## Abstract

Freeform (FF) optical elements offer new opportunities in many fields of applications since they allow the generation of tailored irradiance distributions, which would be difficult to achieve using conventional optical elements. However, FF optical elements are very challenging with respect to mastering and replication, for which techniques like diamond turning, milling, grinding and polishing are common methods for the manufacturing of the tools required for injection molding, which is a common method for fabrication. In contrast, the possibility of mastering FF micro-optical elements (FF-MOEs) with a laser direct writing method and their large-scale fabrication in roll-to-roll processes offers a cost-effective alternative. Still, for such production techniques, the maximum height of the FF-MOEs must be limited, which requires new design strategies. On the other hand, besides producibility, such ultrathin FF-MOEs also allow for new strategies for the integration of optical elements in miniaturized systems and products. Here, we present an improved design concept for extremely thin FF-MOEs with a confined maximal structure height of 50 µm. They allow for uniform illumination in ultrathin direct-lit based luminaire systems with a DHR (distance between the LEDS : height of the system) ratio of 3 by seamlessly stitching the hexagonal-shaped irradiance distributions provided by the individual FF-MOEs. The applicability of the as-designed FF-MOEs is demonstrated by optical ray-tracing simulations of a simple direct-lit luminaire consisting of an array of 14 LEDs and a target plane in a distance of 10 mm. The simulation results confirm a very high degree of uniformity for the overall irradiance distribution on the target plane.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Due to their remarkable advantages, light-emitting diodes (LEDs) based luminaires are more and more displacing traditional light sources. LEDs do not only benefit from technical advantages like a long lifetime, energy saving aspects, a high reliability, or their compact size [1–4], the possibility for the precise control of the lighting parameters also allows to exploit the benevolence of non-visual effects of light to support the physiological behavior of humans, plants and even animals [5]. However, the generation of defined light patterns with LEDs poses a challenge: Rectangular light patterns with a uniform light distribution are the most desired ones in several LED applications such as backlight units (BLU) in liquid crystal displays (LCDs) [6–10] as well as in general and architectural lighting applications [11–13]. Contrarily, the intrinsic light patterns of most LEDs are circularly symmetric with a non-uniform irradiance distribution. Therefore, appropriate secondary optical elements are essential in order to achieve the desired light patterns.

There are two main strategies in order to realize uniform irradiance distributions of the BLUs of liquid crystal displays: direct-lit LED backlights and edge-lit LED backlights. A direct-lit LED backlight, which uses a bottom side arrangement of the LEDs, has in comparison to an edge-lit backlight several advantages such as a scalable size and the ability of 2D local dimming, which allows for an enhanced contrast ratio and a decreased power consumption [14]. Usually, the LED light sources of a direct-lit BLU are arranged in an array and the denser the LEDs are arranged in this array, the higher is the uniformity of the irradiance on a target plane. However, the maximal density of LEDs is limited by the costs (for a large number of individual LEDs) and the available space. Therefore, it would be advantageous to increase the distance between the individual LEDs (and thereby to lower the number of LEDs) while keeping the uniformity of the irradiance distribution high.

The key parameter for LED-to-LED spacing in direct-lit backlights is the distance-to-height ratio (DHR), defined as the ratio of the distance between two adjacent LEDs and the height (or the thickness) of the BLU. Typically, a large DHR value is advantageous [6–11,13,15], which can be achieved by using additional optical elements [6,11].

In this context, the field of freeform (FF) optics is of special interest for direct-lit applications [10,16–18], like BLUs or luminaires for general lighting. The light refracted at the surface of the FF lens can be redistributed into a uniform irradiance pattern on a target plane where a diffusor element is placed which allows for the generation of a uniform radiance distribution by homogenizing the propagation angles of the incident light. For example, in (ref. [9]) a DHR value of 3 could be realized with a very high degree of uniformity (CV(RMSE) = 0.0364). Additional examples in this regard can be found in [19] and [20].

