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Abstract
In this study, we present a smart design concept for extremely thin free-form (FF) micro-optical elements (MOEs) for uniform illumination in direct-lit LED luminaire systems with a large distance to height ratio (DHR). A design example for a FF-MOE with a height of only 50 µm that allows for a DHR value of more than 3 is presented. In particular, it is also shown that the arrangement of the FF-MOE with respect to the substrate (bottom side or top side) is of essential relevance in order to achieve large DHR values. The applicability of these FF-MOEs is demonstrated by comparing the simulation and experimental results of a demonstrator box mimicking a thin direct-lit luminaire.
© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Albeit still young in years, the 21st century has witnessed a remarkable replacement of conventional lighting sources by light emitting diodes (LEDs). The reason for this is the enormous number of advantages of LEDs as a light source, including their long lifetimes, energy saving aspects, their ease for system integration and so on [1,2]. In addition, their compact sizes favor their implementation in sophisticated optical systems and allow for new concepts of luminaire design aiming on backlights for displays and signage, automotive lighting, general lighting and architectural lighting.
For example, luminaire designs allowing for a homogeneous luminance in combination with a large illuminating surface and an overall low height are very appealing for backlight units (BLU) in liquid crystal displays (LCDs) [3–7] as well as in general and architectural lighting applications [8–10]. Such luminaires, which allow for a uniform illumination of a target plane, e. g. a diffusing film, are in most cases realized on the base of edge-lit or direct-lit approaches.
A typical edge-lit system is composed of a light guide plate (LGP) made of glass or a polymer like poly(methylmethacrylate) (PMMA) and is sandwiched between a reflector film on the bottom side and a diffusing film on the top side of the LGP. The LEDs are placed at the edges of the LGP where from the light is coupled into the LGP with the help of optical elements with the help of total internal reflection. Tailored micro-optical structures allow for the out-coupling of the light from the LGP in a way that a uniform irradiance distribution on the diffusing film on the top side of the BLU can be realized. One of the main obstacles of this approach is the wavelength dependent variation of the refractive index and the absorption coefficient of PMMA resulting in color differences between the central area and the edges of displays, in particular of those, which have larger sizes [5].
Contrarily, in a direct-lit system the LEDs are located at the bottom of the BLU and illuminate the diffusing film directly. Therefore, the LEDs are typically arranged in an array and the packing density of the LED dice determines the quality of the irradiance on the diffusing film, in particular its uniformity. In suchlike systems the ratio of the diameter of the illuminated area (in case of one single LED) or the distance between the LEDs(in case of an array of LEDs) and the height of the optical system (DHR) is an important parameter because in an array of LEDs it determines the minimum LED-to-LED spacing for a given height of the luminaire system or the minimal height of the luminaire system for a fixed LED spacing, respectively, in order to allow for a uniform irradiance on the diffusing film. Usually, direct-lit systems suffer from the need of a higher number of LEDs and a comparable larger height in comparison with edge-lit systems [5].
In [3,8,11], the maximal LED-to-LED distances, which are allowed for a uniform irradiance of a target plane in a certain distance to the light sources for different LED array configurations were investigated. Using the formula from [3] shows that for LED light sources having Lambertian radiation patterns and which are arranged in hexagonal arrays, the maximum DHR value to allow for a uniform irradiance distribution without the use of any additional optical element is ~1.154. An increase of this value is highly desirable and is a field of intense research [3,6–8,10,11]. Larger DHR values would allow a reduction of the overall number of LEDs, which is advantageous both from technical and economic point of views (lower heat generation, lower costs…), or to reduce the overall thickness of the luminaire, which may be advantageous for design and luminaire integration purposes. This can be achieved by using different approaches like, e.g., increasing the angle of view of the LEDs with the help of optical elements placed on top of the LEDs [3,8], or by using freeform (FF) lenses which convert the initial radiant intensity distributions emitted from the LEDs into arbitrary radiation patterns [6,7,10]. Also brightness-enhancement-films which were designed to collimate incident light were reported to be beneficial in this regard [7].
