## Abstract

The freeform optical system for an extended source was constructed by partially overlapping a few numbers of point-source freeform surfaces (PFSs) and extracting their contour. Each PFS redistributed the Lambertian emission of a point source into the prescribed light distribution or more frequently into a modified distribution. By adjusting the relative positions of the PFSs and the pattern of the modified light distribution, the optimized freeform surface could be obtained. As an example, an optical system with a height only four times the source radius is designed for achieving a uniform-illuminance distribution on the target. The optimized freeform surface was formed by two PFSs. The virtue-point-sources of the PFSs were located symmetrically on the extended source with a distance of a quarter of the source diameter from each other. Each PFS achieved an increasing-illuminance distribution. The illumination uniformity of this model can be improved by 55.4%, while the optical efficiency within the target area is maintained above 80%.

©2013 Optical Society of America

## 1. Introduction

The lighting source based on large-sized integrated packaging light emitting diodes (LEDs) is becoming an important development trend for semiconductor lighting, due to its low cost and easy adaptation to modular applications. To obtain lighting effects comparable to that of the traditional lighting sources, the optical system with an LED source, which usually exhibits a Lambertian emission pattern, must be designed according to the application circumstances. Meanwhile, to ensure substantial energy-saving advantage, its optical efficiency should be commonly higher than 80% at the same time. Furthermore, the optical system should be compacted so as to reduce the manufacturing and material costs. Under these circumstances, the large-sized integrated packaging LEDs must be considered as an extended source [1]. And the optical system for a Lambertian extended source can hardly be designed by the already mature free-form surface designing methods for a zero-etendue source (e.g. point source) [2–6].

To achieve exact illumination distribution, the design of freeform surface for a Lambertian extended source faces two technical barriers. Firstly, it is unable to obtain an accurate one-to-one mapping relationship between the energy distribution on the target area and that of the Lambertian extended source, especially for a large-sized source. Secondly, any facet on the optical freeform surface has only one main normal vector, which makes it difficult to control all the incident light rays according to the pre-designed regulation precisely [7]. There are two major categories of freeform-surface designing methods for Lambertian extended source at present. One is the Simultaneous Multiple Surface (SMS) method [8–10] based on the edge-ray principle [11]. It uses two freeform surfaces to control the wave fronts emitted from the two edge points on the extended source, respectively. The other is the feedback compensation method [7, 12, 13], which improves the simulated illumination distribution on the target plane through the modification of the freeform surface by feedback and iterations. However, these methods are generally applied for the 1mm^{2} extended sources, and the ratio of the optical system height to the source radius ($h/{r}_{s}$) is approximately 8 to 12 [7, 9, 10, 13, 14]. The method for designing a more compact optical system, for example $h/{r}_{s}=4$, has seldom been studied up to now.

It is worth noticing that the freeform surface can be constructed by the envelopes of families of quadrics [15, 16]. However, this method was applied for the single point source or parallel light [15, 16]. In this paper, we propose a method for designing the optical system for extended light source. The optical system is built by overlapping and optimizing a few numbers of point-source freeform surfaces (PFSs) with different locations, and extracting the contour of these PFSs. This proposed method is simple and easy to operate. The designed optical system can control the incident light field from a large-sized extended source to form a light distribution on the target area close enough to the prescribed pattern. Compared with the traditional methods based on point source, it can achieve a higher uniformity for a uniform illumination distribution system.

## 2. Ideal point-source’s freeform surfaces overlapping method

#### 2.1 Principle of ideal point-source’s freeform surfaces overlapping method

An extended source can be regarded as equivalent to a collection of virtual-point-sources. In this method, the freeform surface for extended source is constructed by partially overlapping two or more point-source’s freeform surfaces (PFSs). Each PFS individually redistributes the Lambertian emission of the virtual-point-source on the extended source into the prescribed light distribution, or more frequently, it redistributes the emission into a modified distribution. Figure 1(a) shows the construction of the extended-source’s freeform surfaces (EFSs) based on overlapping two, three and five PFSs, respectively.

As shown in Fig. 1(a), the positions of the outmost virtual point sources are the same in the three cases. In this study, the multiple PFSs were combined by extracting external contour or internal contour to form the EFS. Figure 1(b) exhibits the external contour situation, and the differences between the three EFSs are mainly within the center region (illustrated in the red dashed box in the picture). Figure 1(c) shows the internal contour condition, and the variation of the PFSs number has no effect on the shape of the EFS.

In addition to the factors mentioned above (i. e. the number and positions of the virtual-point-sources, and the combination approach of the PFSs), the light distribution for the ideal point source is another key factor to enhance the lighting effects. As shown in Fig. 2, *A* is the light distribution on the screen from a point source, and *B* is the light distribution achieved on the screen from an extended source. The light distribution of *A* can be modified differently from the prescribed light distribution *G*, so as to make the light distribution of *B* as closely as possible to *G*. The pretreatments of the light distribution of the virtual point sources include the following: decreasing the radius of light distribution *A* according to the prescribed light distribution *G*; changing the light distribution pattern while maintaining the radius of *A*; and integrating the two techniques mentioned above.

