Energy feedback freeform lenses for uniform illumination of extended light source LEDs 1 1 2 1 1,* ZONGTAO LI, SHUDONG YU, LIWEI LIN , YONG TANG, XINRUI DING, WEI 1 1 YUAN, BINHAI YU 1 Key Laboratory of Surface Functional Structure Manufacturing of Guangdong High Education Institutes, South China University of Technology, Guangzhou 510640, China 2 Department of Mechanical Engineering, University of California, Berkeley, California 94720, United States *Corresponding author: [email protected] Received XX Month XXXX; revised XX Month, XXXX; accepted XX Month XXXX; posted XX Month XXXX (Doc. ID XXXXX); published XX Month XXXX Using freeform lenses to construct uniform illumination systems is important in light-emitting diode (LED) devices. In this paper, the energy feedback design is used for freeform lens (EFFL) constructions by solving a set of partial differential equations that describe the mapping relationships between the source and the illumination pattern. The simulation results show that the method can overcome the illumination deviation caused by the extended light source (ELS) problem. Furthermore, a uniformity of 95.6% is obtained for chip-on-board (COB) compact LED devices. As such, prototype LEDs manufactured with the proposed freeform lenses demonstrate significant improvements in luminous efficiency and emission uniformity. © 2016 Optical Society of America OCIS codes: (220.4298) Nonimaging optics; (220.2945) Illumination design; (230.3670) Light-emitting diodes. sources (PLSs). For LED applications, the brightness of these LEDs is 1. Introduction not sufficient. Although a module with multiple sources makes it possible to obtain sufficient luminous flux, Kuo found that such a Solid-state lighting devices, especially light-emitting diodes (LEDs), solution would lead to the “multi-shadow” phenomenon, which causes have gained attention recently as a result of their outstanding features the eyes to feel tired [9]. [1], which include low energy consumption, ultra-fast response, For high-quality illumination, chip-on-board (COB) LEDs achieve a environmental friendliness, and tunable emitting spectra. These high-power output from one source. In such components, the large size features have made LEDs penetrate deeply into our daily life, in forms of the luminous area has greatly worsened the performance of the such as large-scale backlights, high-performance projectors, freeform lens system [4, 10], which causes the so-called “extended light automobile headlamps, and especially general lighting systems. At source (ELS) problem”. The simultaneous multiple surfaces (SMS) present, the angular intensity patterns of most LEDs are circularly method [11], in which two freeform surfaces are simultaneously symmetric, with the highest energy output lying in the normal designed to refract or reflect a set of input/output ray bundles, direction (i.e., Lambertian distributions), which cause glare upon provides a viable solution. Nevertheless, for compact packages in human eyes. On the other hand, LED lighting systems are always which only single freeform lenses are available, the SMS method is not installed at a certain height, and only the downward-emitting light suitable. On the other hand, optimizations based on tailoring method fulfills the illuminating function, whereas the light emitted at large also demonstrated potentials for solving the COB ELS problems [5], but viewing angles leads to energy losses. Therefore, specially designed calculation errors would be introduced due to the pre-discrete optical lenses are required in order to collect and guide the light into feedback, lowering the design accuracy and efficiency. uniform illumination patterns. In this study, we further develop the PDE freeform lens design method, The freeform lens is one of the most popular optical devices for present the concept of “effective rectification function” for COB performing this job owing to its distinct advantages, such as compact compact sources based on a set of energy feedback functions. size and precise light controlling [2–5]. Compared with the traditional Additionally, an engineering example is discussed, and the trial and error method [6] or the tailoring method [7], solving a set of performance of the lens is experimentally investigated. The results partial differential equations (PDE) [2,8] that describe the mapping show that the improved design method is capable of handling the ELS relationships between the source and the illumination pattern is of problems, and that it exhibits excellent system efficiency. The findings higher accuracy and extensibility for different illumination patterns of this study will offer a valuable reference to LED optics researchers. (such as rectangle or octagon). More