LED system and method

Abstract

Embodiments provide an LED comprising a quantum well region operable to generate light and a substrate having an interface with the quantum well region, wherein light generated by the quantum well region traverses the interface to enter the substrate and exit the LED through an exit face of the substrate. The exit face may be opposite from and a distance from the interface, with some portion or all of this LED being shaped to optimize the light extraction efficiency of the device. The exit face can have at least 70% of a minimum area necessary to conserve brightness for a desired half-angle of light. Sidewalls of the LED may be positioned and shaped so that rays incident on a sidewall reflect to the exit face with an angle of incidence at the exit face at less than or equal to a critical angle at the exit face.

Claims

1 . An LED comprising: a quantum well region operable to generate light; and a shaped substrate having an interface with the quantum well region, wherein light generated by the quantum well region traverses the interface, the shaped substrate comprising: an exit face opposite from and a distance from the interface, wherein the substrate is shaped so that a portion of the light entering the shaped substrate through the interface will exit the shaped substrate through the exit face and wherein the exit face has at least 70% of a minimum area necessary to conserve radiance for a desired half-angle of light projected from the shaped substrate; and a set of sidewalls, wherein each sidewall is positioned and shaped so that at least a majority of rays having a straight transmission path from the interface to that sidewall reflect to the exit face with an angle of incidence at the exit face at less than or equal to a critical angle at the exit face. 2 . The LED of claim 1 , wherein the quantum well region is shaped in conformance with the substrate. 3 . The LED of claim 1 , wherein the exit face has at least the minimum area necessary to conserve radiance. 4 . The LED of claim 1 , wherein the interface is rectangular in shape and wherein the effective solid angle whereby light leaves the exit surface is determined to account for the shape of the interface. 5 . The LED of claim 1 wherein the distance is within 50% of a minimum distance so that all rays with a straight transmission path from the interface to the exit face have an angle of incidence that is less than or equal to the critical angle at the exit face. 6 . The LED of claim 1 , wherein each sidewall is positioned and shaped so that at least 80% of rays with a straight transmission path from the interface to the sidewall are reflected to the exit face with an angle of incidence on the exit face at less than or equal to the critical angle at the exit face. 7 . The LED of claim 1 , wherein the shapes of the sidewall are selected to create a desired light output profile. 8 . The LED of claim 1 , wherein the exit face is parallel to and rotationally aligned with the interface and wherein the exit face has the same shape and aspect ratio as a shape of the interface. 9 . The LED of claim 1 , further comprising a layer of photo-luminescent material capable of emitting desired wavelengths to create white light. 10 . An LED comprising: a quantum well region operable to generate light; a shaped substrate having an interface with the quantum well region, the shaped substrate comprising: an exit face opposite from and a distance from the interface, wherein the substrate is shaped so that a portion of the light entering the shaped substrate through the interface will exit the shaped substrate through the exit face and wherein the exit face has at least 70% of a minimum area necessary to conserve radiance for a desired solid angle of light projected from the shaped substrate; a set of sidewalls, wherein each sidewall is positioned and shaped so that at least a portion of rays having a straight transmission path from the interface to that sidewall reflect to the exit face with an angle of incidence at the exit face at less than or equal to a critical angle at the exit face; wherein the area of the exit face, distance and sidewall shapes are selected to project light with a half angle of between 10 to 60 degrees. 11 . The LED of claim 10 , wherein the interface has a rectangular shape and wherein the effective solid angle whereby light leaves the exit face is determined to account for the area of the shape of the interface. 12 . The LED of claim 10 , wherein the distance is within 50% of a minimum distance so that all rays with a straight transmission path from the interface to the exit face have an angle of incidence that is less than or equal to the critical angle at the exit face. 13 . The LED of claim 10 , wherein each sidewall is positioned and shaped so that at least 80% of rays with a straight transmission path from the interface to the sidewall are reflected to the exit surface with an angle of incidence on the exit surface at less than or equal to the critical angle at the exit surface. 14 . The LED of claim 10 , wherein the exit face has the same shape and aspect ratio as a shape of the interface and wherein the exit face is parallel to and rotationally aligned with the interface. 15 . The LED of claim 10 , further comprising a layer of photo-luminescent material capable of emitting desired wavelengths to create white light. 16 . The LED of claim 10 , wherein the quantum well region is shaped in conformance with the substrate. 17 . The LED of claim 10 , wherein the shapes of the sidewall are selected to create a desired light output profile. 18 . The LED of claim 10 , wherein the exit face has at least the minimum area necessary to conserve radiance. 19 . An LED comprising: a quantum well region operable to generate light; a shaped substrate having an interface with the quantum well region, the shaped substrate comprising: an exit face opposite from and a distance from the interface, wherein: the substrate is shaped so that a portion of the light entering the shaped substrate through the interface will exit the shaped substrate through the exit face; the exit face has an area that is within 30% of a minimum area defined by Φ 2 ⁢ n 1 2 ⁢ A 1 ⁢ Ω 1 Φ 1 ⁢ n 2 2 ⁢ Ω 2  D 2 n 21  wherein Φ 1 is the light flux traversing the interface; Φ 2 is the light flux exiting the exit face and is equal to Φ 1 ; Ω 1 is the effective solid angle whereby light traverses the interface and Ω 2 is the effective solid angle whereby light leaves the exit face; A 1 is the area of the interface; n 1 is the refractive index of the shaped substrate and n 2 is the refractive index of the medium external to the shaped substrate; the distance is at least a minimum distance so that all rays with a straight transmission path from the interface to the exit face have an angle of incidence that is less than or equal to a critical angle at the exit face; and a set of sidewalls, wherein each sidewall is positioned and shaped so that at least a portion of rays having a straight transmission path from the interface to that sidewall reflect to the exit face with an angle of incidence at the exit face at less than or equal to the critical angle at the exit face. 20 . The LED of claim 19 , wherein the area of the exit face, distance and sidewall shapes are selected to project light with a half angle of between 10 to 60 degrees with at least 70% efficiency and a desired light output profile. 21 . The LED of claim 19 , wherein the quantum well region is shaped in conformance with the substrate. 22 . The LED of claim 19 , wherein the interface has a rectangular shape and wherein the effective solid angle whereby light leaves the exit surface is determined to account for the square shape of the interface. 23 . The LED of claim 19 , further comprising a layer of photo-luminescent material capable of emitting desired wavelengths to create white light. 24 . The LED of claim 19 , wherein the shapes of the sidewall are selected to create a desired light output profile. 25 . An LED comprising: a quantum well region operable to generate light; and a shaped substrate having an interface with the quantum well region, wherein light generated by the quantum well region traverses the interface, the shaped substrate comprising: at least two exit faces, wherein the substrate is shaped so that a portion of the light entering the shaped substrate through the interface will exit the shaped substrate through the at least two exit faces and wherein the at least two exit faces have a combined area that is at least 70% of a minimum area necessary to conserve radiance; and a set of sidewalls, wherein each sidewall is positioned and shaped so that at least a majority of rays having a straight transmission path from the interface to that sidewall reflect to an exit face of the two or more exit faces with an angle of incidence at that exit face at less than or equal to a critical angle at that exit face.
