## Abstract

A combined feedback design method based on variable separation mapping is proposed in this paper to design free-form optical systems for an extended LED source with complicated illumination patterns. In this method, macro energy division and micro illuminance distribution feedback modifications are carried out according to the deviation between the simulated illumination results and the target requirements. The free-form optical system is then regenerated, and the deviation could be minimized through multiple iterations. Results indicate that free-form optical system designed by this method could achieve precise energy distribution, high regional illuminance uniformity (89.7%), and high light output efficiency (94.9%) simultaneously.

©2011 Optical Society of America

## 1. Introduction

Free-form optical systems designed by nonimaging optics [1–8] are usually adopted on LED modules to improve the lighting performance by controlling the light distribution.

The variable separation mapping design method has been widely used to design free-form optical systems with point sources [9,10]. Here rectangular coordinates (*x,y*) and spherical coordinates (*u,v*) are employed to denote target cells and source energy divisions, respectively. Correspondence between (*u,v*) and (*x,y*) can be established according to the variable separation energy mapping between source energy divisions and target cells [9]. Then, the free-form surface of the optical system can be constructed. However, for extended source with dimensions comparable with the optical system, a real lighting result will deviate from the target requirement [1].

Recently, a feedback modification method was introduced to minimize this deviation [1], where the target plane is equidistantly separated into sufficiently small rectangular cells, and the source energy divisions are adjusted to perform negative feedback modification. Repeated iterations lead to negligible deviation. Although the optical system designed by this method is able to distribute, in theory, the energy into different regions in the target plane precisely, it’s difficult to reach high regional illuminance uniformity and light output efficiency due to insufficient accuracy during construction, and the energy distribution would not be accurate as expected either.

Another way to achieve higher construction accuracy is the feedback modification method, which equally divides the source angle (*u,v*) into small angles and adjusts the area of target cells to perform feedback modification. The optical system designed by this method could achieve high regional illuminance uniformity and light output efficiency, but it could not achieve precise energy distribution of different regions in the target plane.

The methods above could be used for simple illumination pattern, where the illuminance distribution is a continuous function of (*x*,*y*), but not the complicated one, where the illuminance distribution is a discontinuous function of (*x*,*y*). A combined feedback design method is proposed in this paper for complicated patterns. It contains two feedback processes: macro energy division and micro illuminance distribution feedback modifications. Results show that the optical system designed by this method can achieve precise energy distribution, high regional illuminance uniformity, and high light output efficiency simultaneously.

## 2. Feedback modification method

#### 2.1 Feedback principle

To facilitate the optical system design, we first introduce two matrixes: energy matrix ** E**, denoting the source energy division weight with mesh grid (

*u,v*), and area matrix

**, denoting target cells area weight with mesh grid (**

*A**x,y*). The design process is then simplified to the adjustment of these two matrixes instead of complicated calculation of the correspondence between (

*u,v*) and (

*x,y*). This makes the feedback process easier and clearer. Relationship of

**~(**

*E**u,v*) and

**~(**

*A**x,y*) are shown in Eqs. (1)–(4):

**and**

*E***are**

*A**m*×

*n*matrixes with (

*m*+ 1) × (

*n*+ 1) points on the surface, u

_{k}_{ + 1}is the

*u*angle of the (

*k*+ 1)th curve C

_{k + 1}, and ${v}_{s+1}^{k+1}$ is the

*v*angle of the (

*s*+ 1)th point on the (

*k*+ 1)th curve, so as

*x*and ${y}_{s+1}^{k+1}$.

_{k + 1}*J*(

*u*),

*J*(

*v*) and

*J*(

*x*),

*J*(

*y*) are the Jacobian factors. Suffixes min and max reveal the variation range of the corresponding parameter. The relationship is shown in Fig. 1 , where

*P*is the constructed point on the free-form surface and

*Q*is the mapping point on the target plane.