In particular the FF approach offers a lot of advantages. The light refracted at the FF surface of the lens can be redistributed into a uniform irradiance pattern on the target plane. The related homogenization of the propagation angles of the incident light allows for a direct-lit luminaire with large DHR values. For example, in [6] a DHR value of 3 was reached with a very high degree of uniformity (CV(RMSE) = 0.0364, see below).
In comparison with the LED die itself, the FF optical elements are quite voluminous having large widths and large heights, for their high volume manufacturing techniques like injection molding are applied [12]. The required tools for injection molding can be fabricated with techniques like diamond turning, milling, grinding, and polishing. Still, the demands on the shapes of the FF optical elements are very challenging [13–15] and the use of such voluminous optics increases the weight and the costs of the direct-lit luminaires, especially in case that every single LED die requires its own FF optics. Furthermore, a too large height of the voluminous FF-optics again counteracts any effort to minimize the overall height of the luminaire.
In some recent publications [16,17] we introduced the concept of artificially intersected FF-micro-optical elements (FF-MOEs) which excel by their extremely thin heights in the range of several 10 µm in the context of direct-lit applications. Besides the possibility to keep the overall height of the luminaire comparably thin, such a low thickness of the FF-MOEs also opens new strategies for their time- and cost-effective fabrication, e.g., by using mask-less laser direct write lithography for mastering [18] and roll-to-roll processing for large area manufacturing of ultrathin FF-MOEs on foil substrates. Still, in these publications, the DHR values were below 2.
In the following we will therefore give a more detailed discussion on the theoretical background of the ray mapping process of these MOEs and show that by an appropriate conceptual design and arrangement of such FF-MOEs with a maximal height of 50 µm a DHR value of 3.41 can be realized in a direct-lit luminaire. Furthermore, to experimentally demonstrate the correctness of the theoretical consideration, an array of such FF-MOEs was fabricated using mask-less laser direct write lithography and imprinting techniques for replication [18]. The MOE array is used as part of a demonstrator box that mimics a direct-lit luminaire.
2. Freeform design
The design procedure for the FF MOEs is based on a two-dimensional approach that consists of three consecutive steps: 1) a ray mapping step to calculate the required modifications of the radiant intensity distribution of the source IS(θ) in order to obtain the pre-defined irradiance distribution on the target plane, 2) a geometry step to calculate an adequate FF curvature capable of fulfilling the required modifications by refracting the light emitted from the source and 3) the conversion of the FF curvatures into 3-dimensional FF-MOEs [16–18], which, in case of rotationally symmetric irradiance distributions, can be done by rotating the FF curvature around the perpendicular z axis for 360°.
Fig. 1 Schematic illustration of relationships between the radiant intensity distribution IS(θ) of the source and the transmitted radiant intensity distribution IL(θ) of the FF element.
Thereby, the rays are refracted towards the same points on the target plane as they would hit if they would have the polar angle θL (see Fig. 1). θS_MAX expresses the maximal polar propagation angle captured by the FF-MOE and correlates directly with the maximal diameter of the FF-MOE. θL_MAX = tan−1(DHR/2) is the maximal polar propagation angle of rays from the source that would hit the area of the desired irradiance distribution on the target plane without FF-MOE. Two remarkable parameters are on the one hand the DHR value of the FF-MOE, which is defined by the ratio of the diameter of the pre-defined irradiance distribution on the target plane (D) and the distance between the source and the target plane (H) and on the other hand V, which represents the ratio of the distance between the source and the FF element plane (h) and the distance between the source and the target plane (H). The letters R and B represent points on the target plane or on the FF element plane to facilitate the explanation of the ray mapping process.
In the ray mapping step the required modifications of the polar propagation angles θ of the rays are calculated. Thereby, the given radiant intensity distribution of the source IS(θ) is modified into a radiant intensity distribution IL(θ), which generates the desired irradiance distribution onto the target plane [16,17]. In other words, the polar propagation angles θS of the rays emitted by the source have to be changed into the corresponding angles θL (see Fig. 1).