importantly, this method is high efficient, and fast [2]. However, existing studies are limited to LEDs with small emitting areas (single chip component) – e.g., point light 2. Modeling Figure 1 shows the lighting model of a freeform lens designed for flux of the specific area on the target plane to the total luminous flux on uniform illumination. The light source O is located at the origin of the the target plane equals that of the luminous flux of the specific solid coordinate system. The target plane T is located at a distance z0 from angle to the total luminous flux: the origin, and is perpendicular to the z-axis. The unit normal vector of θ 2 22 2cossinπθθθId the target plane T is =(0, 0, −1), and the points on plane T can be Eyyπ()− θ 0 021= 1 expressed as (, , ). The points on the freeform lens P can be θ , (5) π 2 max expressed in spherical coordinates as (,,(,)), where γ is the ER0 2cossinπθθθId 0 0 angle between the x-axis and the projection of on the x-y plane, and θ is the angle between the z-axis and . The unit normal vector at where is the angle corresponding to the maximum light emission, and R is the radius of the illumination pattern. Equation (5) is the so- point P is , and ρ(γ, θ) is the module of . Let be the unit vector called “energy mapping function.” This mapping function is very of , and be the direction vector of . According to Snell’s law: important in the PDE freeform lens design method, which contains information regarding the illumination distribution on the target plane. 22+− ⋅ = − nnoI2 nnOINnOnI00 I() p I. (1) Altering the energy mapping function can alter the designed illumination distribution. Therefore, Eq. (3) can be transformed into: z − ρ cosθ z θθ0 −− sin (nn0 I cos ) T（x，y，z0） θρθ−+−22 ρ θ (()fz sin) (0 cos) f ()θρθ− sin cosθθ (nn− sin ) , (6) 0 θρθ−+−22 ρ θ I uuur ∂ρ (()fz sin) (0 cos) N = ρ p − ρθ ∂θ z0 cos (γ，θ，ρ(γ，θ)) cosθθ (nn−+ cos ) P 0 θρθ−+−22 ρ θ I (()fz sin) (0 cos) f ()θρθ− sin O sinθ (nn− sinθ ) 0 θρθ−+−22 ρ θ I y (()fz sin) (0 cos) x By solving Eq. (6), the profile of the freeform lens can be obtained, as shown in Fig. 2. Fig. 1. The lighting model of the freeform lens for uniform illumination. The relationship between the incident light and the refracted light can be derived: ∂∂ρρ 2 −+sinγθθγρθγ sin cos cos − sin cos ∂∂γθ nO− nI = 0 xIx ∂ρ 2 nO− nI −−sinθρ sin θ cos θ 0 z Iz ∂θ , (2) ∂∂ρρ cosγθθγρθγ+− sin cos sin sin2 sin ∂∂γθ nO− nI Fig. 2. Molding of freeform light-emitting diodes (LEDs). = 0 yIy ∂ρ 2 nO− nI −−sinθρ sin θ cos θ 0 zIz ∂θ where and are the refractive indices of the lens material (which is set as 1.54) and air, respectively. In order to simply the discussion, we focus on a uniform circular illumination pattern, and the specific value of = 90° s selected. We therefore obtain: ∂ρ sinθθ (nO−− nI ) cos ( nO − nI ) = ρ 00z Iz x Ix , (3) ∂−+−θθ θ cos (nO00z nIIz ) sin ( nO x nI Ix ) which can be solved as ordinary differential equations in this specific situation. In order to solve Eq. (3), an additional condition = () should be introduced. The illuminance E0 on the target plane is constant since Fig. 3. The simulation results of the illumination distribution for the uniform illumination is expected. Therefore, the total luminous flux on case of a point light source (PLS). the target plane can be written as: - φφγθ===ES I(, ) d Ω, (4) For the convenience of discussion, the ratio of the lens central height total0 l v to the source diameter is defined as the L-S factor. Figure 3 shows the , where is the illumination area, (, ) is the luminous intensity at simulated (Monte Carlo ray tracing (MCRT) typical wavelength of 450 the viewing angle (, ) from the light source, and is the unit solid nm is utilized) results for a PLS, from which it can be seen that the angle relative to the light source. Generally, angular intensity patterns illumination on the target plane is quite uniform; this is rather suitable of LEDs are Lambertian distributions.Considering the energy loss for LEDs, especially concerning indoor applications. However, as the L- during light traveling, it can be derived that the ratio of the luminous S factor decreases (i.e., as the PLS gradually transforms into an ELS), the light spots become bright in the center and dark along the edge; Let = () be the input illumination distribution function (IIDF) of this is referred to as PLS-ELS deviation, and is illustrated in Fig. 4. the circular light spot on a given target plane. The energy mapping Obviously, the freeform lens designed by the PLS method is not function is therefore: suitable for the case of an ELS. Further modifications are required in θ y22 order to obtain uniform illumination. 2()ππθθθf yydy 2()sin I d y θ 11= (8) R θ 2()ππθθθf yydymax 2()sin I d 00 To solve Eq.