RELATED APPLICATIONS [0001] This application claims the benefit of priority under 35 U.S.C. 119(e) to U.S. Provisional Patent Application No. 60/827,818, entitled “SHAPED LIGHT EMITTING DIODES”, to Dung T. Duong et al., filed Oct. 2, 2006, and to U.S. Provisional Patent Application No. 60/881,785, entitled “SYSTEM AND METHOD FOR A SHAPED SUBSTRATE LED”, to Dung T. Duong et al., filed Jan. 22, 2007, both of which are hereby fully incorporated by reference herein. TECHNICAL FIELD OF THE INVENTION [0002] This disclosure regards light emitting diodes (“LEDs”). More particularly, embodiments of systems and methods described herein relate to increasing or controlling light output of an LED. Even more particularly, embodiments relate to increasing or controlling light output of an LED by shaping portions of the LED substrate. BACKGROUND [0003] Light emitting diodes (“LEDs”) are ubiquitous in electronics. They are used in digital displays, lighting systems, computers and televisions, cellular telephones and a variety of other devices. Developments in LED technology have led to methods and systems for the generation of white light using one or more LEDs. Developments in LED technology have led to LEDs that generate more photons and thus more light than previously. The culmination of these two technological developments is that LEDs are being used to supplement or replace many conventional lighting sources, e.g. incandescent, fluorescent or halogen bulbs, much as the transistor replaced the vacuum tube in computers. [0004] LEDs are produced in a number of colors including red, green and blue. One method of generating white light involves the use of red, green and blue LEDs in combination with one another. A lighting source that is made of combinations of red, green and blue (RGB) LEDs will produce what is perceived as white light by the human eye. This occurs because the human eye has three types of color receptors, with each type sensitive to either blue, green or red colors. [0005] A second method of producing white light from LED sources is to create light from a single-color (e.g. blue), short wavelength LED, and impinge a portion of that light onto phosphor or similar photon conversion material. The phosphor absorbs the higher energy, short wavelength light waves, and re-emits lower energy, longer wavelength light. If a phosphor is chosen that emits light in the yellow region (between green and red), for example, the human eye perceives such light as white light. This occurs because the yellow light stimulates both the red and green receptors in the eye. Other materials, such as nano-particles or other similar photo-luminescent materials, may be used to generate white light in much the same way. [0006] White light may also be generated utilizing an ultraviolet (UV) LED and three separate RGB phosphors. White light may also be generated from a blue LED and a yellow LED and may also be generated utilizing blue, green, yellow and red LEDs in combination. [0007] Current industry practice for construction of LEDs is to use a substrate (typically either single-crystal Sapphire or Silicon Carbide), onto which is deposited layers of materials such as GaN or InGaN. One or more layers (e.g. GaN or InGaN) may allow photon generation and current conduction. Typically, a first layer of Gallium Nitride (GaN) is applied to the surface of the substrate to form a transition region from the crystal structure of the substrate to the crystal structure of doped layers allowing for photon generation or current conduction. This is typically followed by an N-doped layer of GaN. The next layer can be an InGaN, AlGaN, AlInGaN or other compound semiconductor material layer that generates photons and that is doped with the needed materials to produce the desired wavelength of light. The next layer is typically a P doped layer of GaN. This structure is further modified by etching and deposition to create metallic sites for electrical connections to the device. [0008] During the operation of an LED, as in a traditional diode, extra electrons move from an N-type semiconductor to electron holes in a P-type semiconductor. In an LED, photons are released in the compound semiconductor layer to produce light during this process. [0009] In a typical manufacturing process, the substrate is fabricated in wafer form and the layers are applied to a surface of the wafer. Once the layers are doped or etched and all the features have been defined using the various processes mentioned, the individual LEDs are separated from the wafer. The LEDs are typically square or rectangular with straight sides. This can cause significant efficiency losses and can cause the emitted light to have a poor emission pattern. A separate optical device, such as a plastic dome, is often placed over the LED to achieve a more desirable output. [0010] In nearly all LED applications, it is desirable to maximize light output for a given power input, a quantity often expressed in lumens per watt (lm/W) for white light and longer wavelength light, or milliwatts per watt (mW/W) for shorter wavelength light such as blue. Existing LED technologies may attempt to increase this ratio, typically referred to as “overall efficiency” or “wall-plug efficiency.” However, existing LED technologies still suffer poor overall efficiency and low extraction efficiency. SUMMARY [0011] This disclosure regards an LED having a substrate shaped to achieve greater efficiency than previous LEDs. The sidewalls of embodiments of the LED may be shaped so as to maximize the light output of the LED using total internal reflection and can be shaped to achieve a desired intensity distribution. Additionally, the exit face of the LED may be selected to conserve radiance. [0012] One embodiment of an LED includes a quantum well region operable to generate light and a shaped substrate having an interface with the quantum well region. Light generated by the quantum well region traverses the interface between the quantum well region and the shaped substrate to enter the shaped substrate. The shaped substrate has an exit face opposite from and a distance from the interface. A portion of the light entering the shaped substrate through the interface exits the shaped substrate through the exit face. According to one embodiment, the exit face is at least 70% of a minimum area necessary to conserve radiance for a desired half-angle of light exiting the shaped substrate. Furthermore, the LED can have a set of sidewalls, wherein each sidewall is positioned and shaped so that at least a majority of rays having a straight transmission path from the interface to that sidewall reflect to the exit face with an angle of incidence at the exit face at less than or equal to a critical angle at the exit face. Another embodiment of the LED comprises an LED comprising a quantum well region operable to generate light and a shaped substrate having an interface with the quantum well region. The shaped substrate can comprise an exit face opposite from and a distance from the interface. The substrate can be shaped so that a portion of the light entering the shaped substrate through the interface will exit the shaped substrate through the exit face. The exit face can have at least 70% of a minimum area necessary to conserve radiance for a desired solid angle of light projected from the shaped substrate. The shaped substrate can further comprise a set of sidewalls, wherein each sidewall is positioned and shaped so that at least a portion of rays having a straight transmission path from the interface to that sidewall reflect to the exit face with an angle of incidence at the exit face at less than or equal to a critical angle at the exit face. The area of the exit face, distance and sidewall shapes can be selected to project light with a half angle of between 10 to 60 degrees. [0013] Yet another embodiment can include an LED comprising a quantum well region operable to generate light and a shaped substrate having an interface with the quantum well region. The shaped substrate can comprise an exit face opposite from and a distance from the interface. The substrate can be shaped so that a portion of the light entering the shaped substrate through the interface will exit the shaped substrate through the exit face. [0014] The exit face can have an area that is within 30% of a minimum area defined by Φ 2 ⁢ n 1 2 ⁢ A 1 ⁢ Ω 1 Φ 1 ⁢ n 2 2 ⁢ Ω 2 wherein Φ 1 is the light flux traversing the interface; Φ 2 is the light flux exiting the exit face and is equal to Φ 1 ; Ω 1 is the effective solid angle whereby light traverses the interface and Ω 2 is the effective solid angle whereby light leaves the exit face; A 1 is the area of the interface; n 1 is the refractive index of the shaped substrate and n 2 is the refractive index of the medium external to the shaped substrate. Furthermore, the distance is at least a minimum distance so that all rays with a straight transmission path from the interface to the exit face have an angle of incidence that is less than or equal to a critical angle at the exit face. The substrate can also comprise a set of sidewalls, wherein each sidewall is positioned and shaped so that at least a portion of rays having a straight transmission path from the interface to that sidewall reflect to the exit face with an angle of incidence at the exit face at less than or equal to the critical angle at the exit face. [0015] Yet another embodiment can include an LED comprising a quantum well region operable to generate light and a shaped substrate having an interface with the quantum well region, wherein light generated by the quantum well region traverses the interface. The substrate can comprise at least two exit faces, wherein the substrate can shaped so that a portion of the light entering the shaped substrate through the interface will exit the shaped substrate through the at least two exit faces. The at least two exit faces have a combined area that is at least 70% of a minimum area necessary to conserve radiance. The shaped substrate can also comprise a set of sidewalls, wherein each sidewall is positioned and shaped so that at least a majority of rays having a straight transmission path from the interface to that sidewall reflect to an exit face of the two or more exit faces with an angle of incidence at that exit face at less than or equal to a critical angle at that exit face. [0016] Embodiments of systems and methods provide an LED that provides technical advantages over the prior art by projecting light with a desired half-angle and light output profile, while conserving brightness. Embodiments can provide, for example, light in 10 to 60 degrees half angle (or other half angles) with high efficiency. [0017] Embodiments provide yet another advantage by providing for light output with uniform or near uniform intensity distributions. [0018] Yet another advantage provided by embodiments is that light can be projected with a desired shape and light output profile. BRIEF DESCRIPTION OF THE FIGURES [0019] A more complete understanding of the embodiments and the advantages thereof may be acquired by referring to the following description, taken in conjunction with the accompanying drawings in which like reference numbers indicate like features and wherein: [0020] FIGS. 1A-1B are diagrammatic representations of embodiments of LEDs; [0021] FIG. 2 is a diagrammatic representation of a set of rays traveling from a point to surfaces at different distances from the