrees. 
     A representation of the three dimensional transformation of the light distribution provided by the substrate  800  having a refractive index of 1.5 is shown in  FIGS.  9 A and  9 B . As illustrated, the hybrid micro-pyramid may provide enhanced performance when compared to a “simple” pyramid, such as the pyramid  612  described above. In this embodiment, light energy is not only steered farther away from the 0 degree emitting direction, but also pushed toward four directions approximately 45 degrees from the primary X and Y axes as shown. Along those directions, light typically travels the longest path length reaching the target area where stronger intensity is desired.  FIG.  9 C  is a 2D polar plot of the light intensity distributions represented by  FIGS.  9 A and  9 B . 
       FIG.  9 D  is a 2D polar plot of a representation of the light intensity distribution provided by the substrate  800  having a refractive index of 1.6 with the hybrid micro-pyramids  812  having the top portion  814  with a roof angle α (see  FIG.  3   ) of 85 degrees, and the bottom portion  816  with a roof angle α of 70 degrees. A comparison of  FIGS.  9 C and  9 D  shows the influence the refractive index has on the batwing spreading performance of the substrate  600 . 
     Pyramid roof angles α for the top portion  814  and the bottom portion  816  may be adjusted to optimize the output distribution. Textures may be added to the second surface  820  of the substrate  800  to fine tune the distribution profile and to enhance the optical transmission efficiencies. Although  FIGS.  9 A- 9 C  illustrate a hybrid pyramid that has two portions and sharp edges and transitions between the two portions, it is contemplated that the hybrid pyramid may have more than two portions and/or facets of the hybrid pyramid may be curved, thereby adding flexibilities for further transformation fine tuning and performance optimizations. For example,  FIG.  10    illustrates a three-section hybrid micro-pyramid  1000  that may be used for the microstructure  812  of FIG.  8 A, and  FIG.  11    illustrates a curved-facet hybrid micro-pyramid  1100  that may be used for the microstructure  812  of  FIG.  8 A . 
       FIG.  12 A  illustrates a light transmissive substrate  1200  that has an array of microstructures  1212  on a first surface  1210  thereof. In this embodiment, the microstructures  1212  are in the form of an array of “folded” micro-pyramids that are placed above an LED light source  1230  that outputs light in a Lambertian distribution. As illustrated in  FIG.  12 A , the light enters the substrate  1200  via the first surface  1210  having the array of folded micro-pyramids  1212  and exits a second surface  1220  on an opposite side of the substrate  1200  as the first surface  1210 .  FIG.  12 B  illustrates the folded micro-pyramid  1212  in further detail. As illustrated, the folded micro-pyramid has a frustum or base section  1214  and a recess  1216  having the shape of a micro-pyramid in the base section  1214 , thereby giving the pyramid a configuration that looks as though the tip of a simple pyramid was pressed downward and into the base section  1214  or “folded” into the base section  1214 . Both the base section  1214  and the recess  1216  may have roof angles α (see  FIG.  3   ) of 90 degrees. 
     Folded pyramids may enhance the manufacturability of the light transmissive substrate to overcome a restriction on the height of the microstructures in the Z-direction (represented by ‘h’ in  FIG.  3   ) for many microstructure fabrication processes. Folded pyramids also offer possibilities of achieving functionalities of structures of larger heights (h in  FIG.  3   ) than the fabrication process may allow. In an embodiment, the height h of the microstructures may be in the range of about 10 micrometers to about 50 micrometers. Pyramid roof angles may be adjusted to optimize the output distribution. Textures may be added to the second surface  1220  of the substrate  1200  to fine tune the distribution profile and to enhance the optical transmission efficiencies. 
     A representation of the three dimensional transformation of the light distribution provided by the substrate  1200  is shown in  FIGS.  13 A and  13 B . In this embodiment, light energy is not only steered away from the 0 degree emitting direction, but also pushed toward four directions approximately 45 degrees from the primary X and Y axes as shown. Along those directions, light typically travels the longest path length reaching the target area where stronger intensity is desired.  FIG.  13 C  is a 2D polar plot of the light intensity distributions represented by  FIGS.  13 A and  13 B . 