The correlation between θS and θL can be derived by using the equations:
The first equation expresses the energy conservation by connecting the radiant intensity captured by the FF-MOE with the radiant intensity redistributed into the desired irradiance distribution on the target plane and is used to calculate adequate normalizing factors kS and kL for the radiant intensity distribution functions IS(θ) and IL(θ). Solving Eq. (1) for a LED source with a Lambertian radiant intensity pattern (IS(θ) = cos(θ)) and for a radiant intensity distribution function IL(θ) = 1/cos3(θ), yields normalizing factors kS = 1/πsin2(θS_MAX) and kL = 1/πtan2(θL_MAX). Solving Eq. (2) with these functions, and considering their normalizing factors and tan(θL_MAX) = D/2H results in the expression:
which provides a link for the the polar propagation angles θS and θL in dependence of θS_MAX and the DHR value.The FF-MOE is placed in a distance h from the light source at the FF element plane. This means that a ray emitted by the source propagates with the polar propagation angle θS until it hits the point B on the FF element plane and becomes refracted at the surface of the FF-MOE into a modified polar propagation angle θL’, which is connecting the point B = (tan(θS),h) on the FF element plane with the point R on the target plane (see Fig. 1 the blue pointed arrow connecting the source and R = (tan(θL),H) and compare it with the red line connecting the source and B together with the blue arrow connecting B and R). By using the approximation that the surface of the FF-MOE is close to the position of the FF element plane (a very low height of the FF-MOE), the modified angles θL’ can be determined by calculating the angle between the vectors BR and ez:
In Fig. 2
Fig. 2 Plot of the functions for the modified angles θL’ in dependence of θS for θS_MAX = 63.43°, V = 0.2 and for different DHR values (1, 2, 3, 3.41 and 4).
The dotted black line corresponds to the case of θL’ = θS, which means that there is no change of θS. In case that the functional values are located below above or below the black line, the polar propagation angles of the incident rays have to be increased (θL’ < θS) or decreased (θL’ > θS) respectively at the corresponding points on the FF-MOE surface with respect to the above mentioned diameters.
Depending on the profiles of the calculated functions and by using the vector form of Snell’s law for the refraction of the rays at the FF-surface and a subsequent planar surface (surface of the substrate), the normal vectors of the FF curvature and the corresponding slope of the FF curvature can be calculated, but with the limitation given by the approximation that the surface of the FF-MOE is in close distance to the position of the FF element plane (a very thin height of the FF-MOE) as depicted in Fig. 1.
Please note that the calculated FF curvatures in Fig. 3b
Fig. 3 Plot of the slopes for the top side (a) and the the bottom side (c) placed FF-MOEs and their corresponding FF curvatures (b), (d).
In Fig. 3 the calculated FF curvatures and their corresponding slopes are plotted for two different cases: the surface of the FF-MOE is either located at the top (top side FF-MOE Fig. 3a,b) or located at the bottom (bottom side FF-MOE Fig. 3c,d) of the substrate with the assumption that the LED light source is always facing the bottom side of the substrate. For the calculations the refractive index of the surrounding medium nAir was chosen to be 1, the refractive index nFF was chosen to be 1.512, corresponding to the refractive index of the material the FF MOE was fabricated from in the experimental part of this paper.
As one can conclude from the differences of the FF curvatures for the different DHR values in Fig. 3b,d the orientation (top side or bottom side) of the FF-MOE has a strong influence on the curvature of the FF curves, changing the sign of the slopes (e.g. from positive to negative slopes). This can easily be explained by regarding the sequence of the refractive indices at the respective optical surfaces and Snell’s law of refraction. In case of placing the FF-curve on the bottom side, the rays are passing from air (nAir = 1) into the FF material (nFF = 1.512), refracting the propagation direction of the rays towards the spatial orientation of the normal vector of the surface. In the other case the refractive indices are switched, refracting the propagation direction of the rays away from the spatial orientation of the normal vector of the surface.