In our previous studies [21–24], we discussed the usability of artificially intersected FF micro-optical elements (FF-MOEs) for direct-lit applications. The restricted thickness of the FF-MOE structure emphasizes the use of cost-effective manufacturing methods like laser direct write approaches for mastering [23] and roll-to-roll processing for large area manufacturing of such optical elements, countering the problems of fabrication and space requirements of voluminous FF-optics in suchlike applications. In addition, the compact size of the FF-MOEs also allows keeping the overall height of direct-lit luminaires comparably low. Based on a detailed insight into the theoretical background of the ray-mapping process of such FF micro-optics, we could show that by an appropriate conceptual design and arrangement of FF-MOEs with a maximal height of 50** **µm, a DHR value of 3.41 is achievable in a direct-lit luminaire [24]. Unfortunately, the simulation results indicated a decrease of the uniformity of the irradiance distribution of a direct-lit luminaire system with an array of FF-MOEs in comparison to the irradiance distribution of a single FF micro-optical element. This deterioration of the homogeneity for the luminaire was due to the circularly shaped nature of the irradiance distributions of the individual FF micro-optical elements and the intended overlapping of the irradiance distribution of nearby elements.

In this study, we use optical ray-tracing simulations and their analysis by MATLAB to derive a method for the design of FF micro-optical elements, which show enhanced uniformity of the irradiance distributions also in direct-lit luminaire applications. The overlapping of the irradiance distributions of nearby LEDs is avoided by designing FF micro-lenses allowing for hexagonal radiation patterns, which can be seamlessly stitched together to create ultrathin and large sized direct-lit luminaires with large DHR values and uniform irradiance distributions.

## 2. Methods and setting

#### 2.1. Simulation methods

The calculations of the geometrical designs of the FF-MOEs and the analysis of the uniformity of the generated irradiance distributions were performed with the commercial program MATLAB. For the optical simulations of the FF-MOEs, ray-tracing simulations were conducted with ASAP (Breault Research Organization). For each of the ray-tracing simulations, the FRESNEL AVE option for averaging the polarization of the rays and the SPLIT MONTECARLO option, which selects the direction of each ray after intersecting an object randomly in accordance with the optical properties of the object, were applied. The probability for such a selection of a direction is proportional to the flux that actually would reflect, refract, or scatter in that direction. The design of the FF-MOE and all simulations were performed for a wavelength of 550** **nm. For the active area (emitting surface) of one LED die a square with a size of 0.5** **mm x** **0.5** **mm was used. It was assumed that the rays emitted by the LEDs correspond to a Lambertian radiation intensity distribution. The target plane and the LED dice are located in an air (n = 1) environment, the material of the substrate carrying the FF-MOE structures was chosen to be PMMA with a refractive index of n** **=** **1.4903 at the wavelength of 550** **nm. The material of the FF-MOE structure itself was chosen to be of polyurethane acrylate, a resin material suitable for UV-based imprinting, with a refractive index of n = 1.512 at the wavelength of 550** **nm.

During the ray-tracing simulations, any ray impinging on the target plane was stopped and their respective fluxes, positions and propagation directions were recorded. The resulting ray data files were imported and evaluated in MATLAB. The target area was divided into M x N pixels, where x_{m} and y_{n} are the coordinates for the x- and y-positions of the pixels with the indices (m = 1, 2,** …**.,M and n = 1, 2,** …**.,N). By assigning the rays in dependence of their positions to their respective pixels (x_{m},y_{n}) and cumulating their fluxes, an array of intensity values I(x_{m},y_{n}) at the target area can be generatred, which represents the irradiance distribution on the target plane.

In order to evaluate the degree of uniformity of the irradiance distribution on the target plane, the coefficient of the root mean square error (the CV(RMSE) value) was calculated by using the following equation:

_{Mean}is the arithmetically averaged intensity value calculated from all I(x

_{m},y

_{n}) values. 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 [6].

In some evaluations, we used a pixel averaging function to suppress statistical errors in the irradiance distributions caused by the finite number of rays in the ray-tracing simulation. The subsequent equation is averaging the intensity values of the pixels with the intensity values of adjacent pixels and is used interactively. An iterative step of the averaging function can be written as:

For all irradiance distributions, for which the average function was used, the CV(RMSE) values were calculated for the pixel averaged and non-averaged irradiance distributions to determine the influence of the average function on the respective CV(RMSE) values.