point; [0022] FIG. 3 provides a diagrammatic representation of a top view of an embodiment of an LED; [0023] FIG. 4A is a diagrammatic representation of a cross-section of a model of an LED for determining sidewall shapes; [0024] FIG. 4B is a diagrammatic representation of an embodiment of a portion of a sidewall of an LED; [0025] FIG. 4C is a diagrammatic representation illustrating that the facets for a sidewall can be defined using a computer program; [0026] FIG. 4D is a diagrammatic representation of one embodiment of an LED with sidewalls shaped to cause TIR so that rays are reflected from the sidewalls to the exit surface; [0027] FIG. 5 is a diagrammatic representation of one embodiment for estimating effective solid angle; [0028] FIGS. 6A-6E are diagrammatic representations describing another embodiment for estimating effective solid angle; [0029] FIG. 7 is a diagrammatic representation of one embodiment of a LED; [0030] FIG. 8A-8B are diagrammatic representations of embodiments of an array of LEDs; [0031] FIGS. 9A-9C are diagrammatic representations of embodiments of retaining devices; [0032] FIG. 10 is a functional representation of an embodiment of etching; [0033] FIG. 11 is a functional representation of an embodiment of laser ablation; [0034] FIGS. 12A-12B are functional representation of an embodiment of sawing; [0035] FIGS. 13A-13B are diagrammatic representations of an embodiment of a wafer having had substrates shaped through sawing; [0036] FIG. 14 is a diagrammatic representation of one embodiment of a LED; and [0037] FIGS. 15A and 15B are diagrammatic representations of another embodiment of a wafer. DETAILED DESCRIPTION [0038] Embodiments are illustrated in the FIGURES, like numerals being used to refer to like and corresponding parts of the various drawings. [0039] Embodiments of shaped substrate LEDs may be shaped so as to increase or shape the light emission from the LED. According to one embodiment, the substrate is shaped so that all or a supermajority of the light generated by the quantum well region of the LED is transmitted out the exit face of the substrate of the LED. To this end, the exit face can be sized to take into account principles of conservation of radiance. In one embodiment, the exit face may be the minimum size that allows all or a supermajority of the light entering the substrate through the interface between the quantum well region and the substrate to exit the exit face, thereby combining the desire to conserve radiance with the desire to reduce size, particularly the size of the exit face. Additionally, the sidewalls of the substrate may be shaped so that reflection or total internal reflection (“TIR”) causes light beams incident on substrate sidewalls to reflect towards the exit face and be incident on the exit face with an angle less than or equal to the critical angle. Consequently, light loss due to TIR at the exit face is reduced or eliminated. In a further embodiment, to insure that light striking a sidewall is reflected within the substrate and does not pass through the sidewall, a sidewall or sidewalls of a substrate may also be coated with a reflective material that reflects light to prevent the exitance of light through the sidewall. [0040] While theoretically 100% of the light generated by the quantum wells of the LED exits the exit face, various embodiments may cause lesser amounts of light to exit the exit face while still providing significant improvements over prior LED light emissions. For example, light emitted from the exit surface of the LED may be emitted from the exit surface with a cone half angle of 10-60 degrees with approximately 79% efficiency (there is approximately a 21% efficiency loss due to fresnel losses for a silicon carbide substrate material of 2.73 index of refraction) with a desired intensity profile, exitance profile or other light output profile. [0041] Fresnel losses (e.g. losses at the interface between two mediums such as at the exit face of an LED and air or other medium) occurs when light traverses from a medium of higher index to a medium of lower index. Normal incident fresnel losses are described by the equation: (N 1 −N 2 ) 2 )/((N 1 +N 2 ) 2 ), wherein N 1 and N 2 are the indices of refraction of the two mediums. As an example, for an LED having a silicon carbide substrate, N 1 =2.73 (approximate IOR of silicon carbide), N 2 =1 (approximate IOR of air), yielding Fresnel losses of approximately 21.5%. If the LED utilizes GaN in the quantum well region, Fresnel losses at the interface between the quantum well region (N 1 =2.49) and the silicon carbide substrate (N 2 =2.73) will be 0%. Fresnel losses at the exit face to air interface may be reduced or overcome with anti-reflective coatings. [0042] As described above, the exit face of an LED substrate can be shaped to conserve radiance. The passage of light along an optic path, either within a single medium or from one medium to another, is governed by the law of Conservation of Radiance, also referred to as the Brightness Theorem, which is expressed by the Etendue equation: Etendue Equation: Φ 1 N 1 2 ⁢ A 1 ⁢ Ω 1 = Φ 2 N 2 2 ⁢ A 2 ⁢ Ω 2 [ EQN .   ⁢ 1 ] Φ 1 =light flux (lumens) of region 1 N 1 =IOR of medium of region 1 A 1 =area of entrance to region 1 Ω 1 =solid angle (steradians) that fully contains the light of region 1 Φ 2 =light flux (lumens) of region 2 N 2 =IOR of medium of region 2 A 2 =area of entrance to region 2 Ω 2 =solid angle (steradians) that fully contains the light of region 2 [0043] The area of the exit face of a shaped substrate can be selected to conserve radiance of light entering the substrate from the quantum wells for a desired half angle. Consequently, light can be emitted in a desired half angle with high efficiency. This is unlike traditional LEDs that both emit light with a half angle that is undesirable for many applications, therefore requiring additional optical devices to shape the light; and, emit a significant percentage of light through the sidewalls because the exit face is not large enough to conserve radiance; while also suffering absorption losses due to the light never escaping the substrate. [0044] Furthermore, the passage of light from a medium of one index of refraction to a medium of a different IOR is governed by Snell's Law. Snell's law defines the relationship between the angle of approach of a light ray as measured from the normal to the interface surface, and the angle of departure of that ray from the interface, as a function of the indices of refraction of both media. Snell's Law: N 1 sin(Θ 1 )= N 2 sin(θ 2 )  [EQN. 2] Θ 1 =angle of incidence of ray approaching interface surface N 1 =IOR of medium 1 Θ 2 =angle of refraction of ray departing interface surface N 2 =IOR of medium 2 [0045] In the case of the passage of light from a medium of higher IOR to a medium of lower IOR, the maximum angle at which a light ray may strike the interface surface between the media and still pass through the interface is called the critical angle. Fundamentally, light originating from the medium of higher IOR must approach the media interface at angles not exceeding the critical angle if the light is to pass through the interface and into the medium of lower IOR. For example, in an LED comprised of a substrate and a quantum well region, the substrate medium and the quantum well medium may form an interface that light generated by the quantum well regions traverses. Rays that approach at angles greater than the critical angle will be reflected back within the medium of higher IOR at the interface between the media and will not pass into the medium of lower IOR. This is referred to as total internal reflection (“TIR”). [0046] In a typical LED, the quantum well region has an IOR of approximately 2.49. When these layers are constructed on a sapphire substrate with an IOR of 1.77, the light that can be transmitted into the sapphire is inherently limited by the application of Snell's law and the Brightness Theorem. For LEDs with a substrate of silicon carbide, which may have an IOR of approximately 2.73, the quantum well region has a lower IOR (e.g. approximately 2.49) than the silicon carbide, and therefore Snell's law does not prohibit any of the generated light from passing into the silicon carbide. [0047] In traditional LEDs, a significant portion of light encountering a substrate to air interface will be trapped in the substrate due to TIR. In some cases, a separate optical device (e.g. a solid plastic dome or lens) is used to increase the IOR of the medium into which light passes from the substrate, reducing TIR in the substrate. These separate optical devices may still suffer from losses due to TIR, and the extraction efficiency of domes remains relatively low. Moreover, the use of a dome requires additional steps in manufacturing after the LED is formed. Embodiments of shaped substrate LEDs, on the other hand, can be shaped to minimize or eliminate light loss due to TIR at the exit face of the substrate. According to one embodiment, the exit face of the substrate can be spaced from the interface with the quantum well region by a distance so that none of the rays with a direct transmission path to the exit face experience TIR at the exit face. Additionally, the sidewalls can be shaped to reflect rays encountering the sidewalls to the exit face with an angle of incidence at the exit face that is less than the critical angle, thus allowing all internally reflected rays to exit the exit face of the LED substrate as well. [0048] FIG. 1A is a diagrammatic representation of one embodiment of a LED 20 including a substrate 10 and quantum well region 15 (that may comprise one or more layers or regions of doping). Quantum well region 15 includes a light emitting region 25 , typically a compound semiconductor such as InGaN or AlInGaP or AlGaN. Photons from quantum well region 15 may enter substrate 10 through interface 50 . LED 20 can be a wire bond, flip chip or other LED known or developed in the art. Both substrate 10 and quantum well region 15 form sidewall 60 , sidewall 65 or other sidewalls. In other words, quantum well region 15 is shaped in conformance with substrate 10 . LED 20 further includes exit face 55 that may be substantially the same shape as, substantially parallel to and substantially rotationally aligned with interface 50 within the tolerance of the manufacturing process. [0049] The area of exit face 55 can be chosen to conserve brightness for a desired half angle according to the conservation of radiance (sometimes called the conservation of brightness) equation: Φ 2 ⁢ n 1 2 ⁢ A 1 ⁢ Ω 1 Φ 1 ⁢ n 2 2 ⁢ Ω 2 = A 2 [ EQN .   ⁢ 1 ] [0050] Φ 1 =light flux traversing interface 50 ; [0051] Φ 2 =light flux exiting exit face 55 , Φ 1 =Φ 2 for conservation of brightness; [0052] Ω 1 =effective solid angle whereby light traverses interface 50 ; [0053] Ω 2 =effective solid angle whereby light leaves exit face 55 ; [0054] A 1 =area of interface 50 ; [0055] A 2 =area of exit face 55 ; [0056] n 1 =refractive index of material of substrate 10 ; [0057] n 2 =refractive index of substance external to substrate 10 (e.g. air or other medium). [0058] A 2 represents the minimum surface area of exit face 55 such that light is conserved per the above equation. Assume, for example: quantum well region 15 forms a 1 mm square so that interface 50 has an area approximately 1 mm square, n 1 =1.77, n 2 =1, Ω 1 =3, Ω 2 =1, then A 2 must be at least 9.3987 mm 2 to conserve radiance (i.e. the minimum size of exit face 55 so that all of the light traversing interface 50 can be emitted from exit face 55 for a desired half angle). While in this example the effective solid angles are given, methods for determining Ω 1 and Ω 2 for a desired half angle are discussed below in conjunction with FIGS. 6A-6E . It should be noted that the square profile is a rectangular profile with sides of equal length. [0059] A 2 according to EQN. 1 is the minimum possible size for a given output cone angle or Emission Half Angle to conserve radiance. Consequently, to conserve radiance, A 2 should be at least the size determined from EQN. 1, but may be larger. For example, A 2 may be made slightly larger to compensate for tolerances in the manufacturing process, errors in the size or shape of quantum well region 15 or other factors. [0060] In the case where A 2 is made larger than the value determined by equation 1, flux will be conserved, but exitance (defined as flux per unit area) may be reduced from the maximum attainable value. [0061] To reduce the area of the exit face, however, it may be preferable that A 2 be as small as possible. For example, A 2 may be within 5% of the minimum area needed to conserve radiance. If some light power (luminous flux) may be sacrificed, A 2 can be smaller than the size dictated by the conservation of radiance. As one example, for one embodiment having a 1 mm by 1 mm square interface 50 , exit face 55 can be 2.5 mm 2 to 5 mm 2 (e.g., 4.62 mm 2 ). As another example, for an embodiment having 0.3 mm×0.3 mm interface 50 , exit face 55 can be 0.2 mm 2 to 0.5 mm 2 (e.g., 0.42 mm 2 ). It should be noted, however, the size ranges provided in the previous examples are provided by way of example only, and various embodiments can have a variety of sizes smaller than or greater than the example ranges. Preferably, however, A 2 is at least 70% of the value as determined by EQN. 1. Furthermore, the shape of exit face 55 may be different than that of interface 50 . [0062] The distance between interface 50 and exit face 55 of substrate 10 —referred to as the “height” herein, though the distance may extend in other directions than the vertical—may be selected to reduce or minimize TIR of light rays traveling directly from interface 50 to exit surface 55 . TIR occurs when light is incident on the surface with an angle of incidence greater that critical angle, which is defined by: n 1 sin(θ c )= n 2 sin(90)  [EQN. 2] where n 1 =IOR of substrate 10 ; n 2 =IOR of the medium external to the exit face of substrate 10 (e.g., air or other substance); and θ c =the critical angle. [0063] For example, if n 1 =1.77 and n 2 =1, then θ c =34.4 degrees. Accordingly, the height of substrate 10 can be selected to limit the critical angle of rays incident on exit surface 55 to a range between normal to exit surface 55 and less than or equal to the critical angle. [0064] Referring to FIGS. 2 and 3 , FIG. 2 is a diagrammatic representation of a set of rays traveling from point 57 incident on a surface 55 (represented as surfaces 55 a , 55 b and 55 c at different distances from point 57 ). In the example of surface 55 a , some rays (e.g., ray 56 ) are incident on surface 55 a at greater than the critical angle, causing loss of light due to TIR. In the example of surface 55 b , conversely, some rays that would be incident on surface 55 b at the critical angle or somewhat less than the critical angle (e.g., ray 57 ) will instead be incident on the sidewalls. Preventing loss of these rays, if desired, can cause the complexity of the sidewall design to increase. Moreover, the additional height requires more room to accommodate the LED (i.e., because the LED is taller). Finally, in the case of surface 55 c , rays at or less than the critical angle are incident on surface 55 c while rays that would be greater than the critical angle on exit surface 55 c instead are incident on the sidewalls. TIR or reflection can be used to direct the rays incident on the sidewalls to exit surface 55 c as discussed below. [0065] The limiting ray for selecting height, according to one embodiment, is the ray that travels the longest straight line distance from interface 50 to exit face 55 and is incident on exit face 55 at the critical angle. There may be more than one ray that can be selected as the limiting ray. In a square or rectangular configuration this is the ray that enters substrate 10 at a corner of interface 50 and travels in a straight line to the diagonally opposite corner of exit face 55 such that the ray would be incident on exit face 55 at the critical angle. [0066] FIG. 3 provides a diagrammatic representation of a top view of substrate 10 and of limiting ray 59 for a square configuration. While in the preferred embodiment the height of substrate 10 is selected to limit the critical angle of rays incident on exit face 55 to a range of between normal to exit face 55 and to less than or equal to the critical angle, other heights can be selected, though the use of other heights may decrease the efficiency of LED 20 . In one embodiment, the distance between the interface between the quantum well region and the substrate and the exit face of the substrate may be within 5% of the minimum height that causes all rays with a straight transmission path from the interface to the exit face to have an angle of incidence on the exit face at less than or equal to the critical angle. [0067] Returning to FIG. 1A , with selected boundary conditions of the size and shape of interface 50 , size and shape of exit face 55 , distance between interface 50 and exit face 55 , the sidewalls (e.g., sidewall 60 , sidewall 65 and other sidewalls) of substrate 10 can be shaped to direct light incident on the inner side of the sidewalls to exit face 55 to produce a desired light output profile (e.g., an intensity profile, exitance profile or other light output profile). While for most applications the desired intensity profile is uniform or close to uniform, other distribution profiles can be achieved by varying the height and shapes of the sidewalls. [0068] Broadly speaking, the sidewall shapes are determined so that any ray incident on a sidewall is reflected to exit face 55 and is incident on exit face 55 at the critical angle or less (i.e., so that there is no loss due to internal reflection at exit face 55 ). This is shown in FIG. 1A by ray 70 that has angle of incidence 75 relative to sidewall 65 that is greater than θ c so that ray 70 is reflected to exit face 55 and has an angle of incidence 80 that is less than or equal to θ c . While, in one embodiment, the sidewalls are shaped so that all rays that encounter the inner surface of the sidewalls experience total internal reflection to exit face 55 and are incident on exit face 55 at the critical angle or less, other sidewall shapes that allow some loss can be used. [0069] Turning to FIG. 1B , FIG. 1B is a diagrammatic representation of another embodiment of a LED 20 . LED 20 comprises a substrate 10 and a quantum well region 15 . Quantum well region 15 includes a light emitting region 25 , typically a compound semiconductor such as InGaN or AlInGaP or AlGaN. Photons from quantum well region 15 may enter substrate 10 through interface 50 . In FIG. 1B there may be more losses due to TIR in the quantum well region because the quantum well region is not shaped to appropriately direct light to interface 50 and/or exit face 55 . While in the embodiments of FIGS. 1A and 1B , some sidewall shapes may not direct all the light generated by LED 20 out exit face 55 , the portion of light not exiting exit face 55 will be emitted from sidewalls 65 and may be emitted near exit face 55 , thus allowing for the light generated by LED 20 to be captured usefully. [0070] FIG. 4A is a diagrammatic representation of a cross-section of a model of a LED or a substrate of a LED for determining sidewall shapes. Sidewall shapes can be determined using computer-aided design. A model of the sidewall can be created in a computer-aided design package and simulations run to determine an appropriate sidewall shape. [0071] According to one embodiment, each sidewall can be divided into n facets with each facet being a planar section. For example, sidewall 100 is made of fifteen planar facets 102 a - 102 o rather than a continuous curve. The variables of each facet can be iteratively adjusted and the resulting distribution profiles analyzed until a satisfactory profile is achieved as described below. While the example of fifteen facets is used, each sidewall can be divided into any number of facets, including twenty or more facets. [0072] Each facet can be analyzed with respect to reflecting a certain subset of rays within a substrate. This area of interest can be defined as an “angular subtense.” The angular subtense for a facet may be defined in terms of the angles of rays emanating from a predefined point. Preferably, the point selected is one that will give rays with the highest angles of incidence on the facet because such rays are the least likely to experience TIR at the facet. In a substrate with a square shaped interface area, for example, this will be a point on the opposite edge of the interface. [0073] According to one embodiment, for a selected A 1 , A 2 , and height, the maximum of angle 95 of any ray that will be incident on a given sidewall (e.g., sidewall 100 ) without being previously reflected by another sidewall can be determined. In this example, ray 110 emanating from point 115 establishes the maximum angle 95 for sidewall 100 . If the maximum of angle 95 is 48 degrees and there are 15 facets for sidewall 100 , each facet (assuming an even distribution of angular subtenses) will correspond to a 3.2 degree band of angle 95 (e.g., a first facet will be the area on which rays emanating from point 115 with an angle 95 of 0-3.2 degrees are incident, the second facet will be the area on which rays emanating at point 115 with an angle 95 of 3.2-6.4 degrees are incident, and so on). [0074] For each facet, the exit angle, facet size, tilt angle, or other parameter of the facet can be set so that all rays incident on the facet experience TIR and are reflected to exit surface 55 such that they are incident on exit surface 55 with an angle of incidence of less than or equal to the critical angle. Preferably, the sidewalls are also shaped so that a ray viewed in a cross-sectional view only hits a side wall once. However, there may be additional reflection from a sidewall out of plane of the section. For a full 3D analysis, a ray that strikes a first sidewall near a corner, may then bounce over to a second side wall, adjacent to the first, and from there to the exit face. A curve fit or other numerical analysis may be performed to create a curved sidewall shape that best fits the desired facets. In FIG. 4A , for example, sidewall 105 is curved rather than a set of planar facets. [0075] To optimize the variables for each facet, a simulated detector plane 120 can be established. Detector plane 120 can include x number of detectors to independently record incident power. A simulation of light passing through the substrate may be performed and the intensity and irradiance distributions as received by detector plane 120 analyzed. If the intensity and irradiance distributions are not satisfactory for a particular application, the angles and angular subtenses of the facets can be adjusted, a new curved surface generated and the simulation re-performed until a