The basics of the procedure for the geometry step already have been published elsewhere [16–18]. Briefly it is based on a pointwise calculation of the FF curve by discretizing the radiant intensity distributions into rays with different propagation angles, projecting them onto the target surface to define the target points R and solving the vector form of Snell’s law for every point B of the FF-curve in a sequential process. This calculation method has the advantage that an additional transformation algorithm can be included that allows to limit the maximal thickness of the FF-MOE by defining two threshold values, which specify the minimal and the maximal z coordinates of the points constituting the FF curvature.
Accordingly, the height differences between the upper and the lower threshold values correspond with the maximal height of the FF MOE (see Fig. 4
Fig. 4 Schematic illustration of the transformation algorithm, which generates “artificial edges” in the shape of the FF curvature in order to limit the maximal thickness of the FF-MOE.
Based on the sequence of the refractive indices at the respective optical surfaces and Snell’s law of refraction, a general statement with respect to the conjunction between the angles θL’ (θS, θMAX, DHR, V) (see Fig. 2), the slopes of the FF curvatures (Fig. 3a,c) and the arrangement of the FF-MOE can be drawn: In case that the functional values of the ray mapping function are located on the dotted black line, for which θL = θS, the slope of the FF curvature becomes zero for both the bottom side and the top side placed MOEs, in accordance to the example of an optical plane-parallel plate. For a MOE placed on the top side functional values of the ray mapping function which are located below the dotted black line (θL’ < θS) give reason for negative values of the slope, whereas for a MOE placed on the bottom side a negative slope can be expected for functional values located above the dotted black line (θL’ > θS).
These general statements about the ray mapping function together with the restrictions of the transformation algorithm are forming a set of instructions for the FF-MOE design of [16–18] based on the ray mapping step:
The position of the ray mapping function in relation of the black line determines the favored placement of the FF surface on the substrate, either top side or bottom side. Functions which are above the black line should be designed as bottom side MOEs (e.g., the FF curvature with DHR = 4) and functions below the black line as top side MOEs (e.g. the FF curve with DHR = 1).
In case that the ray mapping function intersects with the black line the slopes of resulting FF curvature will have both positive and negative values, regardless of the arrangement of the FF-MOE on the substrate (e.g. the cases with DHR = 2, 3 and 3.41). In this case the most suitable orientation has to be chosen, which is minimizing the value of the thickness of the FF-MOE caused exclusively by the positive curvature of the FF curvature. When the value is smaller than the pre-defined maximal thickness, the FF-MOE can be designed within the given parameters by applying the transformation algorithm for the negative slopes of the FF curvature.
For a maximal thickness of, e.g., 50 µm and a top side orientation of the MOE this condition is fulfilled for a DHR of 2 while placing the MOE on the bottom side allows for a much larger DHR value of 3.41.
3. Experimental
In order to demonstrate the functionality of the FF-MOE design procedure an array of FF-MOEs allowing for a DHR value of 3.41 (in case that the circular irradiance distributions are not overlapping) was experimentally fabricated and implemented into a demonstrator box mimicking a direct-lit luminaire.
Fig. 5 Plot of the calculated FF curvature (red) arranged on the bottom side of a PMMA plate (Bottom side of the PMMA plate in green, top side in blue) after applying the sequential pointwise FF design procedure. The light source is located at z = 0 and is centrically placed beneath the FF-MOE. The target plane is located at z = 10.