The PFS can be designed by the optical designing methods for point source. As the details of the method used in this study have been described explicitly in [6], it will be introduced briefly. Equation (1) is based on energy conservation, where$\overrightarrow{i}$is the direction of the incident ray; $I(\overrightarrow{i})$is the intensity distribution of a point source, corresponding to a cosine distribution for a Lambertian source; $E(\overrightarrow{r})$is the illuminance at position $\overrightarrow{r}$on the target plane, corresponding to the modified light distribution.

Then the one-to-one mapping of $\overrightarrow{i}$and $\overrightarrow{r}$(Eq. (2)) can be derived from the differential form of the energy conservation equation. The construction of PFS can use the point-by-point way of configurations as illustrated in [6].

#### 2.2 The design process of the ideal point source’s freeform surfaces overlapping method

The procedure of this method is given in Fig. 3. First, the prescribed light distribution *G* on the target plane, the dimensional parameters of the optical system and the material refractive index, etc are determined. Then the pattern of the light distribution *A* for the point source is set (the prescribed light distribution *G* can be used as the initial distribution for *A*), and one of the existent freeform surface construction methods for point source is applied to get the freeform surface *C*1 which can form the light distribution *A* on the target. Then the number of the virtual point sources on the extended source is selected, and *C*1 is emplaced above each selected point sources to precisely control their emission fields individually. The multiple *C*1’s are joined into a new freeform surface *C*2 by a combination approach (e. g. by extracting external or internal contour). The light distribution *B* on the target plane is acquired by simulating the optical field from the extended source through *C*2. Then *B* is compared with the prescribed light distribution *G*. If *B* does not meet the requirements, it is necessary to use the light distribution pretreatment for modifying the light distribution *A*, and re-start the process. Otherwise, the optimized freeform surface *C*2 for the extended source is obtained.

## 3. Design of an optical system and discussions

As an example, a rotational-symmetry optical system with $h/{r}_{s}=4$ was designed by the ideal point source freeform surfaces overlapping method. Figure 4 shows the dimensional parameters of the optical system: the radius of the circular Lambertian source equals 5mm; the height of the freeform surface is 20mm; the refractive index of the freeform lens is 1.49; and the prescribed light distribution *G* is the uniform illumination distribution with 1000mm radius on the screen, which is 1000mm from the extended source.

The optical efficiency is defined as the ratio of the energy within the target area (circular area with 1000mm radius) on the screen to the energy emitted from the LED extended source directly, as shown in Eq. (3). Generally speaking, the optical efficiency of the optical system should be higher than 80%.

Meanwhile, relative standard deviation (RSD) is used to characterize the uniformity of the light distribution on the target area, as shown in Eq. (4):

*N*is the number of meshes in the target area, ${E}_{i}$is the illuminance within each mesh, and $\overline{E}$ is the average illuminance on the target area. RSD values are within the range [0, 1], and a lower RSD value indicates a more uniform illuminance distribution.

As a reference, *C*1* _{ps freeform design}* was the PFS designed by the freeform designing method [6] in two-dimensional case, which could achieve the prescribed light distribution

*G*from an ideal point source accurately. When

*C*1

*was used to directly control the optical field from the 5mm radius circular Lambertian source, the optical efficiency within the target area is 84.87%, and the RSD value is 0.4132.*

_{ps freeform design}The ideal point source’s freeform surfaces overlapping method was then studied and discussed. First, the effect of the PFSs number on the uniformity and efficiency of the optical system was simulated, where the external contour was chosen to form the EFS. Figure 5 shows the efficiency and the RSD values of the optical systems with two, three and five PFS *C*1’s, respectively *(C*1 is same as *C*1* _{ps freeform design}* when light distribution pretreatment is not used). Due to the rotational symmetry of the system, the virtual-point-sources were selected symmetrically on the same diameter of the extended source. The horizontal ordinates in Fig. 5 represent the overlapping modes of the PFSs. For instance, 5F(0,?2.5,?5mm) represents that five

*C*1’s are located above five uniformly-spaced virtual-point-sources, and one of the point source is in the center of the extended source, two are 2.5mm off the source center and the last two are 5mm off the center. In Fig. 5(a), the outmost point source positions are all 5mm off the extended-source center, and the optical efficiency of the three cases are in the vicinity of 55%, while the RSD values have small fluctuations around 0.04. In Fig. 5(b), the outmost positions are all 2.5mm off the center, and the optical efficiency of the systems are all in the vicinity of 75%, while the RSD values are close to 0.135.