satisfactory intensity profile, exitance profile or other light output profile is reached. Additional detector planes can be analyzed to ensure that both near field and far field patterns are satisfactory. Alternatively, the simulation(s) can be performed using the facets rather than curved surfaces and the surface curves determined after a desired light output profile is reached. In yet another embodiment, the sidewalls can remain faceted and no curve be generated. [0076] According to another embodiment, the sidewall shape can be selected based on multiple parabolas with each planer facet representing a linear approximation of a portion of a parabola. For example, FIG. 4B is a diagrammatic representation of a portion 400 of a LED. In FIG. 4B , a hypothetical ray 410 is depicted that emanates from the focus 412 of a parabola 415 and intersects sidewall 420 such that it is reflected off sidewall 420 due to TIR and traverses the substrate to intersect exit face 430 at an exit angle 440 that is less than the critical angle and exits the substrate into air or other medium. As can be seen from FIG. 4B , at the transition from the substrate to air, ray 410 bends as described by Snell's law. Since the tangent point of the sidewall is determined from a parabola and because the ray incident and reflected off the sidewall is in the same medium, the ray will be parallel to the optical axis of the parabola. Thus, light is projected with a half-angle 450 . Angular subtenses defining the shape of sidewall 420 may be adjusted such that hypothetical ray 410 reflects off sidewall 420 such that ray 410 traverses exit face 430 with a desired exit angle 440 or projects light with a desired half angle 450 . [0077] In one embodiment, when fabricating a sidewall or calculating the angular substense of a sidewall, finer substenses may be used towards the base of the sidewall (i.e. nearer the quantum well region) because the effects of the substense are greater or more acute upon reflection near the base, and thus finer subtenses allow for a sidewall with better TIR properties, whereas further from the base, where the effects of the subtenses are less, the subtenses may be coarser. Thus, facets of a sidewall may be numerically greater towards the base of a shaped substrate LED. In one embodiment, a sidewall may have 20 or more facets, with finer facets at the base of the sidewall, wherein the facets approximate one or more subtenses. [0078] A facet can be a linear approximation of a portion 417 of parabola 415 . The parameters of parabola 415 can be adjusted until portion 417 achieves the desired goal of all rays incident on portion 417 reflecting to exit face 430 such that the rays have an exit angle 440 of less than the critical angle. Each facet can be formed from a parabola having different parameters. Thus, a facet for one angular subtense may be based on a parabola rather than an adjoining facet. A 20-facet sidewall, for example, may be based on 20 different parabolas. [0079] FIG. 4C is a diagrammatic representation illustrating that the facets for a sidewall can be defined using a computer program such as Microsoft Excel (Microsoft and Excel are trademarks of Redmond, Wash.-based Microsoft Corporation). The graphing feature in Microsoft Excel can be used to create a graph, shown at 125 , of a sidewall shape. The same general shape can be used for each sidewall or different shapes for different sidewalls. A shaped substrate with the specified sidewall shape (or with a curved sidewall shape based on the specified facets) can be analyzed in, for example, Zemax optical design program (Zemax is a trademark of Zemax Development Corporation of Bellevue, Wash.). A computer simulation can be conducted in Zemax to generate a ray trace and an intensity and irradiance distribution profile. If the resulting intensity and irradiance profile has an unsatisfactory distribution or the transmission efficiency of the shaped substrate is too low, the variables of the various facets can be adjusted and the simulations performed again. This process can be automated through the use of a computer program to automatically adjust facet variables. [0080] More specifically, FIG. 4C depicts a spreadsheet 500 that can be utilized to design a sidewall shape as shown in graph 510 through the specification of angular subtenses. Projected half angle column 550 contains a plurality of angles that correspond to projected half angle 450 of FIG. 4B . Exit angle columns 540 a (in radians) and 540 b (in degrees) contain a plurality of exit angles corresponding to exit angle 440 of FIG. 4B . More particularly, all or a subset of the angles in column 540 a may be angles that are less than the critical angle such that light rays intersecting the exit face at those angles traverse the exit face, exiting the substrate. Columns 540 a and 540 b may be utilized to develop parabola focus column 560 , containing a plurality of foci defining different parabolas. Angular subtense column 565 contains a plurality of angles (in radians) that define the limits of an angular subtense that can be used in conjunction with parabola focus column 560 to define the shape of a sidewall such that a ray from the quantum well region reflects off the sidewall to exit the exit face at less than the critical angle. Using the values contained in parabola focus column 560 and angular subtense column 565 , theta column 570 and radius column 575 can be developed wherein corresponding values in columns 570 and 575 correspond to points on a desired parabola for the angular subtense. In turn, theta column 570 and radius column 575 can be utilized to develop Cartesian coordinates for points on a sidewall (e.g. coordinate transformation columns 577 ) that approximate the parabola for the angular subtense. [0081] For example, a user can specify the LED size (i.e., the area of the interface between the substrate and quantum well region) and the material index. Using the example of an LED having a size of 1, and an index of refraction 1.77, a row in screen 500 can be completed as follows. The user can specify an exit angle in air (assuming air is the medium in which the LED will operate) in column 550 . In the example of the first row, the user has selected 55.3792 degrees. The exit angle in the substrate can be calculated as sin(55.3792/180*π)/1.77 or 0.4649323 radians, column 540 a . Column 540 b can be calculated as a sin(0.4649323)/π*180=27.2058407. The focus of the parabola can be calculated as 1(size)/2*(1+cos(π/2−27.2058407/180*π))=0.732466. Angular subtense column 565 can be calculated based on the number in the next column (representing the relative size of a particular facet) as (90−27.7058047)/20=3.114708. Theta column 570 can calculated using a selected number of facets (in this example 20). For example, in the first row theta is calculated as (90−27.7058407)+3,114708*20=124.5883. The radius of the parabola (column 575 ) for the first facet can be calculated as 2*0.732466/(1+cos(124.5883/180*π)). The contents of coordinate transformation columns 577 can be calculated as follows for the first row:x=−3.3885*cos(124.5883/180*π)=1.923573; y=−3.3885*sin(124.5883/180*π)=2.789594, X=1.923573*cos(27.7058407/180*π)+2.789594*sin(27.7058407/180*π); Y=2.789594*cos(27.7058407/180*π)−1.923573*sin(27.7058407/180*7)−1(size)/2=1.075452 and Y′=−Y. The X, Y coordinates can then be used as data point inputs for a shape fitting chart in Excel. For example graph 510 is based on the data points in the X and Y columns (with the Y column values used as x-axis coordinates and the X column values used as y-axis coordinates in graph 510 ). In addition to the X and Y values a starting value can be set (e.g., 0.5 and 0). The shape from graph 510 can be entered into an optical design package and simulations run. If a simulation is unsatisfactory, the user can adjust the values in spreadsheet 500 until a satisfactory profile is achieved. [0082] In one embodiment, when a satisfactory light transmission efficiency and irradiance and intensity profiles are achieved, a LED with a substrate having the specified parameters can be produced. An example of such a LED is shown in FIG. 4D which provides a diagrammatic representation of one embodiment of a LED having a substrate with sidewalls shaped to cause TIR so that rays are reflected from the sidewalls to the exit surface. The shape of each sidewall, in this embodiment, is a superposition of multiple contoured surfaces as defined by the various facets. While a curve fit may be performed for ease of manufacturability, other embodiments can retain faceted sidewalls. While in FIG. 4D , the area of the quantum well region is shown as being square or rectangular, this is by way of illustration and not limitation. For example, the shape of the area of the quantum well region can be any of a variety of shapes, e.g. circular, rectangular, triangular. Likewise, the shape of the exit face of a LED can be any of a variety of shapes, e.g. circular, rectangular, triangular. [0083] Returning to FIGS. 1A and 1B , as described above with regard to FIGS. 1A and 1B , various boundary conditions, particularly the area of exit face 55 of substrate 10 , are determined for substrate 10 so that light is conserved. The minimum area of exit face 55 can be determined from EQN. 1 above, which relies on various effective solid angles. Typically, the effective solid angle of light is determined based on equations derived from idealized sources that radiate as Lambertian emitters, but that are treated as points because the distances of interest are much greater than the size of the source. The observed Radiant Intensity (flux/steradian) of a Lambertian emitter varies with the angle to the normal of the source by the cosine of that angle. This occurs because although the radiance (flux/steradian/m 2 ) remains the same in all directions, the effective area of the emitter decreases to zero as the observed angle increases to 90 degrees from normal. Integration of this effect over a full hemisphere results in a projected solid angle value equal to π steradians. [0084] Turning to FIG. 5 , assume a sphere 130 of given radius (R) surrounds point source 132 (in this example, point source 132 approximates a Lambertian source at a significant distance). The projected area of a hemisphere of the sphere is πR 2 and the projected area of the full sphere is 2πR 2 . This model can be used to design a LED because an interface between a quantum well region and a substrate can be modeled as a Lambertian emitter such that from any point on a hypothetical hemisphere centered over the interface, a given point on the interface will have the same radiance. The area A 3 can be calculated as the flat, circular surface (e.g., surface 136 ) that is subtended by the beam solid angle of interest using a radius of the circle 134 (R c ) that is the distance from the normal ray to the intersection of the spherical surface. For a given half angle 137 of θ of the beam, R c is the product of R (the radius of the sphere) and the sine of the angle θ, such that R c =R *Sin(θ)  [EQN. 3] The area equals: A 3 =πR c 2 =π( R *Sin(θ)) 2   [EQN. 4A] The area A 3 is the projected area of the solid angle as it intersects the sphere. The area A 3 is divided by the projected area of the hemisphere (A h =Tπ R 2 ) and the quotient is multiplied by the projected solid angle of the full hemisphere (equal to π) to obtain the projected solid angle Ω, such that: Ω=π*{projected area of desired