Fig. 6 a. ,b,c,d) Top side pictures of the different components of the direct lit luminaire demonstrator box. (a) Bottom part of the box, including the circuit board with 14 LEDs arranged in a hexagonal array. b) Spacer component between source plane and FF plane of the box including the PMMA plate with the FF-MOEs (c) Spacer component between FF plane and target plane of the box and the diffusing film. (d) Top side of the Box. (e) Schematic illustration of the arrangement of the different components shown in (a-d)
4. Simulation and experimental results
Ray-tracing simulations were conducted with the commercial software ASAP (Breault Research) using a refractive index of n = 1.512 for the MOE material (resin) and n = 1.4903 for the PMMA substrate for an optical wavelength of 550 nm. The source and the target plane where hosted in air (n = 1), the position of the target plane was chosen to be on the bottom of the diffusing film for monitoring the achieved irradiance distribution. In every simulation we used the FRESNEL AVE option for averaging the polarization of the rays and the SPLIT MONTECARLO option which is selecting the direction of each ray after intersecting an object randomly according to the optical properties of the object, where the probability that a given direction is selected is proportional to the flux that would actually reflect, refract, or scatter into that direction. The resulting ray data (flux, position and direction) of incident rays on the target plane were imported and evaluated in MATLAB. The target area was subdivided into 201 x 201 pixels, where xm and yn are indices for the x- and y-position of the pixels (m,n = 1, 2, …, 201 = M,N). By assigning the rays in dependence of their position to their respective pixels (xm,yn) and accumulating their flux, an array of the intensities I(xm,yn) in the target area can be obtained. In order to evaluate the degree of uniformity of the irradiance distribution on the target area, the coefficient of root mean square error (the CV(RMSE) value) was calculated by using the following equation:
where IMean is the arithmetically averaged intensity value calculated from all I(xm,yn). In this definition a smaller value represents a higher degree of uniformity. CV(RMSE) is a common measure in image processing, in which the reciprocal ratio of the CV(RMSE) value is indicating the signal to noise ratio of an image [3].In the first part of the simulations the optical functionality of a single FF-MOE was investigated. The simulation model was composed of a single 0.5 x 0.5 mm2 LED light source, the PMMA substrate either with or without the FF-MOE and an absorbing surface at the position of the target plane. A sufficient large number of rays (10 millions) was used for the simulations, to minimize the influence of statistical errors on the CV(RMSE) values.
Fig. 7 Simulated irradiance distribution within the target area for different simulation settings: a) Simulation with FF-MOE b) Simulation without FF-MOE, c) Simulation results considering only rays which were refracted at the artificial edges of the FF MOE, d) Simulation result without considering rays which were refracted at the artificial edges.
For the second simulation setting the whole demonstrator box including an array of the FF-MOEs was investigated. All relevant optical parts of the direct lit luminaire box system were included in the simulation model and the whole luminaire box was simulated. For these simulations 2 million rays per light source were used, leading to a total of 28 million rays for the whole simulation setting.
Fig. 8 Simulated irradiance distribution on the bottom side of the diffusing film without considering the effects caused by the artificial edges. a) Irradiance distribution without FF-MOEs, b) Irradiance distribution with FF-MOEs.
As one can see from the simulation results, the reference simulation without the FF MOE (Fig. 8a) causes a smaller CV(RMSE) = 0.511 value compared to the case of the reference simulation of Fig. 7b with CV(RMSE) = 0.767. This effect is caused by the overlapping of the irradiance distributions of the individual LED light sources arranged in an array, which is increasing the degree of uniformity in the target area. The efficiency of the optical system, defined by the percentage ratio of the radiant flux emitted from the light sources and the radiant flux that hits the target, was determined to be 93.2% in case that no MOEs are used.
On the other hand the irradiance distribution caused by the FF–MOEs (Fig. 8b) now has a larger CV(RMSE) value of 0.135 compared to the case of a single FF-MOE (Fig. 7b) with CV(RMSE) = 0.043. The reason for this is again the overlap between the different circular irradiance distributions, which creates areas of higher and lower intensities in the intermediate areas that in this case worsen the CV(RMSE) value. As one can see from Fig. 8b, the tradeoff between areas of higher and lower intensities in the intermediate areas was chosen in a way that the aberrations from the mean intensities for areas with lower and higher intensities are minimized for both. However, the efficiency of the luminaire box system was determined to be 86.2% when using the PMMA plate with the FF-MOEs. The observed decrease of the efficiency compared to the reference simulation can be explained by the fact that without the FF-MOEs a lower amount of rays are reflected towards the circuit board and the side walls of the box, which are partially absorbing.
