## Abstract

A feedback method combined with fitting technique based on variable separation mapping is proposed to design freeform optical systems for an extended LED source with prescribed illumination patterns, especially with uniform illuminance distribution. Feedback process performs well with extended sources, while fitting technique contributes not only to the decrease of pieces of sub-surfaces in discontinuous freeform lenses which may cause loss in manufacture, but also the reduction in the number of feedback iterations. It is proved that light control efficiency can be improved by 5%, while keeping a high uniformity of 82%, with only two feedback iterations and one fitting operation can improve. Furthermore, the polar angle $\theta $and azimuthal angle$\phi $is used to specify the light direction from the light source, and the *(*$\theta ,\text{\hspace{0.17em}}\phi $)-(x,y) based mapping and feedback strategy makes sure that even few discontinuous sections along the *equi-*$\phi $plane exist in the system, they are perpendicular to the base plane, making it eligible for manufacturing the surfaces using injection molding.

©2013 Optical Society of America

## 1. Introduction

Light-emitting diodes(LEDs), distinguished for its long life, high reliability, environmental friendly working process and energy saving potential, are used more and more widely not only in areas of special lighting, such as architectural lighting, stage lighting, traffic lighting and sign display, but also in areas of general indoor and outdoor lighting [1, 2]. However, traditional encapsulated LED devices have a Lambertian light distribution, which could not meet various practical requirements of lighting applications. Therefore freeform nonimaging optical systems are developed to redistribute the light rays emitted for LED devices to form prescribed illuminance or luminance distribution on the target area. Research on design method for freeform nonimaging optical systems has been an active field during the last two decades [3–16]. Amongst them the variable separation mapping method is widely used to design freeform optical systems with point sources [8]. Based on energy conservation and Snell’s law, it uses numerical techniques to establish separately the correspondence between variables on the light source and the target plane, ensuring that the optical surfaces could redirect the incident light rays to their corresponding target points. Recently feedback design methods based on variable separation mapping are developed for extended LED sources with complicated illumination patterns [9, 10]. These are simple but super effective method to design freeform optical systems for point or extended LED sources. Nonetheless, there still remain some problems calling for a better solution, including:

- (1)
*Tradeoff between accuracy and speed.*Accuracy in light control adds to the difficulty in the design process. Typically, the freeform surfaces are made with many facets, whether smooth or unsmooth, with each little facet controlling only a proportion of the energy emitted from the source. Hence, calculating the large amount of points on the surfaces in order to obtain a satisfying result becomes challenging, even with the help of advanced softwares. For example, the previous iterative illuminance compensation approach, using the (simulated illuminance)/(desired illuminance) ratio to compensate the distribution, requires several to dozons of feedback iterations for a lens or reflector surface to reach the final desired distribution. Researchers have to figure out the solution to a quick design and overcome the error in freeform surface construction process. - (2)
*Difficulty of manufacturing the surfaces using injection molding.*In the variable separation mapping method and the feedback modification methods based on it, (*u*,*v*)-(*x*,*y*) mapping is commonly used wherein the (*x*,*y*) Cartesian coordinates is used to specify the position on the irradiance plane, while (*u*,*v*) coordinates is used to specify the light direction from the light source, with*u*being the angle between the light ray and the*y*axis parallel to the emitting plane of the LED chip, and*v*the angle that the plane containing the light ray and the*y*axis forms with the*z*axis perpendicular to the emitting plane [8–10]. In order to limit the deviation produced in the surface construction process, a normal vector correction mechanism is employed, and as a result discontinuities are introduced onto the lens surface. As the discontinuous section is not perpendicular to the base plane [8, 10], it can prevent the mould from escaping the lens during injection molding process, as shown in Fig. 1(a) .

- (3)
*Manufacturing defects.*Even if the discontinuous section is perpendicular to the base plane, as shown in Fig. 1(b), manufacturing defects such as surface roughness and obtuse transition between two discrete sub-surfaces exist in discontinuous freeform lenses, which often leads to loss on light control ability, sometimes with up to 30% decrease in uniformity [11]. Though some work has been done to obtain a quite uniform result using sub-surfaces mechanism method with hundreds of pieces of discrete sub-surfaces, it is not a good option for mass production. - (4)
*Actual light control efficiency.*We define light control efficiency (LCE) as the ratio between the energy projected to the target zone and the total energy emitted from the source. It is different from light output efficiency, which includes rays hitting outside the target zone, which ought to be a waste of energy. However, most of the previous reports didn’t provide the LCE value. Their efficiency, sometimes reaching above 90%, is actually light output efficiency.