solid angle}/(projected area of hemisphere)  [EQN. 4B] Ω=(π)[{π( R *Sin(θ)) 2 }/(π R 2 )]  [EQN. 4C] =π*Sin 2 (θ)  [EQN. 5] [0085] For interface 50 of FIG. 1 , for example, θ is 90 degrees, leading to a projected solid angle of π*Sin 2 (90)=π, and for the desired half angle of 30 degrees, the projected solid angle is π*Sin 2 (30)=*π/4. Using these values for Ω 1 and Ω 2 for EQN. 1, A 2 can be determined for any half angle. [0086] In the above example, the solid angle is determined using equations derived from a Lambertian source modeled as a point source. These equations do not consider the fact that light may enter a substrate from a quantum well region through an interface that may be square, rectangular, circular, oval or otherwise shaped. While the above-described method can give a good estimate of the solid angle, which can be later adjusted if necessary based on empirical or computer simulation testing, other methods of determining the effective solid angle can be used. [0087] FIGS. 6A-6E describe another method for determining the effective solid angle for a substrate of an LED. FIG. 6A is a diagrammatic representation of one embodiment of an interface 150 and an exit face 155 of a shaped substrate 160 (shown in FIG. 6B ) and a hypothetical target plane 156 onto which light is projected. FIG. 6A illustrates examples for a position of an effective source origin 152 , central normal 153 and effective output origin 154 . For purposes of further discussion, it is assumed that the center of interface 150 is at 0,0,0 in a Cartesian coordinate system. Target plane 156 represents the parameters of the resulting pattern (e.g., size and half angle used by other optics). According to one embodiment, the half angle at the diagonal (shown as α 1 in FIG. 6B ) is the starting point. For example, if the desired light at target plane 156 has a maximum half angle of 30 degrees, α 1 for a square- or rectangular-faced substrate is 30 degrees. The half-angle within shaped substrate 160 (labeled β 1 and also shown in FIG. 6C ) can then be determined according to: n 2 Sin(α 1 )= n 1 Sin(β 1 )  [EQN. 6] where n 1 is the IOR of shaped substrate 160 ; n 2 is the IOR of the material (typically air) into which the light is projected from shaped substrate 160 ; α 1 is the half angle at the exit face in the medium external to the substrate (typically air); β 1 is the desired half angle in the substrate. [0088] For example, if the desired half-angle α 1 is 30 degrees, and a shaped substrate having an IOR of 1.77 is projecting into air having an IOR of 1, then β 1 =16.41 degrees. A similar calculation can be performed for a ray projecting from a point on the long and short sides of entrance face 150 . For example, as shown in FIGS. 6B and 6C , α 2 and β 2 can be determined for a ray traveling from the center of one edge on interface 150 to the center of the opposite edge of exit face 155 . (The critical angle is the same at 16.41, but β 1 is not the same as β 2 . β 2 is determined by the geometry of the sides and the height of the shaped substrate.) [0089] Using the angles calculated, the location of an effective point source can be determined. For a square interface 150 , of length l 1 , the effective point source will be located X=0, Y=0 and Z eps = l 1 2 * tan ⁡ ( β 1 ) [ EQN .   ⁢ 7 ] Where zeps is the distance the effective point source is displaced from entrance face 150 of shaped substrate 160 . [0090] The X, Y and Z distances from the effective point source to points F 1 and F 2 can be calculated assuming F 1 intersects a sphere of unity radius according to: X F1 =cos(ψ 1 )sin(β 1 )  [EQN. 8] Y F1 =sin(ψ 1 )sin(β 1 )  [EQN. 9] Z F1 =cos(β 1 )  [EQN. 10] X F2 =0  [EQN. 11] Y F2 =cos(ψ 2 )*sin(β 1 )  [EQN. 12] Z F2 =cos(β 1 )  [EQN. 13] where ψ 1 is the angle of the diagonal ray in the X-Y plane (45 degrees for a square) and where ψ 2 =90 degrees for a ray projecting from the middle of a side parallel to the X axis as shown in FIG. 6C . As shown in FIG. 6A , because 156 intersects the spherical surface at four points and the magnitude of angle β 2 is less than the magnitude of critical angle β 1 , the values for point F 2 are calculated based on the projection of a diagonal with an angle β 1 onto the plane of the side ray. A similar methodology based on the geometries previously calculated can be used to determine other points (e.g., for example, the location of points T 1 and T 2 can be determined based on the location of points F 1 and F 2 and the desired half angle of light at target plane 156 .) [0091] FIG. 6D illustrates the diagonal rays and one ray from the short side projected onto a sphere 159 for exit face 155 and sphere 161 for target plane 156 . For exit face 155 , the projection of the intersection of the edge rays at the sphere 159 onto the plane of the exit face 155 , forms elliptical segments. Likewise, the projection of the refracted exit rays at the edge of the target face intersect the sphere 161 . FIG. 6E , for example, points out the circular intersection of the rays lying in the plane formed by the edge 163 of target face 156 intersecting sphere 161 (shown at 162 ), and the projection of that intersection onto the target plane 156 (shown at 164 ). By calculating the area of each of the elliptical segments surrounding the square of the target face and adding that to the area of the target face, we find the total projected area of the target face. The effective solid angle can be determined for the target plane using EQN. 4B Similarly, by using sphere 159 and the elliptical segments formed thereon by rays, the effective solid angle for the LED can be determined. For example, the total projected area is determined as described above and inserted as “projected area of desired solid angle” in EQN. 4B. [0092] As one illustrative example, using the above method to project light with a half-angle of 30 degrees using a LED having a substrate with a square shaped interface and exit face yields an effective solid angle of 0.552 steradians to the target in air. By contrast, the use of the traditional circular projected area with a 30 degree half angle projection specification would yield an effective solid angle of 0.785 steradians. When these values are then used in EQN. 1, for given IORs and flux, the traditional (circular) calculation yields a required exit area that is undersized by about 30%. If one were to design a system using this approach, the applicable physics (i.e. the conservation of radiance) would reduce the light output by 30% over the optimum design. Conversely, using the corrected effective solid angle described above calculates an exit face area that will produce 42% more light output than is achievable with the circular calculation. [0093] Although particular methods of determining the effective solid angle for a LED are described above, any method known or developed in the art can be used. Alternatively, the minimum surface area to conserve light can be determined empirically. Moreover, while the minimum surface area calculations above assume light is entering the substrate across the entire surface of the interface between the quantum well region and the substrate, in physical devices, light may not enter the substrate in an even distribution across the entire surface of the interface. The calculations of the minimum area of the exit face can be adjusted to account for the actual distribution of light traversing the interface, rather than being based entirely on the size of the area of the interface. In one embodiment, the actual area of the interface through which light enters the substrate can be used as A 1 . [0094] Embodiments of LEDs can project light into a desired cone angle of 10-60 degrees with a theoretical efficiency of up to 89% (meaning that 89% of the light entering the substrate is emitted in the desired half-angles with 11% fresnel loss) depending on substrate material and Fresnel losses. The efficiency can be 100% without fresnel losses. Even at only 70% efficiency, embodiments of LEDs provide greater efficiency than other LED technologies, while also allowing for uniform or near uniform intensity distributions at both near and far fields. [0095] Fresnel losses at the substrate to air (or other medium) interface can be overcome by the application of anti-reflective coatings to the exit face of the substrate. Anti-reflective coatings that can be used are any that would be known to one of ordinary skill in the art and include single layer MgO or MgF, multilayer coating or other anti-reflective coatings. Through the utilization of anti-reflective coatings, Fresnel losses can be reduced or eliminated, increasing the light output efficiency of a LED. [0096] An embodiment of a LED may have more than one exit face. For example, a shaped substrate may allow substantially all the light generated by the LED to exit the LED, but through more than a single exit face. FIG. 7 is a diagrammatic representation of an example of a LED 700 with more than one exit face. In FIG. 7 , exit faces 710 a and 710 b of LED 700 are shown. A LED having more than one exit face might emit light into a solid angle greater than a hemisphere. To maximize the light exiting the exit faces, the sidewalls of a substrate with more than a single exit face may have multiple curved or faceted surfaces. [0097] For a LED with two or more exit faces, it is possible for the solid angle of emission of the LED to be greater than a hemisphere (and the projected solid angle to be greater than pi). An example of this would be if instead of a single planar exit face, the LED had a four sided pyramidal set of exit faces. If the sidewalls of the substrate of the LED are shaped to direct light entering the substrate through the interface to one of the four exit faces so as to strike the exit face at an angle not greater than the critical angle, then all the light entering the substrate may exit the LED through one of the four exit faces. [0098] Since the faces of the pyramid are not in a plane, but rather are at angles to each other, any ray that strikes an exit face at the critical angle to that exit face will refract to an exit angle of 90 degrees. The total solid angular space defined this way would then be a function of the angular relationship of the four exit faces. To satisfy the etendue equation, the four exit faces in this example would have to have a total surface area at least equal to the calculated value using the effective solid angle for that construction. [0099] This multi-exit face construction may still be constructed in such a way as to conserve radiance. That is, by making the total projected exit face area equal to the calculated value, and by designing the sidewalls to provide uniform distribution of the light to each portion of the exit faces, radiance can be conserved. If the exit faces are made larger than the required value, then light entering the substrate may exit through the exit faces, with a corresponding reduction in luminous intensity. [0100] A further embodiment of a shaped substrate with multiple exit faces is one in which the sidewalls of the shaped substrate are themselves exit faces. Depending on a point of entrance of a given light ray, it may strike a given sidewall at an angle not greater than the critical angle, and pass through that sidewall, or it may strike at an angle greater than the critical angle and be internally reflected to another face or sidewall. [0101] If the sidewall exit faces and sidewalls are