As illuminance = luminous flux / cell area, if the illuminance of a target cell is lower than expected, the cell area can be reduced or its received luminous flux increased to increase the illuminance of the cell, and vice versa. Thus, once the simulated illuminance distribution matrix ** P** is obtained, energy matrix

**and area matrix**

*E***can be calculated separately with inverse and direct proportion to perform negative feedback.**

*A*#### 2.2 Feedback method with fixed area matrix

In this method, energy matrix ** E** is adjusted when the area matrix

**is maintained to perform negative feedback modification. The process can be expressed as Eq. (5):**

*A**E*is the energy matrix of the

_{k}*k*th feedback iteration and

*E*is the energy matrix of the initial design. ${P}_{pre}^{k}$ and ${P}_{sim}^{k}$ are the prescribed illuminance distribution matrix and the simulated illuminance distribution matrix interpolated with the mesh grid of the area matrix of the

_{0}*k*th iteration, respectively.

Owing to the Lambertian intensity distribution property of the LED source, there will be large *u* and *v* angle divisions near the edge of the free-form surface. This may lead to low construction accuracy, because the accuracy of the approximation between infinitesimal plane and curved surface decreases when the plane enlarges. Furthermore, it would also cause energy leakage and low uniformity of regional illuminance, as shown in Fig. 2
. The surface *P _{1}P_{2}P_{3}P_{4}* constructed with smaller source angle

*α*in Fig. 2(a) has a better accuracy of construction than

_{1}, α_{2}, α_{3}*P*with larger angl

_{1}P_{2}'*e β*in Fig. 2(b). As a result, ray ${\overrightarrow{r}}_{1}$ is controlled more accurately than ray ${\overrightarrow{r}}_{2}$.

#### 2.3 Feedback method with fixed energy matrix

In this method, area matrix ** A** is adjusted when the energy matrix

**is maintained. The process can be expressed as Eq. (6):**

*E**A*is the area matrix of the

_{k}*k*th feedback iteration and

*A*is the area matrix of the initial design. ${P}_{pre}^{k}$ and ${P}_{sim}^{k}$ are the prescribed and simulated illuminance distribution matrixes interpolated with the mesh grid of the area matrix during the

_{0}*k*th iteration, respectively.

Source angles *u* and *v* are equally divided into small angles, which could maximize the accuracy of the surface construction according to the drawer principle. But since the area matrix is not fixable, there will be a mesh grid mismatch between the area matrix and the prescribed illuminance distribution matrix, or the boundaries of the regions divided by the area matrix will not cover the boundaries of the regions divided by the prescribed illuminance distribution matrix, as shown in Fig. 3
. It would cause nonaccurate energy distribution.

This feedback method is especially suitable for designing free-form optical systems with uniform illumination distribution pattern for an extended source.

#### 2.4 Combined feedback method

Owing to the disadvantages mentioned above, either of these two methods could not deal with complicated illumination patterns very well. Hence, a combined feedback method is proposed in this paper for designing optical systems with complicated illumination patterns. Two feedback processes are used: one is used to adjust the macro energy distribution through ** E_{w}**, and the other is used to adjust the micro regional illuminance distribution through

**, so it can ensure precise energy distribution of different regions, high illuminance uniformity, and high light output efficiency at the same time. This method is performed as follows:**

*A*_{s}- 1. Divide the prescribed illumination pattern into specified regions with uniform illuminance distribution or other simple illuminance distribution.
- 2. Obtain the prescribed macro energy matrix
*Ew*according to the prescribed illuminance distribution of each region._{0} - 3. Perform the feedback process to adjust the energy weight of the divided regions according to
- 5. Divide the source angle (
*u*,*v*) equally in each region. Obtain the initial regional area matrix ${A}_{s}^{0}$ according to the prescribed regional illuminance distribution. Perform the feedback process to adjust the illuminance distribution of each region according to$${A}_{s}^{k}(i,j)=\frac{{P}_{{}_{sim}}{}_{s}^{k-1}(i,j)}{{P}_{{}_{pre}}{}_{s}^{k-1}(i,j)}\u2022\frac{{P}_{{}_{sim}}{}_{s}^{k-2}(i,j)}{{P}_{{}_{pre}}{}_{s}^{k-2}(i,j)}\u2022\cdot \cdot \cdot \u2022\frac{{P}_{{}_{sim}}{}_{s}^{0}(i,j)}{{P}_{{}_{pre}}{}_{s}^{0}(i,j)}\u2022{A}_{s}^{0}(i,j),i=\mathrm{1...}{m}_{s},j=\mathrm{1...}{n}_{s},s=\mathrm{1...}{m}_{w}\times {n}_{w},$$ - 6. Build the regional curves as shown in Fig. 4(b). Generate the free-form optical system model by integrating all the curves, then use the actual LED source dimensions to obtain the simulated illuminance distribution matrix of
*k*th iteration. - 7. Repeat steps 3–6 until the performance of the free-form optical system satisfies the requirement.