This study mainly focuses on the application of the FF-MOEs in an array arrangement, for which the superposition of the resulting irradiance distribution of one FF-MOE with the irradiance distributions of adjacent FF-MOEs is an essential parameter for the overall uniformity of the irradiance distribution on the whole target plane. Ideally, it should be possible to assemble the individual irradiance distributions from each of the FF-MOEs seamlessly on the target plane. In some of the simulations, the ray data of the simulated irradiance distributions were additionally duplicated six times and laterally shifted to the center positions of adjacent FF-MOEs on the target plane. The superposition of these adjacent irradiance distributions on the target plane mimics an optical simulation of the irradiance distribution of a FF-MOE that is located in the center of a hexagonal arrangement of seven FF-MOEs. This method makes it possible to save computer resources and to use a higher number of rays in the ray-tracing simulations in order to investigate and optimize the irradiation distribution of a single FF-MOE taking into account the superposition with adjacent FF-MOEs.

The results of a simulation of the overall system considering an LED array consisting of 14 LEDs and their respective FF-MOEs are discussed in the section on array arrangement.

#### 2.2. Settings

The techniques and procedures discussed in this study in order to improve the uniformity of the irradiance distribution of arrays of FF-MOEs in direct-lit luminaires with a large DHR value are in principle universally valid. Still, in order to be able to evaluate the enhancement of uniformity quantitatively, a concrete application case is used in this study, which is similar to the setting of the direct-lit luminaire set-up investigated in (ref. [24]). The use of the CV(RMSE) values as a measure for the uniformity of the irradiance distribution on the target plane, as it still was done in (ref. [24]) allows to compare the results of this study with the previous one directly.

Figures 1(a) and 1(b) show a schematic illustration of the investigated setting in the xy-plane: The FF-MOEs with a maximal diameter of 8 mm are designed to redistribute the light emitted by 0.5 mm × 0.5 mm LED dice into hexagonally shaped irradiance distributions. A single irradiance distribution has a circumradius of 17.378 mm and an inscribed base-circle radius of 15.05 mm on the target plane, which is located at a height of 10 mm above the emitting area of the light source (Fig. 1(a)). The individual irradiance distributions are stitched together in a way that a larger sized surface can be covered seamlessly. The resulting array has a lattice constant of 30.1 mm (distance between the light sources) resulting in a distance to height ratio (DHR) of 3.01 for the whole luminaire (Fig. 1(b)). In comparison with typical direct-lit luminaire setups, the FF-MOE approach allows to realize an extremely thin set-up and large DHR values.

Figure 1(c) shows a schematic illustration along the xz-plane of one sector of a FF-MOE (see Fig. 3) that is used for the luminaire setup. The target plane is located at a height of 10 mm and the bottom side of the 1 mm thick substrate carrying the FF-MOE is located at a height of 2 mm above the top surface of the LED die (location of the emitting area). The height of the FF-MOE is confined to a maximal size of 50 µm and the FF-MOE is located on the bottom side of the substrate, facing the LED die (for the benevolence of such an arrangement see (ref. [24])). The length of the FF-MOE sector L_{S} can vary for the different sectors of the FF-MOE (see Fig. 3). The length of the target irradiance distribution on the target plane L_{L} represents a radial cut of the hexagonally shaped irradiance distribution and varies from 15.05 mm to 17.378 mm for the different sectors of the target distribution (see Fig. 3).

#### 2.3. Flowchart of the design process

Figure 2 shows a design flowchart for the whole design and optimization procedures for the final design of the FF-MOEs.

## 3. FF algorithm

The simulation procedure for the basic design of the FF-MOEs has already been published in detail elsewhere [21–24]. Briefly, it is based on a two-dimensional freeform approach for point sources, composed of three consecutive steps: a ray-mapping step, a geometry step and as the final step the conversion of the FF curves into 3-dimensional structures [21–24].

Figures 1(c) and 1(d) are showing all relevant parameters for the calculations during the ray-mapping step. A point source representing the LED die is located at the source plane and emits light with a specific radiant intensity distribution I_{S}(θ) towards the target plane. The topology of the FF-MOE at a certain point B directs an incident ray with an angle θ_{S}, which is measured at the z-axis, to a corresponding point R. If the point R is connected to the origin point of the light source, the polar angle θ_{L} is obtained (see Fig. 1(d)). In this way the given radiant intensity distribution of the source I_{S}(θ) can be converted into a radiant intensity distribution I_{L}(θ), which generates the desired irradiance distribution on the target plane [21–24].

