## Abstract

A feedback modification method based on variable separation mapping is proposed in the design of free-form optical system with uniform illuminance for LED source. In this method, the non-negligible size of LED source is taken into account, and a smooth optical system is established with single freeform surface regenerated by adding feedback to the lens design for a point light source. More rounds of feedback can improve the lens performance. As an example, a smooth free-form lens with rectangular illuminance distribution is designed, and the illuminance uniformity is improved from 18.75% to 81.08% after eight times feedback.

©2010 Optical Society of America

## 1. Introduction

Light emitting diodes (LEDs) usually can’t be directly applied in general lighting systems with perfect illumination effect and high efficiency because of their Lambertian illuminance distribution. Reflective or refractive optical systems are often used to reshape the light distribution, where LEDs are considered as point light sources for most cases. The main challenge of using point sources to form the desired illuminance distribution with 3-D geometry is to find the solution of the Monge-Ampere elliptical nonlinear partial different equation [1,2], and there are several methods to tackle this problem [3–9]. Among these methods, variable separation mapping design method is a more effective one due to avoid directly solving the complex second-order Monge-Ampere partial different equation; and also in this method, discontinuities are introduced onto the lens surface to restrict the normal deviations [9]. Accurate light irradiation control can be achieved with point light source by this method. However, the discontinuous surface of lens greatly increases the difficulties of mold fabrication, and also increases the total internal reflection loss if the light source can’t be approximated as a point one.

In fact, in the design of compact optical systems where the size of the system is comparable with the dimension of the optical source, the size of the LED can’t be neglected, and the direction of the rays can’t be controlled as precise as in the design with point light source. For example, rays 2 and 3 coming out two edge points on the LED source are refracted into rays 2’ and 3′, which have large offsets from the given direction 1’ (see Fig. 1 ), hence large deviation is produced between the simulation result for the real size LED and that for point light source.

In the design of 3-D optical systems with extended light sources, one approximate solution is to use the Simultaneous Multiple Surface (SMS) method [10], wherein two free-form surfaces are simultaneously designed to control two input and two output orthotomic ray bundles. Unfortunately, the final differential equation of this method is also a complicated Monge-Ampere type; and two free-form surfaces increase the difficulties of mold fabrication and also increase the Fresnel loss.

In this paper, we propose a Feedback modification method. The optical model is first designed with point light source based on variable separation mapping method [9]. Then, the illuminance distribution *E’ _{1}*(

*x,y*) is obtained through simulation with the actual (extended) LED source. Let

*E*

_{0}(

*x,y*) be the desired illuminance distribution, then by employing a Feedback modification function

*η*(

_{i}*x,y*)=

*η*(

_{i}*E’*(

_{i}*x,y*),

*E’*(

_{i-1}*x,y*),…,

*E’*(

_{1}*x,y*),

*E*(

_{0}*x,y*)) wherein i = 1, the free-form surface is regenerated for the new illuminance distribution

*η*·E

_{1}_{0}(

*x,y*). System performance improves with increasing i, the number of iterations. In this method, there is no need to introduce discontinuities to control normal errors. In fact, deviations of illuminance distribution caused by the size of LED source and surface construction can be virtually eliminated after several feedbacks. As an example, a smooth free-form lens with rectangular illuminance distribution is designed, and the illuminance uniformity is improved from 18.75% to 81.08% after eight feedbacks.

## 2. Feedback modification method

#### 2.1 Energy conservation

Let *I*(*u,v*) and *E*(*x,y*) denote the intensity distribution of the light source in the direction (*u,v*) and the desired illuminance distribution in the target plane, respectively. Thus, energy conservation of a lossless optical system can be expressed as
Eq. (1) [9]:

*u,v*)| and |J (

*x,y*)| are the absolute Jacobian factors.

Variable separation mapping method establishes the correspondence between variables in the source and target planes; all rays from the light source are mapped into the target plane as shown in Fig. 2(a) .

Rectangular coordinates (*x,y*) in the target plane can be obtained from the polar coordinates (*u,v*) of the free-form surface expressed as Eq. (2) and Eq. (3):

In fact, we can also obtained (*u,v*) from (*x,y*) expressed as Eq. (4) and Eq. (5), reversely [11],

#### 2.2 Freeform surface construction

The target plane is equidistantly separated into sufficiently small rectangular cells, which specifies all the points **T**(*x,y,z*), wherein z is constant; the freeform surface is also separated into the same number of cells, and the polar coordinates (*u,v*) of these cells’ points **P**(*r,u,v*) can be calculated from Eq. (4) and Eq. (5); Then the freeform surface construction is reduced to solve the lengths *r* of these points to the origin point O.

As shown in Fig. 3
, assuming **P**(*r _{0},u_{0},v_{0}*) on curve

**C**(

*j*) is a known quantity, and it’s corresponding point on the target plane is

**T**(

*x*); Taking

_{0},y_{0},z_{0}**In**(

*u*) as the unit incident vector, the refractive vector

_{0},v_{0}**Out**and normal vector

**N**at point

**P**(

*r*) can be calculated from Eq. (6) and Eq. (7):

_{0},u_{0},v_{0}The length *r* of point **P**(*r,u,v*) on the next curve **C**(*j+1*) can be derived from Eq. (8):

If we know the initial curve C(*1*), then all the points and curves can be obtained from Eqs. (4)~(8); The initial curve is constructed as follows: assuming an initial point, the next point on curve C(*1*) is then calculated as the intersection of the light ray with the tangent plane of the previous point [9]. A smooth freeform surface could be generated by integrating all the curves [10,12]. However, normal errors are produced in this surface construction process, which results in angular deviations of exiting rays [9].