In this paper, a feedback method combined with fitting technique is proposed to effectively address the problems mentioned above. It reduces the feedback iterations from a dozen to a few while achieving higher LCE as well as light distribution accuracy. Besides, it keeps the number of discrete sub-surfaces at a reasonable level and ensures the discontinuous sections perpendicular to the base plane, which is eligible for the injection molding process.

## 2. Feedback method combined with fitting technique

The process of the proposed method is illustrated in Fig. 2 . It is a development of variable separation mapping design method and feedback modification methods. The framework of the feedback part, which includes mapping establishment, surface construction, simulation and feedback, remains the same as conventional methods, while some modifications are applied during implementation. In addition, a newly introduced fitting technique, combined with the modifications on previous feedback framework, ensures that our method is outstanding.

These modifications and the fitting technique are illustrated below:

(1) *(*$\theta ,\text{\hspace{0.17em}}\phi $)-(x, y) mapping strategy is employed.

In the proposed method, the *(*$\theta ,\text{\hspace{0.17em}}\phi $) coordinates is used to specify the light direction from the light source, with *θ* to be the polar angle between the light ray and the symmetry axis of the light source, and *φ* to be the azimuthal angle between the light ray and *x* axis parallel to the emitting plane of the LED chip. Traditional *(*$\theta ,\text{\hspace{0.17em}}\phi $) *-(x, y)* mapping strategy mostly deals with rotational illuminance requirement [17]. In the proposed method, the target plane could have complicated illumination patterns rather than only a rotational one. In the following, a rectangular region with uniform illuminance distribution is taken as an example, and this region is sampled into cells as shown in Fig. 3
. Take the first quadrant for example, points are equally spaced along x direction below the diagonal line (correspondingly, equally spaced along y direction above the diagonal line), since this scheme can minimize the statistical error of the simulation software through our experiment. *(*$\theta ,\text{\hspace{0.17em}}\phi $) division is employed to construct the lens surface with *equi-*$\theta $ curves on lens corresponding to horizontal lines and vertical lines in Fig. 3 (green dashed lines), and *equi-*$\phi $ curves on lens corresponding to oblique lines with different slope in Fig. 3 (red dashed lines).

After dividing the target plane into cells with specified energy (defined as prescribed energy in later paragraphs), the target-to-source mapping is calculated with two steps, both of which are based on the energy proportion.

Firstly, *equi-*$\phi $ curves are determined. Let ${E}_{prescribed}(i,j)$ be the prescribed energy of the ${j}_{\text{th}}$ cell on the *i th* stripe (space between the *i-th equi-*$\phi $ line and the (*i* + 1)- *th equi-*$\phi $ line on target plane, shown in Fig. 3), ${E}_{stripe}(i)$ the sum of all the cell energy along the *i-th* stripe, while $\phi \left(i,j\right)$and $\phi \left(i+1,j\right)$stand for the $\phi $ value of the two edges of *equi-*$\phi $ stripe on target plane (red dashed lines in Fig. 3), respectively. Then $\phi \left(i,j\right)$ is obtained according to the following equation:

Secondly, $\theta $ values in each *equi-*$\phi $stripe are determined. $\theta $division is more complicated than $\phi $division, since the energy integration of specified solid angle is a quadratic term of $\mathrm{cos}\theta $, expressed as:

This equation is simplified by taking out the *φ-*related term and calculating the integration, to obtain the energy ratio:

*equi-*$\phi $stripe it is on, numerical method can be applied to work out the corresponding $\theta $ value.

(2) A new feedback strategy is employed.

The feedback strategy used here belongs to fixed area matrix method [10]. In the fixed area matrix method, the division of *x or y* on the target plane is fixed during the feedback process. What varies along with the feedback iterations is the prescribed energy assigned into these cells, which corresponds to different *θ* and *φ* angle. Let ${E}_{initial}$ denote the initial prescribed value matrix for each cell as well as the desired energy at last, ${E}_{prescribed\_pre}$and ${E}_{prescribed}$ stand for the previous iteration and current iteration prescribed energy, respectively, ${E}_{simu}$stand for the current energy after simulation, corresponding to ${E}_{prescribed\_pre}$. A strategy function *f(.)* is needed to assign ${E}_{prescribed}$ in the next iteration according to simulation illuminance result,

The feedback function *f(.)* can be expressed in the following form:

Differed from traditional feedback process [9, 10, 15], the desired energy of each cell is varied in each feedback iteration according to light output efficiency. In feedback function (5), the initial energy of each cell is calculated in the initial iteration, assuming that all the energy can be projected to the target plane. In the next several feedback iterations, this initial energy remains unchanged, indicating that the sum of all the energy within target zone ought to be equal to the energy emitted from light source. However, actual LCE changes with each feedback iteration, and always stays less than 1. Hence, the desired energy should change accordingly in order to achieve real uniformity. It can be expressed as mathematical equation by reassigning ${E}_{initial}$ according to the actual efficiency in each feedback iteration:

It is worth noting that feedback iteration doesn’t start from scratch. The initial iteration gives a not so bad simulation result under extended-source simulation, although corresponding to point source case. Thus the prescribed value only needs to be slightly changed, especially after we introduced fitting technique which can lower the illuminance requirement of preceding feedback result. Previous feedback strategy discussed in [18], wherein ${E}_{prescribed}$ is set inversely proportional to ${E}_{simu}$, easily causes sharp oscillation of ${E}_{prescribed}$. It is proved that our feedback function which results in only mild change of ${E}_{prescribed}$ uses only 2 or 3 feedback iterations to get a similar or better result than the previous method.

(3) A fitting technique is employed.

The advantage of *(*$\theta ,\text{\hspace{0.17em}}\phi $)-(x, y) mapping compared with (*u*, *v)*- (*x*, *y*) mapping is that even if discontinuous facet exists, it will be perpendicular to the base plane and hence to be easy-molding. However, according to the work done in [11], discontinuous sections will cause the problem of losing uniformity and efficiency. For lens shape whose corresponding target distribution reaches the requirement, we use fitting method to reestablish the *(*$\theta ,\text{\hspace{0.17em}}\phi $)-(x, y) mappings to smoothen the lens surface and to decrease the number of truncation surfaces.

It is necessary to state that our fitting method is to act not on the three dimensional spatial points but on the *(*$\theta ,\text{\hspace{0.17em}}\phi $)-(x, y) mapping [15]. After obtaining the mapping between *(*$\theta ,\text{\hspace{0.17em}}\phi $)-(x, y), we actually have a data sets containing limited amount of sample points. Taking x or y as dependent variable and *(*$\theta ,\text{\hspace{0.17em}}\phi $) as two dimensional independent variables, then fitting process is implemented for the *(*$\theta ,\text{\hspace{0.17em}}\phi $) -x or *(*$\theta ,\text{\hspace{0.17em}}\phi $) –y data sets, expressed as *x = Fitting(*$\theta ,\text{\hspace{0.17em}}\phi $) and *y = Fitting(*$\theta ,\text{\hspace{0.17em}}\phi $). The algorithm can be referred to [19]. After fitting, a surface *x = f _{1}(*$\theta ,\text{\hspace{0.17em}}\phi $) and

*y = f*$\theta ,\text{\hspace{0.17em}}\phi $) is obtained. The results before and after fitting technique are shown in Fig. 4 .

_{2}(If similar values of *(*$\theta ,\text{\hspace{0.17em}}\phi $) are mapped to very different *x* values in the *(*$\theta ,\text{\hspace{0.17em}}\phi $) -x or *(*$\theta ,\text{\hspace{0.17em}}\phi $) -y mapping, discontinuous sections will appear on the lens surface. Aforementioned fitting process can fit a new surface, which may not pass any of the original points but tend to the mean value. It is just due to this reason that reduction effect in numbers of discontinuous sections can be feasible. As a result, the *equi-*$\theta $ and *equi-*$\phi $ curve on target plane becomes smoother after fitting. In theory, it ought to leverage the light control ability since deviation exists between design and construction. However, through experiment we proved the effect was minor because of compensation effect of feedback method. Instead, the deviation we found out, might lead to decrease in feedback iterations which will be described in the next paragraph.

It is worth noting that the lens surface assumes the role of transforming the light distribution of the LED source to the desired distribution on the target plane. However, the lens surface is constructed using the discrete points created by the mapping [8].Therefore, more points there are in the mapping, higher accuracy of the lens surface and more efficient to achieve the distribution transform. Certainly we can divide the target plane into more cells at the beginning, but it would be time-consuming to construct so many points in every feedback iteration. As an alternative, interpolation at more grid points of *(*$\theta ,\text{\hspace{0.17em}}\phi $) in the fitting after completing the feedback process, can be employed to obtain similar effect. In our work, we inserted 3 times of points along $\theta $ and 4 times along $\phi $ direction.