The correlation between θ_{S} and θ_{L} for modifying the radiant intensity distribution of a source with a Lambertian radiant intensity pattern I_{S}(θ) = cosθ hitting the circular area of the FF-MOE with a radius r = L_{S} into a radiant intensity distribution I_{L}(θ) = 1/cos^{3}θ, which generates an uniform irradiance distribution within a circular area with a radius r = L_{L} on the target plane, has already been published in (ref. [22]) and can be expressed as:

_{S_MAX }= arctan(L

_{S}/h) expresses the maximal polar propagation angle captured by the respective FF-MOE and θ

_{L_MAX }= arctan(L

_{M}/H) expresses the maximal polar propagation angle of a ray from the source that would hit the area in which the irradiance is distributed on the target plane without FF-MOE.

For the calculation of the FF-MOEs discussed in this study, a more general formulation is needed. This formulation has to be able to calculate the modification of a radiant intensity distribution of a source hitting a circular ring area within the area of the FF-MOE with an inner radius r_{S} = h·tanθ_{S1} and an outer radius R_{S} = h·tanθ_{S2} into a radiant intensity distribution, which generates an uniform irradiance distribution within a circular ring area with inner radius r_{L} = h·tanθ_{L1} and an outer radius R_{L} = h·tanθ_{L2} on the target plane:

Equation (4) expresses the energy conservation by connecting the radiant intensity hitting the area on the FF-MOE with the respective area on the target plane and is used to calculate adequate normalizing factors k_{S} and k_{L} in mutual dependence for the radiant intensity distribution functions I_{S}(θ) = cosθ and I_{L}(θ) = 1/cos^{3}θ. Solving Eq. (5) with the calculated normalizing factors k_{S} and k_{L} yields the functional correlation between θ_{S} and θ_{L} for the considered angular ranges of θ_{S1} ≤ θ_{S }≤ θ_{S2} and θ_{L1} ≤ θ_{L }≤ θ_{L2}:

The basics of the procedure for the geometry step have already been published elsewhere [19–22]. 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. The pointwise determination of the FF curve allows the definition of two threshold values representing the minimal and the maximal z-coordinates of the points constituting the FF curve in order to confine the maximal height of the FF structure to a defined value. When a point of the FF curve falls below the lower threshold, it is artificially shifted towards the higher threshold value during the geometry step. By doing so, an artificial edge is created in the FF curve, which may lead to deviations of the pre-defined irradiance distribution on the target plane. In (ref. [22]) a peak of a bit higher intensity was observed in the center of the irradiance distributions, which was caused by these artificial edges in the FF curves.

In (ref. [24]), the conversion of the FF curve into a 3-dimensional (3D) structure was done by rotating the curve around the perpendicular z-axis for 360°. By doing so, the generated FF-MOEs can only generate circularly shaped irradiance distributions.

For the generation of FF-MOEs with, e.g., square or hexagonally shaped irradiance distributions, the area of the target distribution is sliced into individual sectors with different lengths. The lengths of the sectors depend on the shape of the pre-defined target distribution (Fig. 3(a) the distance between the respective positions on the envelope of the irradiance distribution and the center of the distribution). For each of the sectors separate FF curves are calculated (Fig. 3(b), (c)), see (ref. [21]). The 3D FF-MOE is generated by extruding the FF curves rotationally around the perpendicular z axis by the sectorizing angle φ and stitching the resulting FF sectors together to a 3D FF-MOE (Fig. 3(d), (e)). By using rotationally extruded geometries the shape of the resulting irradiance distribution shows minor deviations from the desired form (see Fig. 3(a) red, blue, green and yellow areas). This error is governed by the sectorizing angle or the number of sectors used for the generation of the FF-MOE and choosing an adequate number of sectorizings (e.g. 96 sectors) ensures a satisfying result of the irradiance distribution observed by the eye (see Fig. 4(b)).