The results indicate that the uniformity and efficiency of the optical systems are markedly affected by the outmost positions of the selected point sources, and the increasing number of PFSs has no significant influence on the results. The reason might be that the shape of the EFS *C*2 was mainly affected by the positions of the outermost point sources, and when the external contour was extracted, varying the number of the PFSs only influences the center region of *C*2, which accounts for a very small proportion of the entire *C*2. Therefore, in the following calculations only two virtual-point-sources are considered so as to simplify the optimization procedure.

The decision of whether to adopt external contour or internal contour as the combination approach is discussed as follows. Figure 6 shows the optical efficiency and RSD values of the optical systems by extracting the external contour and the internal contour with different *S,* respectively. *S* stands for the distance that the two symmetrical point source positions deviate from the center of the extended source. If *S* = 0, it means that the proposed overlapping method is not used, and the freeform surface *C*2 coincides with the PFS *C*1. According to Fig. 6, as *S* increasing, no matter the external or the internal contour is extracted, the efficiency of the optical system declines and the value is lower than that of the case *S* = 0. However, the light distribution uniformity is significantly improved with the increase of *S* when the external contour is used; but the uniformity deteriorated in the internal contour case. Consequently, to obtain better light distribution uniformity, extracting external contour is adopted as the combination approach of the freeform surfaces *C*1’s.

Table 1 shows the efficiency and the RSD values of the optical systems with *S* varying from 0 to 5mm, assuming the external contour condition. It reveals that an increased efficiency of the optical system is often accompanied by a reduction of the uniformity (RSD value gets larger) when *S* varies, and vice versa. In addition, it is noticed that when *S* = 1.25 mm, the efficiency is 81.95% and the uniformity of the light distribution is increased by 36.8% compared with the situation for *S* = 0. Therefore, the following optimizations would be based on a pair of symmetrical point source positions with *S* = 1.25 mm firstly, and the light distribution pretreatment was then employed to continue reducing the RSD value while maintaining the efficiency above 80%.

Figure 7 depicts the axial cross-sectional view of the original light distribution *A* and part of the light distribution pretreatments. The line marked with circles portrays the original light distribution *A* which is the 1000mm radius uniform illumination distribution (the same as the prescribed distribution *G*). Considering the scope within [0mm, 1000mm], the solid line and the dotted line portray the linear-decreasing (LD) illuminance distribution (Eq. (6)) and the cosine-decreasing (CD) illuminance distribution (Eq. (7)) respectively; the dash-dot line is the linear-increasing illuminance distribution (Eq. (6)); the dashed line is the hollow illumination distribution (Eq. (8)) in which the minimum illuminance is at the midpoint of the radius; the line marked with dots is the light distribution with a shortened radius. In addition, simultaneously reducing the radius and changing the light distribution are included in light distribution pretreatments. All the axial cross-sectional lines in Fig. 7 are represented with the same total luminous flux on the screen.

The equations of the light distributions depicted in Fig. 7, which are used to substitute $E(\overrightarrow{r})$ in Eq. (1), are as follows:

*R*is the radius of the prescribed light distribution; $r\subseteq \left[0,R\right]$ refers to the distance from the center of the target area; ${E}_{0}$is the illuminance at the center of the light distribution;

*t*in Eq. (6) and Eq. (7) is the ratio of the illuminance at the edge of the light distribution to the illuminance at the center of the light distribution;

*h*in Eq. (8) is the ratio of the minimum illuminance to the illuminance at the center (or the edge) of the light distribution.

For the case *S* = 1.25 mm, the effect of decreasing illuminance distribution pretreatment on the results was first studied. Linear-decreasing (LD) illuminance distribution and cosine-decreasing (CD) illuminance distribution were analyzed and compared. Figure 8 shows the variations of the efficiency and RSD values of the systems with *t* varying from 0.5 to 1.0. It indicates that the efficiency of these optical systems is higher than that of the uniform distribution case (*t* = 1.0), where the decreasing illuminance distribution is used. However, the uniformity suffers significant deterioration. Therefore, the decreasing illuminance distribution pretreatment was improper when the prescribed light distribution for extended source was uniform illuminance.

Then the linear-increasing illuminance distribution pretreatment was analyzed with different *t* values. The radius of the light distribution *A* remained 1000mm. In Fig. 9, *t* = 1.0 represents the condition in which the light distribution pretreatment is not used. As the value of *t* increases from 1.0 to 2.3, the optical efficiency drops slightly, but the RSD value decreases discernibly, indicating significant improvement of the light distribution uniformity.

Figure 10 shows the simulations that use the hollow illumination distribution pretreatment with different *h* values. The radius of the light distribution *A* was still 1000mm. In Fig. 10, *h* = 1.0 represents the condition when the light distribution pretreatment is not used. As *h* decreases from 1.0 to 0.5, the optical efficiency declines a little faster than that of the systems using the linear-increasing illuminance distribution, and the RSD declines slowly.

