## Abstract

An ultra-compact rotational symmetric lens with double freeform surfaces based on the edge-ray principle is designed in this paper. The lens redistributes light emitting from a Lambertian LED light source to achieve uniform illumination within the target area. The initial design is optimized for optics compactness under structural constraints and illumination requirement using the genetic algorithm. A design for the double-freeform-surface lens with a height of the optics system *h* = 12.56 mm for a circular LED source with a diameter *D* = 10 mm is demonstrated for uniform illumination within 45° and thus achieves optics compactness *h/D* = 1.256, which is half of that achieved by the single-freeform-surface lens. The Monte-Carlo ray-tracing result shows an illumination circular area with a clear cut-off at *R* = 1000 mm at the target plane in a distance *H* = 1000 mm. The uniformity within the target illumination area is greater than 0.9 and the light output efficiency is as high as 0.9865.

© 2015 Optical Society of America

## 1. Introduction

In the present lighting market, light emitting diodes (LEDs) have great advantages such as long lifetime, high efficiency and compact size compared to other lighting sources [1]. Secondary optics like freeform-surface lens is often employed to redistribute light from the LED source to meet specific requirement for illumination. Design methods of freeform-surface optics such as Ordinary Differential Equation, Surface Tailoring, Simultaneous Multiple Surfaces and Supporting Paraboloids have been presented for generating a prescribed intensity distribution as a function of angle or position on the target plane [2–13
]. Reflective or refractive optics surfaces can be designed by the point-to-point mapping method when a light source is small enough to be approximated as a point source, i.e. at less than 1/10 of the size of the optics [14]. However, for extended light sources such as chip-on-board LEDs at a size of several centimeters or even more [15], design based on point source approximation would result in bulky systems which are not cost effective for manufacturing and application. In contrast, the edge-ray principle [16] works well for designing non-imaging optics of extended light sources, in which only the rays from the edge of the source are traced and thus the design process is greatly simplified. Mao, et al investigated the design of a single freeform-surface rotational symmetric immerse lens for uniform illumination based on the edge-ray principle, resulting in excellent compactness (*h/D* = 2.5:1) [17].

In this paper, we investigate the design of a rotational symmetric lens with double freeform surfaces. The formalism is developed based on the edge ray principle to obtain an initial design of the lens. The design is then optimized for optics compactness using the genetic algorithm and achieves *h/D* = 1.256:1. A highly uniform illumination with a clear cut-off can be realized with the light output efficiency as high as 0.9865.

## 2. Initial design

The principle of “luminance engineering” proposed by Elmer [18] is illustrated in Fig. 1(a)
, where *I*
_{θ}, *D*
_{θ} and *L*
_{θ} are the far-field intensity, the projection length of a light source and the luminance in the *θ* direction, respectively. The far-field intensity *I*
_{θ} equals the product of *D*
_{θ} and *L*
_{θ}. A Lambertian light source has equivalent *L*
_{θ} in all directions such that *I*
_{θ} is proportional to *D*
_{θ} in the direction of observation. The optics system is used to modify *D*
_{θ} in different directions in order to realize the illumination with prescribed intensity or illuminance distribution.

For optics with rotational symmetry, the design can be implemented simply in 2D configuration, whereas the 3D lens model can be obtained by rotating the 2D profile afterwards. For generating a uniform illuminance distribution *E*
_{2D}(*x*) *= E*
_{0}, the optics for a Lambertian light source should satisfy [17],

_{0}is the projection length of the light source in the axial direction.

The design process to determine the initial structure of the lens with thickness *t* is illustrated in Fig. 1(b), where both curves *A*
_{0}
*A*
_{n} and *B*
_{0}
*B*
_{m} are quadratic polynomial curves expressed by

*i*= 1,2 representing the two surface profiles. Ray

*LA*

_{0}

*B*

_{0}is emitted from point

*L*at the left edge of the light source and refracted at points

*A*

_{0}and

*B*

_{0}at the two surface profiles. While

*A*

_{0}is arbitrarily chosen in the second quadrant, the relationship between directional angles

*φ*

_{0}and

*φ*

_{1}is further made [19]where

*γ*is the correlation coefficient to be preset for a design. For an interface, there exists the following expression for the normal vector

**e**

_{I}and

**e**

_{O}are the unity directional vectors of incidence and refraction, respectively, and

*n*

_{I}and

*n*

_{O}are the refractive indices of the incident and the refractive media, respectively. In this design, we assume that the lens is made from PMMA (

*n*= 1.4935). As the slope at endpoint

*A*

_{0}is known from its normal vector, i.e. ${k}_{{A}_{0}}={({N}_{y}/{N}_{x})}_{{A}_{0}}$, parameters

*a*

_{1}and

*b*

_{1}of the quadratic polynomial curve

*A*

_{0}

*A*

_{n}can be derived asThe ray after refraction at endpoint

*B*

_{0}will become parallel to the

*y*axis. Therefore, for an assigned thickness

*t*, parameters

*a*

_{2},

*b*

_{2}of the curve

*B*

_{0}

*B*

_{m}can be derived similarly according to the above process.