In order to get an impression on the performance of the FF-MOE, Fig. 9
Fig. 9 Photos of the illuminated top surface of the direct-lit luminaire box system: a) with PMMA plate and without the FF-MOEs) PMMA plate containing the FF-MOEs.
For the setting in Fig. 9a a PMMA plate without FF-MOE was used. As one can see by comparing Fig. 9a and Fig. 8a the experimental results are qualitatively in a very good accordance with the simulation results.
For the photo in Fig. 9b a PMMA plate with FF-MOEs was used to illuminate the top surface. Except of some minor deviations, a qualitative good agreement between experimental and simulation results (Fig. 8b) can be observed. One can clearly see the functionality of the FF-MOE in accordance to the simulation results of Fig. 8b, redirecting the light of the respective light sources into circular irradiance distributions on the target plane. The minor deviations between simulation and experimental results are mainly concerning the intensity peaks in the center of the irradiance distributions caused by the respective FF-MOEs and a small variation of the diameter of the different circular intensity distributions. In order to investigate the reasons for these deviations further, besides studies on the impact of the artificial edges also studies on tolerance analysis were performed. They highlighted that also minor deviations of the distance between the source and the FF plane (h) or the distance between the source and the target plane (H) of the demonstrator may be apparent reasons for these effects.
5. Summary and discussion
In this study we have shown the applicability of FF-MOEs with a height of 50 µm for direct-lit applications with high DHR values. The FF-MOEs were designed for redistributing the light emitted by the LEDs into circular shaped homogenous irradiance distributions on a diffusing film at the top surface of a demonstrator set-up mimicking a direct-lit luminaire. By using this approach it was possible to lower the CV(RMSE) value of the irradiance distribution of a single LED on the diffusing film from 0.767 to 0.043 with a decrease of the maximal intensity peak of approximately 65%, indicating a very homogenous irradiance distribution for a DHR value of 3.41. Comparing this value with others from literature for voluminous FF-lenses (CV(RMSE) = 0.0364) [6], the result is appealing.
Furthermore the implementation of the presented FF-MOEs in a direct-lit luminaire box system with stackable components was presented. Simulation results of the system indicated a decrease of the inhomogeneity of the irradiance distribution on the diffusing film located at the top surface of the box to a CV(RMSE) value of 0.135 with an optical efficiency of 86.2% compared to CV(RMSE) = 0.511 and an optical efficiency of 93.2% for the reference case without FF-MOEs. Photos of the illuminated top surface of the direct-lit luminaire box showed a good qualitative agreement between experimental and simulation results. The deterioration of the homogeneity of the irradiance distribution in the direct-lit luminaire box system with FF-MOEs in comparison to the irradiance distribution of a single FF-MOE is caused by the overlapping of the circular shaped irradiance distributions of the different FF-MOEs inside the direct-lit luminaire. One possible approach to overcome this weakness is further research for realizing flat FF-MOEs with non-circular shaped irradiance distributions that can be placed one next to the other without overlapping areas. In this case also for the LED-to-LED distance the same value as for the diameter of the irradiance distribution can be chosen since no overlapping parts are required.
Nevertheless, the proposed approach of flat FF-MOEs has a huge potential for improving the DHR value of direct-lit applications and to replace conventional or voluminous FF optics in suchlike systems, which is an important step towards the realization of direct-lit luminaires with reduced heights and low weights.
The authors would like to thank L. Kuna, C. Auer, and M. Postl for helping with the fabrication of the experimental set-up.
Funding
Austrian BMWFW, Research Studio Austria program of the Austrian Research Promotion Agency (FFG), project “Green Photonics” (844742).
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