       FIG.  14 A  illustrates an embodiment of a light transmissive substrate  1400  with an array of microstructures  1412  in the form of corner cubes having square shaped faces that may be used in place of the light transmissive substrates described above. A representation of the three dimensional transformation of the light distribution provided by the substrate having a refractive index of 1.5 and the microstructures  1412  is shown in  FIG.  14 B .  FIG.  14 C  is a 2D polar plot of the light intensity distribution represented by  FIG.  14 B . 
       FIG.  15 A  illustrates an embodiment of a light transmissive substrate  1500  with an array of microstructures  1512  in the form of corner cubes having triangular shaped faces. A representation of the three dimensional transformation of the light distribution provided by the microstructures  1512  having a refractive index of 1.5 is shown in  FIG.  15 B .  FIG.  15 C  is a 2D polar plot of the light intensity distribution represented by  FIG.  15 B . 
       FIG.  15 D  is a 2D polar plot of a representation of the light intensity distribution provided by the microstructures  1512  having a refractive index of 1.6. A comparison of  FIGS.  15 C and  15 D  shows the influence the refractive index has on the batwing spreading performance of the substrate  1500 . 
     The light transmissive structures according to any of the embodiments described herein may be created using many techniques known in the art. For example, in an embodiment, the shape of the microstructures may be cast onto a substrate using a suitable master mold, and thermally-curing polymer or ultraviolet (UV) light curing polymer, or the shape may be impressed into a thermoplastic substrate through compression molding or other molding, or may be created at the same time as the substrate using extrusion-embossing or injection molding. The microstructures may be produced by replicating a master. For example, an optical diffuser may be made by replication of a master containing the desired shapes as described in U.S. Pat. No. 7,190,387 B2 to Rinehart et al., entitled “Systems And Methods for Fabricating Optical Microstructures Using a Cylindrical Platform and a Rastered Radiation Beam”; U.S. Pat. No. 7,867,695 B2 to Freese et al., entitled “Methods for Mastering Microstructures Through a Substrate Using Negative Photoresist”; and/or U.S. Pat. No. 7,192,692 B2 to Wood et al., entitled “Methods for Fabricating Microstructures by Imaging a Radiation Sensitive Layer Sandwiched Between Outer Layers”, assigned to the assignee of the present invention, the disclosures of all of which are incorporated herein by reference in their entirety as if set forth fully herein. The masters themselves may be fabricated using laser scanning techniques described in these patents, and may also be replicated to provide diffusers using replicating techniques described in these patents. 
     In an embodiment, laser holography, known in the art, may be used to create a holographic pattern that creates the desired microstructures in a photosensitive material. In an embodiment, projection or contact photolithography, such as used in semiconductor, display, circuit board, and other common technologies known in the art, may be used to expose the microstructures into a photosensitive material. In an embodiment, laser ablation, either using a mask or using a focused and modulated laser beam, may be used to create the microstructures including the indicia in a material. In an embodiment, micromachining (also known as diamond machining), known in the art, may be used to create the desired microstructures from a solid material. In an embodiment, additive manufacturing (also known as 3D printing), known in the art, may be used to create the desired microstructure in a solid material. 
     For any of the embodiments of the light transmissive substrate described herein, roof angles of the microstructures may be adjusted, and or textures may be added to the second surface of the substrate to fine tune the distribution profile and to enhance the optical transmission efficiencies. As described above, the refractive index of the microstructures also has an influence on the batwing spreading performance and may be adjusted to optimize performance. 
     The embodiments described herein represent a number of possible implementations and examples and are not intended to necessarily limit the present disclosure to any specific embodiments. Instead, various modifications can be made to these embodiments, and different combinations of various embodiments described herein may be used as part of the invention, even if not expressly described, as would be understood by one of ordinary skill in the art. 
     For example, although four-sided pyramids have been described, it is contemplated that other geometries, such as microstructures having 3, 5 or 6 sides or circular (cone) geometries may be used. Also, it is contemplated that the surfaces of the microstructures may have variations and either in a pattern or random variations or combinations thereof. In some embodiments, the microstructures may have asymmetrical instead of symmetrical shapes and arrays of microstructures may include microstructures having different shapes and/or sizes, either in a pattern or random variations or combinations thereof. Any such modifications are intended to be included within the spirit and scope of the present disclosure and protected by the following claims.