The rest part of the meridian profile of the lens can be extrapolated from the part expressed by the polynomial curves. Now with known *D*
_{0}, Eq. (1) defines *D*
_{θ}, which is the distance between the edge ray pair for output in *θ* direction, *LA*
_{1}
*B*
_{1} and *RA*
_{n + 1}
*B*
_{m + 1}. A tiny slope change *δ* is set as illustrated in Fig. 1(b) so that *B*
_{m + 1}’s coordinates can be derived from *B*
_{m}’s. With the output vector in *θ* direction and the normal vector at *B*
_{m + 1} both known, the incident vector along *A*
_{n + 1}
*B*
_{m + 1} can be obtained using Snell's law

*A*

_{n + 1}is on the tangent line at

*A*

_{n}, we can simply get

*A*

_{n + 1}’s coordinates, while the normal vector at

*A*

_{n + 1}is calculated using Eq. (4). The extrapolated part of the profile is completed by repeating these steps till the left edge ray outputs at

*B*

_{m}, where the output angle

*θ*

_{max}is the maximum angle for uniform illumination and has to meet the illumination requirementwhere

*R*is the radius of the uniform illumination area and

*H*is the illumination distance.

By presetting values of *x _{A}*

_{0},

*y*

_{A}_{0},

*γ*,

*t*and

*δ*, e.g. 4, 8, 0.1, 6, and 0.05, respectively, we can obtain an initial design of the lens profile. One example is illustrated in Fig. 2 , noted with the lens size, i.e. the height

*h*and the radius

*r*. It is worth noting that the edge points of the two surface profiles are separated by a gap

*d*. Its reduction will lead to smaller lens size. Moreover, there exists an angle of elevation

*β*, within which light cannot be collected by the lens. Its reduction will lead to higher output efficiency.

## 3. Optimization using genetic algorithm

As discussed above, an initial design can be obtained simply with a set of preset parameters, whereas further optimization is necessary [20, 21
]. To optimize compactness of the optics, i.e. minimal *h* and *r*, merit functions of the multi-object optimization problem can be expressed as:

*d*

_{T},

*β*

_{T}and

*θ*

_{T}are target restrictive parameters determined by requirements of the lens structure and illumination distribution, and Δ

*is an accuracy control parameter for angular deviation. The second part of MF*

_{θ}_{1}or MF

_{2}will be either 0 if the restrictive condition are satisfied or α otherwise. We set

*α*= 1000 which is far greater than the dimension of the lens for this design. To demonstrate the optimization of the lens structure, we design a rotational symmetric double–freeform-surface lens with target restrictive parameters given in Table 1 .

The genetic algorithm is used for solving the multi-object optimization problem. Figure 3
shows the Pareto front to demonstrate the tradeoff relationship between *h* and *r*. The average Pareto distance *ε* is used to depict how the non-dominated solutions converge to global optimal solutions for a multi-object optimization problem and smaller *ε* means better convergence. In this demonstration, the genetic algorithm ran for 500 generations with a population of 200 individuals and *ε* is as small as 0.00048. Each point in the figure represents one structure which is preferably determined by the structural requirement.

One optimized profile of the lens is shown in Fig. 4(a)
and has *h* = 12.56 mm, so the optics compactness *h/D* has achieved 1.256:1. The simulation result of illumination in Fig. 4(b) shows that the illumination area has a clear cut-off at the edge and the uniformity within the target area is greater than 0.9. Meanwhile, the light output efficiency of the illumination optics is as high as 0.9865. Therefore, an ultra-compact lens design for uniform illumination is obtained.

Compared to a design of single-freeform-surface immerse lens with *h*/*D* = 2.5:1 [17], the double-freeform-surface lens possesses much more compact size and higher uniformity of illumination within the illumination area (see Fig. 5
) due to an extra freeform surface to control refraction of light.

## 4. Conclusion

We have designed a highly compact rotational symmetric lens with double freeform surfaces for uniform illumination. An initial design of the lens is acquired by combining the edge ray principle and the luminance engineering principle. The genetic algorithm is then applied to optimize the initial design to meet the requirements of illumination distribution and structural constraints. As a demonstration, a PMMA lens with *h* = 12.56 mm for an LED extended source with diameter *D* = 10 mm was designed for uniform illumination within an angle of 45° at a distance of 1000 mm. The simulation results shows the circular area of illumination with a clear cut-off at *R* = 1000 mm, within which the illuminance uniformity is greater than 0.9 and the light output efficiency is as high as 0.9865. The ultra-compact double-freeform-surface lens with high illumination uniformity and light output efficiency will bring cost-saving benefit to LED illumination engineering.

## Acknowledgment

The authors acknowledge financial support from the National Natural Science Foundation of China (grant no. 61176085 & 61377055), the Department of Education of Guangdong Province, China (grant no. gjhz1103) and the open-project fund support from the State Key laboratory of Opto-Electronic Material and Technologies (Sun Yatsen University), Guangzhou, China. N. Zhu acknowledges financial support from Natural Science Foundation of China (61308007) and Specialized Research Fund for the Doctoral Program of Higher Education (20134407120012).

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