designed such that any ray entering the substrate from any point on the interface passes through a sidewall exit face, then all of the light entering the substrate will exit the substrate. [0102] Shaped substrate LEDs with multiple exit faces may be appropriate for use in general lighting applications where broad area emission is desired. Such LEDs may be used in conjunction with additional lens or reflector elements that will direct light produced by the LED into a smaller solid angle. [0103] The potential benefit of a shaped substrate with multiple exit faces or in which sidewalls act as exit faces is that the LED may have a smaller volume or may have a shape that is more readily manufactured—such as planar faces instead of curved surfaces. [0104] LEDs can be arranged in an array of LEDs. An array of LEDs can be used to produce a desired amount of light and a desired light pattern. For example, LEDs may be arranged in a square or other shape. Using an array of LEDs to produce the desired amount of light may be more efficient or may consume less space than using a single LED. An array of LEDs can be formed during manufacture. For example, an array of LEDs can be formed from the same wafer. In FIG. 8A , LED array 800 comprises LEDs 810 a - 810 c that are formed from the same wafer. Wafer material 820 is removed to form LEDs 810 a - 810 c . LED 810 a remains attached to LED 810 b at point 830 a . Likewise, LED 810 b remains attached to LED 810 c at point 830 b . Thus, through the selective removal of substrate material, arrays of LEDs may be formed. FIG. 8 represents one method of forming arrays of LEDs and is illustrative and not limiting: other methods for forming arrays of LEDs as would be known to one skilled in the art are within the scope of the invention. [0105] One advantage of using an array of LEDs is that the shaped substrates of the multiple LEDs in the array may be thinner than the shaped substrate for a single LED having the same amount of light output. Additionally, an array of smaller LEDs may be more efficient than a single LED; that is, an array of smaller LEDs that consume a certain amount of input power may produce more light than a single large LED of the same exit face size and input power. [0106] Because some embodiments of LEDs may utilize thicker substrates than traditional LEDs, a retaining device can be used to secure a LED within an LED package. A mechanical attachment device (e.g., of molded plastic, metal or other material) can secure an LED or LEDs within an LED package or other structure and contact an LED to create a normal force to keep the LED in place. Sideways motion can be prevented by frictional force between the attachment device and the LED. The device may have the same IOR as the substrate so that rays exiting the substrate are not deviated as they pass through the attachment device. The attachment device can include secondary optics such as a lens, layer of material or other face through which light exiting the substrate passes. Consequently, the attachment device can additionally act to shape or further define the output beam. FIGS. 9A-9C are diagrammatic representations of various examples of retaining devices 900 a - 900 c that can be used to secure LED 910 . Retaining device 900 b of FIG. 9B includes a lens 920 b that can further focus light from LED 910 . Retaining device 900 c of FIG. 9C includes a reflective focusing device 920 c that can further focus light from LED 910 . In one embodiment, reflective focusing device 920 c is a compound parabolic concentrator. [0107] One or more methods may be used to shape or form a LED or the substrate of a LED. The methods of shaping a substrate as described below are exemplary and comprise a subset of the numerous methods available. The methods described below and other methods used in the LED or optics industries may be used to produce LEDs. Methods that can be used, alone or in combination, to shape a LED or a substrate by removing material include etching, laser ablation, water jet cutting, ultrasonic removal, and mechanical removal. [0108] Etching describes a chemical process of removing substrate material in a highly controlled manner as to yield the appropriate shape. There are typically two types of etching methods: wet etching and dry etching. Wet etching involves using liquid-phase etchants to remove the substrate material. In dry etching, plasma etching and reactive ion etching, ions are created and imparted onto the substrate. There, either based on chemical reaction or particle momentum, material is removed from the substrate. [0109] Starting with a wafer of substrate material (that may further include material comprising the quantum well region), a particular pattern of photoresist can be deposited on a side of the wafer. The wafer is then etched. Locations on the wafer covered with the photoresist would not be etched, whereas places without the photoresist would have material removed. There are many ways to tune the process to achieve the desired contour at the edge of the photoresist. For example, thicker layers of photoresist can be applied and then sacrificially removed during the etching process, or other sacrificial layers can be used in conjunction with the protoresist. These layers are removed over time by the etchant in such a manner as to produce the desired contour of the LED substrate. This can be exploited to accurately etch the wafer so as to produce shaped substrates. Another way is to use multiple resists and multiple etching steps. Each photoresist and etch step can be used to remove a small layer of material. Multiple small steps can be use get the desired 3D shape. [0110] FIG. 10 is a diagrammatic representation of an example of etching. In FIG. 10 , wafer 1000 is etched to form shaped substrates 1030 a - 1030 c . Photoresist 1010 is applied to a surface of wafer 1000 in a desired pattern such that portions of the surface of wafer 1000 are covered with photoresist 1010 . Subsequent to the application of photoresist 1010 , an etchant is applied to the same surface of wafer 1000 that the photoresist was applied. The etchant removes substrate material 1020 , forming shaped substrates 1030 a - 1030 c . Photoresist and etchant can be applied in successive layers and patterns to achieve a desired sidewall shape. [0111] Etching parameters may be based on the substrate material. The etch rate varies depending on the etchant and the substrate. For substrate materials used in LED applications such as sapphire and silicon carbide, the etch rates using reactive ion etching can range from 250 nm to 2.5 um per min, which may be slow for commercial production purposes. Silicon carbide is on the upper end of the above etch rate while sapphire is on the lower end. [0112] Laser ablation is the process of using a high power laser to produce LEDs by removing or ejecting quantum well region or substrate material. Each laser pulse will only remove a minute amount of material. The laser can be translated to remove material with each subsequent pulse. By translating in the X-Y and Z directions, a 3D shape can be removed. Embodiments of laser ablation can be used to shape substrates faster than etching. Using known techniques, laser ablation can remove around 500 um to 1 mm in thickness per minute in silicon carbide and sapphire. [0113] FIG. 11 is a diagrammatic representation of laser ablation. Laser 1110 is applied to wafer 1100 to ablate substrate material 1120 , forming shaped substrates 1130 a - 1130 b. [0114] A water jet may be used to ablate a wafer to form a substrate of the desired shape. In one embodiment of water jet ablation, short pulses of water can be used to ablate a wafer in stages. The process by which pulses of water are used to ablate a wafer may be similar to the process described above in regard to laser ablation. In one embodiment of water jet ablation, the water jet can be used to cut completely through the wafer at a certain angle, then the angle shifted slightly and the water jet used to cut through the wafer at a slightly higher angle, eventually producing a substrate of the desired shape. In a further embodiment, the water jet may be charged with abrasive material (e.g. industrial diamond particles) to increase the rate at which material is ablated. [0115] Another option is to mechanically remove material through grinding, milling, sawing, ultrasonic grinding, polishing, drilling, or other systems or methods of mechanical removal. There are numerous methods for removing material to shape one or more LEDs through mechanical removal. For example, a sawing blade (typically, a diamond grinding wheel) can be used to dice the wafer into multiple LEDs or substrates. The saw blade is prepped to have the inverse shape of one or more sidewalls, taking into account minor material losses that will occur to the blade during the removal process (i.e., the blade may begin slightly oversized). The blade is then used to saw the wafer. The rotation of the saw blade may polish the substrate sidewalls as it cuts. Each cut forms the sidewalls of adjacent substrates. According to one embodiment, the saw will first make multiple cuts in one-axis and then make cuts orthogonally to that axis to form the shaped substrates. The mechanical removal of material may be done in stages. [0116] FIGS. 12A-12B are diagrammatic representations of forming a LED utilizing a sawing blade. In FIG. 12A , a sawing blade 1210 having the inverse shape of a desired sidewall of a shaped substrate is applied to wafer 1200 . FIG. 12B shows sawing blade 1210 when fully impacted into wafer 1200 , forming sidewalls 1220 a and 1220 b. [0117] FIGS. 13A and 13B are diagrammatic representations of top views of wafer 1200 subsequent to the application of sawing blade 1210 to wafer 1200 . Subsequent to the mechanical removal of substrate material, the shaped substrate may be polished to enhance TIR within the shaped substrate. [0118] While the above methods of ablating a wafer of material to form shaped substrates have been described individually, the above methods may be combined. For example, it may be possible to use a combination of mechanical removal and water jet ablation to ensure an appropriately curved sidewall shape. Similarly, various other combinations of methods and techniques for removing substrate material from a wafer to produce LEDs can be used as appropriate depending on the substrate material. In addition, other methods, such as ultrasonic machining can be used to form shaped substrates. Ultrasonic grinding may also be used to shape substrates. In embodiments of ultrasonic grinding, a tool with the inverse shape of one or more LEDs is primed with an abrasive and brought into contact with the substrate material while the tool is ultrasonically vibrated to produce a scrubbing/scuffing action on the substrate material such that material is removed and shaped substrates are produced. [0119] While the above embodiments describe forming LEDs from a wafer of material, shaped substrates utilized to produce LEDs can be formed from a bar of substrate material or may be shaped from substrate material individually. [0120] In one embodiment of a LED, LEDs may be individually produced. For example, an individual shaped substrate may be mounted in a receiving tool such that it is accurately located. The sidewalls may be protected by protectors such as resist materials, shielding structures in the receiving tool or other protectors. The desired epitaxial layers may