The optimized models and results of using the linear-increasing illuminance distribution pretreatment (Model OPT1) and the hollow distribution pretreatment (Model OPT2) are shown in Table 2 for *S* = 1.25mm. The criteria for the optimized models are that the efficiency be higher than 80% and the RSD has the minimized value. Compared with the Model Reference2 in which the light distribution pretreatment was not used, the efficiency of the optimized Model OPT1 declines slightly (2.3%), but the uniformity improves significantly by 29.5%. The improvement in uniformity is about 55.4% when compared with Model Reference1. Consequently, OPT1 was selected as the optimized model in the condition that *S* = 1.25mm.

Moreover, the light distribution pretreatments (including linear-increasing distribution, hollow distribution and radius-decreased light distribution) were used to optimize for *S* = 1.875, 2.5, and 3.125 mm, respectively. Table 3 shows the optimized models and results of the four conditions.

According to Table 3, with the point source positions deviating from the center of the extended source, it is found that *R* should be decreased gradually, so as to concentrate the energy into the target area and enhance the optical efficiency above 80%. With respect to Reference1, the uniformity of the illumination distribution in Model OPT3 and OPT5 is also markedly improved by 45.28% and 45.26%, respectively. The differences among these optimized results are minor. In addition, the optimized models and results in Table 3 indicate that the optimized pattern of the light distribution pretreatment is related to the selection of the point source positions. Therefore in the procedure of optimizations, the positions can be fixed at first, by comparing and ascertaining the ones that closest to the optimization criteria. Then a series of light distribution modifications can be implemented to obtain the optimized results.

Figure 11 gives the light distributions on the target, including that of the Models Reference1, OPT1 (*S* = 1.25 mm), OPT3 (*S* = 1.875 mm), OPT4 (*S* = 2.5 mm) and OPT5 (*S* = 3.125 mm).

The freeform surface contour lines of Models Reference1, OPT1, OPT3, OPT4 and OPT5 are depicted in Fig. 12(a), and the corresponding lenses are shown in Fig. 12(b), while the height of these lenses are all 20mm.

## 4. Conclusions

An optical designing method for extended source was proposed, which is simple and easy to operate. The designed freeform surface can control the light emissions from an extended source to form a prescribed light distribution on the target plane. This freeform surface was constructed by overlapping a few numbers of point-source freeform surfaces (PFSs). Each PFS enabled the virtual point source on the extended source to achieve the prescribed light distribution or a modified light distribution on the target plane. The optimized results can be obtained through optimizations.

As an example, an optical system with 20mm height of the freeform surface was designed for a 5mm-radius circular Lambertian extended source. The optical system was aimed to achieve a 1000mm radius circular uniform illumination distribution on the screen. The optimized freeform surface was constructed by two PFSs which achieved a linear- increasing illuminance distribution (the ratio of the edge illuminance to the center illuminance is 2.1) on the target plane. And the centers of these two PFSs were above two symmetrical point positions which are 1.25mm off the center of the extended source, respectively. Compared with the system designed with the traditional method, the illumination distribution uniformity of the optimized system could be improved by 55.4%, and the optical efficiency in the target area could be higher than 80%.

In addition, although this method relied on the “trial-and-error” strategy, there are the regularities that derived from the calculations and discussions, which might be helpful for other optical designing problems. The regularities are summarized as follows:

1) If the value of $h/{r}_{s}$ is not extraordinary small, the shapes of the optical systems are markedly affected by the outmost positions of the virtual-point-sources. Therefore, only a few numbers of the PFSs are necessary for constructing the optical system.

2) Extracting external contour of the PFS can improve the uniformity of the light distribution of the optical system. In the external cases, increasing the distance of the outmost positions of the virtual-point-sources can get a better uniformity, however the efficiency will have a reduction.

3) There is a trade-off between the efficiency and the uniformity of the optical system usually.

4) Along the directions from the center to the edge of the light distribution, if the modified light distribution has an increasing trend and a relatively higher value at the edge of the distribution, it is profitable to improve the uniformity of the optical system. If there is a decreasing trend near the center of the distribution and a lower value at the edge of the distribution, it is helpful to concentrate the energy into the target area, which can enhance the efficiency.

5) The relative positions of the PFSs and the pattern of the modified light distribution are not independent of each other. Therefore, the positions can be fixed at first, by comparing and ascertaining the ones that closest to the optimization criteria. Then the optimized results will be obtained after a series of light distribution modifications for PFS.

## Acknowledgments

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 Opened Fund of the State Key Laboratory on Integrated Optoelectronics (Grant No. IOSKL2012KF09), 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).

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