#### 2.3 Feedback modification

As shown in 2.1, the correspondence can be obtained between the rectangular coordinates (*x,y*) and polar coordinates (*u,v*) with the desired illuminance distribution *E _{0}*(

*x,y*). Then the optical model is calculated in 2.2, and simulation with the LED source using the optical model is performed, the simulation result

*E’*(

*x,y*) may have a large deviation from

*E*(

_{0}*x,y*) because of the extended light source and the normal errors. The aim of this letter is to design the optical model with a new given illuminance distribution

*E*(

*x,y*), which can make the simulation result approach the desired one,

*E*(

_{0}*x,y*). An effective way named Feedback modification method is introduced to obtain

*E*(

*x,y*). The idea of this method can be described as follows: during the (

*i+1)-th*iteration, the given illuminance distribution

*E*(

_{0}*x,y*) is modified through the feedback function

*η*(

_{i}*x,y*)

*= η*(

_{i}*E’*(

_{i}*x,y*),

*E’*(

_{i-1}*x,y*)

*,…, E’*(

_{1}*x,y*),

*E*(

_{0}*x,y*)), wherein

*E’*(

_{i}*x,y*) is the simulation result of the

*i-th*iteration. The relationship between the new given illuminance distribution

*E*(

_{i + 1}*x,y*) for

*(i+1)-th*iteration and

*E*(

_{0}*x,y*) can be written as Eq. (9):

However, as shown in Eq. (10), if *E’ _{i}* (

*x, y*)=0,

*E’*(

_{i}*x,y*) <<

*E*(

_{0}*x,y*) or

*E’*(

_{i}*x,y*) >>

*E*(

_{0}*x,y*) at somewhere on the target plane, there will be a dramatic change in the value of

*η*(

_{i}*x,y*). To avoid this from happening, a feedback function with lower and upper tolerance based on Eq. (10) is introduced as Eq. (11) and Eq. (12):

*r*and

_{1}*r*are the lower and upper tolerance, respectively.

_{2}With the new given illuminance distribution *E _{i+1}(x,y*), we can get new correspondence between the rectangular coordinates (

*x,y*) and polar coordinates (

*u,v*); New freeform lens model can be regenerated based on this new correspondence.

A flow diagram of compact free-form lens design for LED source is shown in Fig. 4 . The steps of design are generally divided into four parts: requirement analysis, calculating model, system simulation, and feedbacks using Feedback modification functions.

## 3. Design example

As an example of the proposed method, a uniform illuminance distribution of a street lamp is specified over a rectangular target. As shown in Fig. 5 , all rays from the 1mm×1mm LED source are refracted by the freeform lens with the refractive index of 1.59 onto the 10000mm×30000mm rectangular target region, and the mounting height is 10000mm. Assuming the LED source has a Lambertian radiation distribution with full half space directional range. To simplify the calculation, the LED source is assumed to be immersed in the lens. The virtual point light source used in the design of lens model has the same radiation distribution with the LED source.

The rectangular target region is equidistantly separated into 800 grids, and each grid needs to be assigned the same illuminance. The desired illuminance distribution can be considered as a matrix *E _{0}*(

*l,m*),

*l*=1,2,3,…,40,

*m*=1,2,3,…,20. The feedback function shown in Eq. (11) and Eq. (12) can also be written as Eq. (13) and Eq. (14):

*r*=0.5, the upper tolerance

_{1}*r*=2, and

_{2}*l*=1,2,3,…,40,

*m*=1,2,3,…,20. The lower and upper tolerance values are decided by the actual simulation process, and those values that result in the simulation results converging to the desired distribution with the least feedback times should be selected. In this design example, better simulation results could be obtained by setting lower and upper tolerance value as 0.5 and 2, respectively.

The (*i*)*-th* simulation result *E’ _{i}*(

*l,m*) is produced based on Monte Carlo method, and the illuminance uniformity is defined as Eq. (15):

*E’*and

_{imin}*E’*are the minimum and average illuminance value of the matrix

_{iaverage}*E’*(

_{i}*l,m*), l=1,2,3,…,40, m = 1,2,3,…,20.

The illuminance uniformity is 18.75% if using optical model designed with point light source; it is monotonically increasing to 81.08% on the eighth feedback; after nine times feedback, the illuminance uniformity becomes saturated. (See Fig. 6 ).

Figure 7
show the simulation results based on the Monte Carlo method of rectangular illuminance distribution before feedback and after eight feedbacks. Although Feedback modification method can effectively improve the illuminance uniformity in the given region, it does not guarantee all light rays will be transmitted to the given region, a small portion of the light rays are refracted onto the outside region (beyond the given region) and blur the edges (see Fig. 7b). The initial and final lens models are shown in Fig. 8a
and 8b; the dimension (length, width and central height) of these two lenses is 12.59mm×7.87mm×7mm and 14.63mm×8.93mm×7mm, respectively, and their light output efficiency is 94.75% and 94.03%, respectively. The comparisons of the cross-sectional profiles on *xz* plane and *yz* plane are shown in Fig. 8c and 8d, which clearly show that the dimension of the final lens is broadening in *xz* direction and *yz* direction while the central height maintains 7 mm.

## 4. Conclusions

In this paper, we demonstrate a feedback modification method in the design of compact optical systems with uniform illuminance for LED source. The model of lens is corrected with the simulation results by employing feedback function with lower and upper tolerance until the uniformity of the illuminance distribution is satisfied. A rectangular illuminance distribution with the uniformity of 81.08% is specified over a rectangular target after eight times feedback. The light output efficiency is 94.03% for this design. This method can be applied in the design of non-uniform illuminance distribution, and can also used well in the elimination of deviations of the illuminance distribution caused by other aspects.

## 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 and2008AA03A194),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), Open Program of State Key Lab of Integrated Optoelectronics of China.

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