be deposited on the base of the shaped substrate, thus producing a LED. By way of further example, the individual shaped substrate may be made of a moldable material such as high temperature glass. The glass shaped substrate may then be developed into a LED by depositing the desired epitaxial layers on the base using low temperature epitaxial growth methods. [0121] In some cases, it may be desired to generate white light using a LED. This can be accomplished by impinging light from a single-color (e.g. blue), short wavelength LED onto phosphors or other particles that absorb the light and re-emit light in a wavelength that the human eye perceives as white light. Phosphors or other particles can be used with embodiments of LEDs to produce white light. FIG. 14 shows one embodiment of LED 20 . Phosphors or other particles can be placed at interface 50 between quantum well region 15 and substrate 10 . In such a case, the light entering substrate 10 would be white light as seen by the human eye. Phosphors can also be placed subsequent to exit face 55 of substrate 10 . According to one embodiment, a layer of particles can coat exit face 55 . According to other embodiments, particles can be on another piece of material that contacts the exit face or that is offset from the exit face. For example, phosphors can be on an interior face of a lens that encapsulates, but does not contact, exit face 55 of LED 20 . [0122] Coating the exit face or faces of a LED may have fabrication advantages that may allow for the simplification of white light LED fabrication, which may in turn reduce the cost of white light LED fabrication. For example, a side of a wafer out of which shaped substrate LEDs will be formed may be coated with a layer containing phosphors or other particles (i.e. a particle coating) that can be excited to emit white light. The side of the wafer that has not been coated with the particle coating can be ablated. When the wafer has been ablated to produce multiple LEDs, the LEDs will have an exit face having the particle coating necessary to produce white light. Furthermore, because a shaped substrate directs a supermajority of light entering the substrate to a known exit face or faces, coating a particular exit face or faces may be highly effective in the generation of white light. Accordingly, the use of a shaped substrate may eliminate the need to coat the sidewalls or a portion of the sidewalls of an LED with a particle coating. Thus, there will be no need to individually apply a particle coating to each LED. Applying a particle coating to a side of a wafer may be cheaper than applying a particle coating to individual LEDs. The sidewalls of the substrate can be designed so that light scattered back into the substrate by interactions with the particle coating may be partially or fully recycled. Utilizing nano-particles to generate white light in conjunction with a LED allows for minimum deflection of light, thus minimizing backscattered light and maximizing the light emitted from the exit face. [0123] Heat generated in the quantum well region of LEDs causes the degradation of phosphor particles or nano-particles. In current LEDs, phosphors conformally coat the LED chip. The thermal density is highest at the chip, and therefore phosphors are subject to tremendous heat. In embodiments of an LED with a shaped substrate having a phosphor coating on the exit face, because the exit face is removed from the heat generating quantum well region by the height of the substrate, the degradation of the phosphors is reduced. Thus, an LED with a thick substrate and consequently an exit face removed from the quantum well region may generate white light of a desired quality longer than existing designs. [0124] Potential applications for embodiments of LEDs include cell phone display lighting. Present systems typically use three side-emitting blue LEDs with phosphor-filled encapsulant material to generate white light. The sides of the LED are typically opaque and a large percentage of the light generated is absorbed by the sidewalls. This results in over 50% of the light being lost to absorption. In addition, the index change at the interface of the encapsulant to air creates a TIR condition for exit rays striking the interface at greater than the critical angle. This results in approximately 44% loss at the interface. Embodiments of shaped substrate LEDs, can deliver 80% of the generated light to the light guide, resulting in very large system brightness improvements. [0125] Another potential application for embodiments of LEDs is use as a cell phone camera flash. Present systems typically use LEDs with Gaussian energy distributions that produce a very bright area in the center of the image and dark areas at the edges, causing uneven lighting of the subject matter. Moreover, the beam shape of present flash units is circular, while the image captured by the CCD camera is rectangular. Additionally, the index of refraction change at the interface of the encapsulant to air creates a TIR condition for exit rays striking the interface at greater than the critical angle. This results in losses at the interface that are a function of the exit solid angle. Embodiments of LEDs, on the other hand, can deliver a rectangular or square flash, with 80% of the light entering the substrate of the LED being provided to the image area in a uniform distribution. This results in more uniform scene illumination and higher levels of illumination as opposed to prior art LED flash systems. [0126] Another potential application for embodiments of LEDs is for liquid crystal display (“LCD”) backlighting. Traditional LCD systems use a linear array of red, green and blue LEDs. The light from the LEDs is directed into a mixing light guide to provide uniformity of color and intensity. Typically, the LEDs have a dome placed over the LED and light is captured by elliptical reflectors to direct the light to the light guide. While elliptical reflectors work well for point sources, LEDs are not point sources and some of the rays will not get to the focii inside the light guide. Moreover, since some light from a dome encapsulant is emitted at greater than 180 degrees, some of the light is absorbed by the substrate, PCB board and other components. Furthermore, because the dome is large with respect to the size of the cavity in the dome, a certain percentage of light typically gets refracted. Because these losses are multiplicative, only a percentage of the light originally emitted from the LED actually gets to the light guide. [0127] Embodiments of LEDs, on the other hand, can provide up to 80% of the light entering the substrate of the LED to the light guide (assuming Fresnel losses) in the desired cone angle. Consequently, lower power LEDs can be used to achieve the same results as are possible in current systems or more light can be delivered at the same power consumption level. Indeed, in some embodiments, the light guide may not be required and arrays of LEDs may be used to directly backlight LCDs. [0128] Another potential use for embodiments of LEDs is in car headlights, flashlights, digital light processing (“DLP”) systems and other devices. The shape of the LED can be selected so as to provide the desired projection cone and beam profile. Furthermore, the combination of a LED and a condenser lens or other optical device, such as the Primary Optic Device (“POD”) described in U.S. patent application Ser. No. 11/649,018, entitled “SEPARATE OPTICAL DEVICE FOR DIRECTING LIGHT FROM AN LED”, allows for the emission of a narrow solid angle (on the order of 0.1 steradians or less) while conserving radiance and doing so in a very small volume. Such a combination is applicable for flashlights, spot lights or any other narrow beam application. [0129] In U.S. patent application Ser. No. 11/649,018, entitled “SEPARATE OPTICAL DEVICE FOR DIRECTING LIGHT FROM AN LED”, a Primary Optic Device (“POD”) is described. Embodiments of systems and methods described herein differ from and provide significant advantages over the POD device. The POD device is a passive optic element used in conjunction with LEDs to focus light emitted from the LED. The POD device itself is unable to generate light and directs light generated by other sources. The POD is typically made of materials having an IOR of approximately 1.5. Thus, the amount of light that can enter the POD device from an LED is limited by the change in IOR from the LED material to the POD device material. By contrast, this disclosure regards a LED that itself collects and directs the light generated in its own quantum wells. All of the light in the substrate of the LED can theoretically be captured and directed. If the substrate material has a higher IOR than or equal IOR to the IOR of the quantum well region material, then all of the light generated by the quantum wells of the LED may be captured and directed. With the POD device, this may not be possible. Shaping an LED to capture and direct the light generated by the quantum wells of the LED itself results in a single LED that may be used in many applications without the need for additional optic components. [0130] FIGS. 15A and 15B are a diagrammatic representations of another embodiment of a layout of multiple die being made in a single wafer 1500 . In this case, the die are packed more closely together and the sidewall shape is formed by any of the methods already described, except the depth of cut of the sidewall is greatly reduced. For example for a 3 mm thick wafer in the previous embodiment, the sidewall shape would be cut 2.75 mm deep with 0.25 mm remaining to hold all the die together. In the example of FIG. 15 , the depth of cut is only about 1 mm, leaving about 2 mm to hold the die together. This creates a much stronger wafer for handling in manufacturing, and allows the die to packed more closely together. This results in more die per wafer on the order of 1.5 to 2.4 times as many die per wafer. After all the operations are completed on the wafer, the die are separated for example by diamond sawing them with a thin sawblade (on the order of 0.1 mm wide. The surface produced by the sawblade may closely approximate the intended sidewall shape in that region, and may provide an adequately polished surface, such that no further operations are needed. Alternatively, subsequent operations may be performed to finish shaping and polishing that portion of the sidewalls, on the individual die. [0131] While this disclosure describes particular embodiments, it should be understood that the embodiments are illustrative and that the scope of the invention is not limited to these embodiments. Many variations, modifications, additions and improvements to the embodiments described above are possible. For example, the various ranges and dimensions provided are provided by way of example and LEDs may be operable within other ranges using other dimensions. By way of example, while shaped substrates have been described in regard to sapphire and silicon carbide, other substrates that allow the passage of light may be used. For example, substrates may be made of glass or diamond. In one embodiment, substrates may be molded from moldable glass, providing a cost effective and easily shaped substrate. It is contemplated that these variations, modifications, additions and improvements fall within the scope of the invention as detailed in the following claims.

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