## 4. Problems with the approach

The different lengths of the individual sectors cause an obvious problem, since the lengths of the calculated FF curves L_{S} = h·tan *θ*_{S_Max} are determining the amount of captured radiant flux from the LED die (see Fig. 1(b) and 1(c). When all FF curves have the same length (Fig. 3(b), (d)), the sectors of the FF-MOE (Fig. 3(d) red and blue areas) are redistributing the same amount of radiant intensity onto different sized areas on the target plane (Fig. 3(a) compare red and blue area), causing non-uniformities within target area of the irradiance distribution. On the other hand, the use of a two-dimensional approach for calculating the FF-curves is excluding a reallocation of the radiant intensity between the different irradiance sectors because the FF-MOEs alter only the polar propagation angle θ but not the azimuthal propagation angle θ of the rays. One approach to address this problem is to adapt the radiation intensity captured by each of the FF-MOE sectors to the size of the areas of their corresponding sectors within the target irradiance distribution by an adjustment of *θ*_{S_Max}, the maximal polar angle captured by the FF curve for each sector of the FF-MOE:

Figure 4 shows the simulated irradiance distribution at the target plane (Fig. 4(b),(e)) for two sectored FF-MOEs, composed of 96 sectors, with an adjustment factor F = 0 (Fig. 4(a)) and an adjustment factor F = 1 (Fig. 4(d)). As one can see from Fig. 4(b), the irradiance in the target area is not uniformly distributed because the different sectors within the irradiance distribution have different intensity levels. Figure 4(c) shows the irradiance distribution on the target area superimposed with irradiance distributions of identical neighboring FF-MOEs, see chapter 2.A, Simulation. The resulting irradiance distribution shows a CV(RMSE) value of 0.07 without averaging and a value of 0.061 when applying the averaging algorithm 4 times (see chapter 2.A). As one can see from Fig. 4(e), the irradiance in the center of the target area looks more uniformly distributed in comparison with Fig. 4(b), however strong intensity peaks can be observed at the boarders of the target area. This means the length reduction of some sectors of the FF-MOE causes a higher proportion of unguided light, which is not affected by the individual sectors of the FF-MOE and overlaps with the desired uniform irradiance distributions on the target plane. Figure 4(f) shows the irradiance distribution of the target area superimposed with irradiance distributions of identical neighboring FF-MOEs (see chapter 2.A). The resulting irradiance distribution shows a CV(RMSE) value of 0.124 after applying the averaging algorithm 4 times, which states an even worse result in terms of homogeneity compared to the FF-MOE with F = 0. However, by using values between 0 and 1 for F, the amount of unguided light, which is redistributed with additional FF geometries in the following, can be controlled.

## 5. Redistribution of unguided light

The problem of the large proportion of unguided light can be addressed by redistributing the unguided light with additional FF geometries. The concept of this approach is based on the implementation of additional structures in order to direct light into secondary target areas outside the primary target area. These additional FF geometries allow to distribute the otherwise unguided light within these secondary target areas. By choosing suitable geometries for these additional target areas and using periodically arranged FF-MOEs, it is possible to achieve an area-wide superposition of the secondary target areas with the primary target areas of adjacent FF-MOEs in order to achieve an overall uniform irradiance distribution.

Figure 5 shows a schematic representation of this concept for FF-MOEs with hexagonally shaped primary irradiance distributions: The sectored FF-MOE generates a hexagonal intensity distribution (Fig. 5(b) red area). Further FF-structures redistribute the light that is not captured by the angular adjustment of the individual sectors of the FF-MOE into triangular shaped secondary irradiance distributions (Fig. 5(b) blue areas). The chosen form of secondary target distributions has the advantage that the equilateral triangles of adjacent FF-MOEs can be superimposed seamlessly within the primary hexagonal irradiance distribution (see Fig. 5(c) yellow area). This approach even allows the use of non-uniform irradiance distributions within the hexagonal and the triangular target areas, since only by the superposition of both distributions is intended to generate a uniform irradiance distribution on the target plane.