## 3. Design example

The example is to perform a complicated illuminance distribution pattern in a 30m × 30m quadrate region with the source mounted at a height of 7m. Design parameters and prescribed illuminance distribution pattern are shown in Fig. 5 , where region 1, region 3, and region 5 are three rectangular regions with uniform illuminance distribution. The total energy ratio between the white region and the blue region is 8:1. Dimension of the LED source immersed in the optical system is 1mm × 1mm. The central height and the refractive index of the optical system are 5mm and 1.59, respectively.

Simulation results of the initial and the final optical models designed by three feedback methods are shown in Fig. 6
for comparison. Results of the initial designs are far from the prescribed illuminance distribution pattern, while the results of the final designs are much better after several iterations of feedback modification. Besides, the initial results are different because energy matrix and area matrix obtained by different feedback design method would lead to different construction results. Furthermore, a final result of the feedback method with fixed ** A** shows low regional illuminance uniformity and severe energy leakage. On the other hand, final results of the other two methods show good control on energy leakage and regional illuminance distribution.

Figure 7
shows the comparisons between these simulated results on three parameters. As shown in the figure, improvements in total energy ratio and illuminance uniformity are obtained at the expense of decrease in light output efficiency. The low construction accuracy and energy leakage of the feedback method with fixed area matrix ** A** cause a deviation in the total energy ratio, as well as low regional illuminance uniformity and low light output efficiency. The feedback method with fixed energy matrix

**has a good performance in both regional illuminance uniformity and light output efficiency, but its total energy ratio doesn’t satisfy the requirement. The final optical system model designed by the combined feedback method has the best results on all three parameters shown in Fig. 7, which has a total energy ratio of 7.92:1, an average regional illuminance uniformity of 89.7% and a light output efficiency of 94.9% (Fresnel loss ignored).**

*E*Profiles of the final optical system models designed by three feedback methods are shown in Fig. 8 . Although the dimensional differences between these models are small, the illumination results are quite different due to comparable dimensions of the source and the optical system. The 3-D geometry of the final optical system designed by combined feedback method is shown in Fig. 9 . The dimensions (length, width, and central height) of this model are 13.71mm × 12.8mm × 5mm.

## 4. Conclusions

A combined feedback modification method is proposed in this paper to design free-form optical systems for an extended LED source with complicated illumination patterns. This method contains two feedback modification processes: macro energy division and micro illuminance distribution feedback modifications, which could keep high accuracy of the free-form surface construction and have good control on the energy distribution.

The design example using the proposed method achieved precise energy distribution, high regional illuminance uniformity (89.7%), and high light output efficiency (94.9%) simultaneously.

## Acknowledgments

This work was supported by the National Natural Science Foundation of P. R. China (under Grant Nos. 60536020, 60723002, 50706022, and 60977022), the “973” Major State Basic Research Project of China (Nos. 2006CB302800 and 2006CB921106), the “863” High Technology Research and Development Program of China (Nos. 2007AA05Z429 and 2008AA03A194), Beijing Natural Science Foundation (No. 4091001), and the Industry, Academia and Research Combining and Public Science and Technology Special Program of Shenzhen (No. 08CXY-14).

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