The method proposed in this paper not only inherits the advantages of feedback process and fitting technique, but also has the advantages of decreasing the feedback iterations and raising the LCE. Fitting technique can cause deviation from original pure feedback result. This deviation, if properly dealt with, can be beneficial to the simplification of the design process as well as improvement on uniformity and efficiency. In our fixed area matrix feedback type, for a specific region on the lens surface, the denser original points locate, the smaller the solid angle they cover. In other words, the less energy can be assigned to the corresponding cell on target plane. Fitting method will change less in the denser points area, thus cell with less energy suffer less distraction effect from the calculated point. On the other hand, for those cells with more energy than the others, fitting methods will change the lens surface more in these parts, correspondingly distracting more energy from the calculated point to spray around it, which can be interpreted as smoothing the illuminance distribution. Thus, there’s no need for the feedback process to get a so good-looking pattern as it was before. The aim of the feedback process is to converge to the degree that there is no obvious isolated macula on the target plane (which means that there is no hot spot around it to spray energy after fitting). The remaining work can be done by the fitting method. As for LCE, we find out that fitting method can raise it, typically by 5% to 10% compared with previous scheme. This happens because the original square edge becomes round and converged inside, as shown in Fig. 5 .

## 3. Case study – Street Lamp with uniform illuminance

In order to specify the proposed method, a street LED lamp is designed for uniform illuminance distribution over a rectangular target. The sketch of the setting is shown in Fig. 6 and the detailed description can be found in Ref [9], except that lens height (the distance from center apex to the LED source) is restricted to 10 mm. The possibility of Fresnel loss is not taken into account in the calculation of efficiency.

Obviously, the source and target distribution has mirror symmetry for each neighborhood quadrant pairs. Thus we can simply construct the lens surface in first quadrant and use mirror technique to complement it into the whole lens. The freeform lens model is obtained using a fitting technique after two feedback iterations. The freeform lens models before and after fitting are shown in Fig. 7 . It is clearly shown that the discontinuous sections are perpendicular to the base plane, and the number of discontinuous sections is reduced from 88 to 28 after fitting, indicating the high performance of fitting technique in reducing discontinuous sections.

The variation of uniformity and efficiency during feedback and after fitting are shown in Fig. 8 and Fig. 9 . It shows that the proposed feedback method could effectively improve uniformity while having little impact on LCE. After two feedback iterations and one fitting, the number of truncation surfaces in the first quadrant is reduced to 8, and the final uniformity is 82%, while LCE is 77%, 5% higher than previous scheme [9]. The total energy projected on the target plane (light output efficiency) is 94%, ruling out the effect of Fresnel Loss. If one more feedback iteration and fitting is added after the scheme stated above, the uniformity can be a little bit higher (which is 85%), and with a LCE of 75%. Since the proposed method is focused on improvement of the speed and the LCE, and the scheme stated above can achieve a sounded statistical result, we don’t prefer to take this extra measure but just to put a figure (see Fig. 10 ) here to show the result.

## 4. Summary

A time-saving method to design easy molding freeform optical system for an extended LED source with prescribed illumination patterns is proposed. The main thread is to firstly use feedback process with normal correction mechanism to create a discontinuous surface providing precise light distribution on the target plane, then employ fitting technique to minimize the discontinuous sections as well as to increase LCE. Compared with previous feedback method to design continuous lens surface, our method is faster and simpler, while keeping high uniformity and light control efficiency. Though discontinuous sections still exist, the number of them has been significantly reduced. More importantly, discontinuous sections in our method are guaranteed to be perpendicular to the base plane thus eligible for injection molding process, which is a huge step towards mass production compared with those designs with too many discontinuous sections that are difficult to be molded.

Finally, by only twice feedback iterations, the design example using the proposed method achieved a LCE of 77% which is 5% higher than previous scheme, and uniformity of 82% which remains the same or even slightly higher.

## Acknowledgment

This work was supported by the National Key Technology Research and Development Program of the Ministry of Science and Technology of China (Grant No. 2011BAE01B07, and 2012BAE01B03), Science and Technology Planning Project of Guangdong Province (Grant No. 2011A081301003), the National Basic Research Program of China (Grant Nos. 2011CB301902, and 2011CB301903), the High Technology Research and Development Program of China (Grant Nos. 2011AA03A112, 2011AA03A106, and 2011AA03A105), the National Natural Science Foundation of China (Grant Nos. 61176015, 60723002, 61176059, 60977022, and 51002085). Thanks to Synopsys for providing a temporary license of the illumination design software LightTools.

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