Figure 6(a) shows a sectored FF-MOE, composed of 96 sectors, with FF-structures (red) for generating a hexagonal shaped primary irradiance distribution and additional FF-structures (blue) for generating triangular shaped secondary irradiance distributions. The secondary FF-structures are calculated with the same algorithms as the primary FF-structures and therefore are based on rotationally extruded FF-curves too. For the calculation of the primary FF-structures of the FF-MOE an adjustment factor of F = 0.55 was used in Eq. (7) to calculate the adjusted angles $\theta _{S\_MAX}^n$, which turned out to be the most favorable value in terms of optimizing the CV(RMSE) value in the subsequent optimization of the FF-MOE. The ray-mapping for the respective sectors n of the primary FF-structure were calculated by using the adjusted angles $\theta _{S\_MAX}^n$ in Eq. (3) for ${\theta _{S\_MAX}}$ and $\theta _{L\_MAX}^n = \textrm{arctan}({L_L^n/H} )$ for ${\theta _{L\_MAX}}$. The ray-mapping for the secondary FF-structure of the FF-MOE was calculated by using Eq. (6) with ${\theta _{S1}} = \theta _{S\_MAX}^n$, $\; {\theta _{S2}} = \theta _{S\_MAX}^{MAX}$, ${\theta _{L1}} = \textrm{arctan}({L_L^n/H} )$, and ${\theta _{L2}} = \textrm{arctan}({L_{L2}^n/H} )$, where $L_{L2}^n$ is the length of the sector from the center to the respective points forming the curve that envelopes the star shaped irradiance distribution.

Figure 6(b) shows the simulated irradiance distributions on the target plane after applying the averaging algorithm 4 times, when using the FF-MOE depicted in Fig. 6(a). For a better understanding of the result, the rays which impinging on the target plane are divided into three categories: Rays, which interact with the primary FF-structure, rays, which interact with the secondary FF-structure and rays, which missed the FF-MOE. The rays of these categories are plotted separately in Fig. 6(d)–(f). As can be concluded from the results, the secondary FF-structure prevents a large part of the unguided light, caused by the shortening of certain sectors of the FF-MOE, from overlapping with the hexagonally shaped primary irradiance distribution on the target plane (compare Fig. 6(b) and Fig. 4(e)). This part of the unguided light is now redistributed into the triangular shaped areas at the boarders of the primary irradiance distribution on the target plane (see Fig. 6(e)) where it overlaps with the light, which is missing the FF-MOE (see Fig. 6(f)).

Figure 6(c) shows the irradiance distribution of the target area generated by the FF-MOE of Fig. 6(a) superimposed with irradiance distributions of identical neighboring FF-MOEs. The resulting irradiance distribution shows a CV(RMSE) value of 0.041 without averaging and a value of 0.031 when applying the averaging algorithm 4 times. From these results, it can be concluded that the additional use of secondary FF-structures significantly improves the homogeneity of irradiance distribution on the target surface.

## 6. Iterative improvement of the uniformity of the irradiance distribution

For a further increase of the uniformity of the irradiance distribution, an iterative ray-mapping process of the primary irradiance distribution is used to adjust the primary and the secondary irradiance distributions. However, due to the rotational extrusion of two-dimensional FF-curves the FF-structures are not capable to change the azimuthal direction of incident rays. For this reason, only the polar irradiance distribution within the different sectors of the irradiance distribution can be adjusted with this approach. One cycle of this iterative algorithm is composed of 3 steps: First, calculating the FF-MOE based on the ray-mapping for every sector. Second, conducting a ray-tracing simulation with the current shape of the FF-MOE and importing the achieved irradiance distribution of each sector in MATLAB. Third, conducting the following calculations to adjust the ray-mapping for the next iterative step. In MATLAB the irradiance distribution is subdivided into the same irradiance-sectors used for the calculation of the FF-MOE. A cross section is placed through the middle of each irradiance-sector, resulting in a function ${I_{org}}(\theta )$ for every irradiance-sector that describes the intensity along the cross section depending on the polar angle *θ*. Subsequently, the functions are converted into step functions $I_{org}^k$ with K equivalent angular steps. Each step is defined by a start $\theta _{L\_start}^k$ and an end $\theta _{L\_end}^k$ angle, with $\theta _{L\_start}^{k = 1} = 0$ deg, $\theta _{L\_end}^{k = K} = arc\; \textrm{tan}({L_L^n/H} )$ and $\theta _{L\_start}^{k = K} = \theta _{L\_end}^{k = K - 1}$, by averaging the intensity enclosed between these ranges for each angular step. The same procedure is conducted with the same steps K for the superimposed irradiance distribution (see the chapter on the methods) to obtain the step functions $I_{imp}^k$ for the irradiance-sectors of the superimposed irradiance distribution.

The following procedure is conducted for every sector n of the FF-MOE and their allocated irradiance sectors. For every step k of $I_{imp}^k$ adjustment factors $A_{iter}^k$ are calculated for the different iteration steps using the following equations:

_{S}and k

_{L}calculated using Eq. (4) with ${\theta _{S1}} = 0$ deg, ${\theta _{S2}} = \theta _{S\_MAX}^n$, with ${\theta _{L1}} = 0$ deg and ${\theta _{L2}} = arc\; \textrm{tan}({L_L^n/H} )$ deg. Equation (10) is used with $\theta _{S\_start}^{k = 1} = 0$ deg to calculate $\theta _{S\_end}^{k = 1}$. This process is sequentially repeated for every angular step k with $\theta _{S\_start}^k = \theta _{S\_end}^{k - 1}$ to obtain the angular steps on the FF-MOE $\theta _{S\_start}^k$ and $\theta _{S\_end}^k$ for their corresponding angular steps $\theta _{L\_start}^k$ and $\theta _{L\_end}^k$ of $I_{imp}^k$. By changing the angular steps of the FF-MOE the angular steps of the irradiance distribution are supplied with either more ($A_{iter}^k$ < 1) or less ($A_{iter}^k$ > 1) intensity, depending on the respective adjustment factors $A_{iter}^k$. Subsequently Eq. (11) is used to calculate the correlation between

*θ*

_{S}and

*θ*

_{L}for every angular step k.

Figure 7 shows the results of the iterative ray-mapping optimization process using ray-tracing simulations with 20 million rays. The change of the shape of the FF-MOE, the change of the resulting irradiance distributions and the change of the superimposed irradiance distributions are investigated for different iterations of the ray-mapping (Fig. 7(a),(b),c: iter. step 0, Fig. 7(d),(e),f: iter. step 1 and Fig. 7(g),(h),i: iter. step 10). The resolution for the plots of Fig. 7(a),(d),g was set to be 201 × 201 pixels resulting in 8755 pixels (area of the primary irradiance distribution) for the plots of the superimposed irradiance distributions and for the CV(RMSE) calculations. The averaging algorithm was applied for every plot of the irradiance distributions 4 times. The scale bars for the plots of the superimposed irradiance distributions were kept identical to give an impression of the increase in uniformity.

Already at iteration step 1 of the iterative ray-mapping optimization process a significant increase of the homogeneity of the superimposed irradiance distribution can be observed (compare Fig. 7(b) with Fig. 7(e)). The peak of higher intensity in the center of the irradiance distribution caused by the artificial edges in the FF curves is lowered significantly. The CV(RMSE) value for the non-averaged distributions is reduced from 0.041 to 0.032 and the CV(RMSE) value for the averaged distributions is reduced from 0.031 to 0.021.

Figure 7(h) shows the superimposed irradiance distribution after 10 iterations. As one can see most of the local non-uniformities of the superimposed irradiance distribution were reduced and has a CV(RMSE) value of 0.024 for the non-averaged distributions and a CV(RMSE) value of 0.012 for the averaged distributions.

## 7. Array arrangement

In order to investigate the improvements of this approach with respect to the uniformity of the irradiance distribution on the target plane, a full ray-tracing simulation of an array arrangement (see Fig. 1(b)) including 14 LEDs and a target plane in a vertical distance of 10 mm was conducted. For this simulation, 42 million rays (3 million per LED) were used.

Figure 8(a) shows a plot of the simulated irradiance distribution with a resolution of 301 × 301 pixels on the target plane for a reference simulation for which no optical elements were applied between the LEDs and the target plane. Due to the finite surface of the target plane only 91.4% of the emitted radiant flux is hitting the target plane. As one can see, the high DHR value of 3.01 gives reason for an extremely non-uniform irradiance distribution on the target plane.

Figure 8(d) shows a plot of the simulated irradiance distribution on the target plane when the optimized FF-MOE of Fig. 7(i) is used. The simulation results indicate that now 82% of the emitted light is hitting the target plane. The loss of intensity compared to the reference simulation of Fig. 8(a) can be explained by Fresnel reflections of rays, which are hitting the optical surfaces of the FF-MOEs and therefore are partially reflected back to the LED light sources. Here, these reflected rays are considered as lost. However, in a real device, efficiency can be further optimized by using a reflective printed circuit board and using reflective surfaces at the outer boundaries of the device to reflect these rays back toward the target plane.

Compared to the reference simulation, the irradiance distribution of Fig. 8(d) indicates an auspicious increase of uniformity of the irradiance distribution, especially in the central region of the target plane. In order to investigate this uniformity more closely, the irradiance distribution on the target plane is divided into different zones by neglecting pixels that are not in the investigated area. (See Fig. 8(b)).

The central region (see Fig. 8(b) region C) is defined by that area on the target plane, which is completely surrounded by adjacent primary distributions of the different FF-MOEs (See Fig. 8(b) regions N). After neglecting pixels which are not in the evaluated area, 12644 pixels are left for the CV(RMSE) evaluation. The yellow region is defined by the area on the target plane where the primary and the secondary irradiance distributions of the different FF-MOEs are overlapping. After neglecting pixels which are not in the evaluated area, 30426 pixels are left for the CV(RMSE) calculation.

Figure 8(e) shows the irradiance distribution inside the yellow area generated from the irradiance distribution of Fig. 8(d). The resulting irradiance distribution already shows a very low CV(RMSE) value of 0.025 when applying the averaging algorithm 4 times and a value of 0.035 without using the averaging algorithm. The evaluation shows that 57.8% of the emitted light is guided into this region. This value seems to be quite low for e.g. lighting applications however the loss of efficiency is depending on the ratio of the yellow region in Fig. 8(b) to the boundary regions. In a real device with a large array of LED dice, this ratio becomes smaller and therefore gives reason for a higher efficiency of the device.

Figure 8(c) depicts the central region of the irradiance distributions of the reference simulation (Fig. 8(a)) and Fig. 8(d) depicts the result of the simulation of the optimized FF-MOE structures. This comparison highlights the increase of uniformity quantitatively. The central region of the reference simulations shows a CV(RMSE) value of 0.400 after using the averaging algorithm 4 times and a value of 0.407 without averaging algorithm. The central region of the simulation with FF-MOE shows a CV(RMSE) value of 0.008 after using the averaging algorithm 4 times and a value of 0.026 without averaging algorithm.

## 8. Conclusion and outlook

In this study, we presented a smart design concept for extremely flat FF-MOEs with a confined maximal structure height of 50 µm for uniform illumination in ultra-thin direct-lit luminaires with very high DHR values. In contrast to our earlier work [22], where FF-MOEs with circular irradiance distribution were used, the algorithms were adapted to enable the calculation of sectored FF-MOEs with hexagonal irradiance distributions. The problem of the higher proportion of unguided light, which is interfering with the generated irradiance distribution, was overcome by the implementation of secondary, triangular target areas outside the primary hexagonal target area, which are overlapping the primary hexagonal irradiance distributions generated by adjacent FF-MOEs. Furthermore, by using an algorithm for an iterative ray-mapping of the primary irradiance distribution, the primary and the secondary irradiance distributions were adjusted to each other in a way that a superposition of the two irradiance distributions is generating a very uniform overall irradiance distribution. Using ray-tracing simulations, it was shown that for LED dice with emitting areas of 0.5 mm x 0.5 mm and the use of a periodical arrangement of hexagonal FF-MOEs CV(RMSE) values of 0.035 for the overlapping areas and values of 0.026 for the central areas can be achieved. Still, these values can be even further reduced to values of 0.025 and 0.008 when an averaging algorithm is used to suppress statistical errors of the irradiance distributions caused by the finite number of rays in the ray-tracing simulation.

The achieved results state an huge improvement in uniformity of the generated irradiance distribution compared with (ref. [24]), where a CV(RMSE) values of 0.135 was found for the use of FF-MOEs with circularly shaped irradiance distributions in the same configuration. Such uniformity values are quite comparable to those which are reported in other studies using voluminous FF-lenses, e.g., (ref. [9]), where a CV(RMSE) value of 0.0364 is reported for a DHR value of 3 or (ref. [25]), where a CV(RMSE) value of 0.03 is reported for a DHR value of 4.

Besides all of this, the presented design concept also highlights a strategy towards ultrathin FF-MOEs in general, which may be a step towards the exploitation of all the benevolence of FF optics also in ultra-miniaturized opto-electronic system set-ups.

## Funding

Austrian BMWFW, Research Studio Austria program of the Austrian Research Promotion Agency (FFG), project “Green Photonics” (844742).

## Disclosures

The authors declare no other conflicts of interest.

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