## Abstract

A two-step optimization method is proposed to design a compact single-surface far-field illumination system, satisfying the requirements of illuminance uniformity and light control efficiency with *h*/*D* less than 3:1. In the first step, the conventional tailored edge-ray design (TED) method is employed to generate prescribed illumination distribution for the rotationally symmetric optical system, and an optimization process is added to reach a balance between illuminance uniformity and light control efficiency. Based on the improved TED method, we can construct an initial optical system more accurate than that obtained by point source assumption. In the second step, an iterative feedback modification process is employed to optimize the initial optical system, so that the degradation of performance due to insufficient control of skew rays is mitigated. Because the initial optical system constructed in the first step is accurate enough, the second-step feedback modification can converge to a satisfactory result within several iterations. As an example, a free-form rotationally symmetric lens with the height of *h* = 25 mm is designed for a discoidal LED source with the diameter of *D* = 10 mm. Both high illuminance uniformity of 0.75 and high light control efficiency of 0.86 are obtained simultaneously. The method can be further used to achieve more complex non-uniform illumination distributions. The design of an optical system with *h*/*D* = 2.5:1 and a circular linear illumination distribution is demonstrated.

© 2014 Optical Society of America

## 1. Introduction

As a new generation of light source, light-emitting diodes (LEDs) have been widely used in different lighting applications due to their high efficiency and long lifetime [1, 2]. Free-form optical systems [3–10] are often employed to improve the lighting performance by reshaping the light intensity distribution of the LEDs, which are taken as point light sources in most cases. However, due to the low power of a single-chip LED (usually about 1W), dozens or even hundreds of LEDs (each with a secondary optical system) are needed to achieve a high flux level required by general lighting applications. This discrete structure results in a huge luminaire dimension, leading to high cost due to considerable waste of materials and inconveniences in assemblage, dismantlement and maintenance of the luminaire.

The above problems can be solved by using the emerging high brightness LED surface light source (e. g. the chip-on-board LED source) [11, 12] instead of conventional single-chip LEDs in the luminaire design. The high brightness LED surface light source, wherein multiple LED chips are directly integrated within a single package, can easily achieve a power of tens to hundreds of watts. As a result, it can obtain a high luminous flux with a relatively small light-emitting area. Therefore, LED surface light source enables an excellent miniaturization and design flexibility of the luminaries, and is considered a very promising technical route.

In fact, point light source or surface light source is a relative concept, as shown in Fig. 1.The ratio of center height *h* of the optical system to the maximum
dimension *D* of the source is used in this paper to approximately describe the
extension of the source as well as the compactness of the optical system. Generally, the source
can be considered as a point source when *h*/*D* is much larger
than 10:1 [3].

A LED surface light source usually has a diameter of over 1 cm [12]. In practical applications, it is desirable to limit the dimension-ratio
*h*/*D* to within 3:1 for system miniaturization and cost
reduction. The extension of the source will have a dramatic impact on the target distribution if
the optical system is based on the point source assumption, as illustrated in Fig. 2.A rotationally symmetric lens with a center height of *h* = 25 mm is
designed adopting the point source approximation. A uniform target distribution is obtained for
a point source. However, obvious deviation occurs when a discoidal LED surface source with
*D* = 10 mm is used.

However, designing a compact optical system for surface light sources is difficult,
because each point on the optical surface can only precisely control one ray emitted from the
light source. As shown in Fig. 3, assuming Ray 1 passing
through a point *P* on the optical surface is precisely refracted into the
direction of Ray 1’, with the normal vector at *P* determined by
Snell’s law, it is found that the refractive directions of other rays (e. g. Rays 2 and
3), emitted from different points across the source and passing through the same point
*P* on the optical surface, are also determined and cannot be controlled as
prescribed.

Most of the existing optical system designing methods are based on point light source assumption. Only a few methods have been reported to cope with the surface light sources. They are mainly classified into two categories: one is inherently taking the extension of the source into account, and the other is still using point source approximation, but with an additional optimization process added. The first category includes the simultaneous multiple surface (SMS) design method [13, 14], the surfaces-overlapping method [15] and the tailored edge-ray design (TED) method [16–19]. The SMS design method is indeed an effective method to deal with extended source, by coupling multiple pairs of incoming and outgoing wavefronts with multiple surfaces designed simultaneously [13]. Both numerical and analytic solutions have been proposed. The SMS design method is a rather effective way to transport the energy of the light source into a prescribed target region. However, as mentioned in Ref [14], it is not easy to directly-calculate or approximate the outgoing wavefront that create a specified irradiance or intensity distribution. As for the surfaces-overlapping method [15], the optical system is constructed by overlapping several optical surfaces constructed using point light sources. The method is simple and easy to operate, yet it relies on a try-and-error strategy. The TED method was proposed in the 1990s and was generally focused on trough reflectors with translational symmetry for strip or tubular sources [16–18]. In the method, trough reflectors were tailored based on the edge-ray principle to control the flashed area and thus the intensity in a given direction. The TED method was extended to design rotationally symmetric reflector by Gordon and Rabl in 1998 [19], in which a quasi-empirical formula for the luminance function was adopted and a uniform illumination distribution was achieved. Recently, based on the concept of the TED method, a double-surface rotationally symmetric lens has been successfully designed for a discoidal LED source by Peter Goldstein, and a uniform angular intensity distribution was generated within a specified range [20]. By adopting two surfaces, part of the light emitted from the LED source can be precisely controlled to some degree, but it increases the difficulties in model fabrication. Besides, the method is introduced to generate prescribed angular intensity distribution rather than the illumination distribution desired for many general lighting applications, such as street lighting and interior lighting. And it is worth noting that all the above-mentioned work on TED usually just takes the lighting uniformity into account, while the efficiency of the optical system, which also plays an important role in illumination design, is to some extent neglected.

Most of the present methods belong to the second category, which firstly construct an initial optical system based on point source assumption, and then introduce some optimization method to improve the performance of the initial system [21–25]. However, these methods are mainly used in the case of *h*/*D* > 5:1, where the simulated distribution does not deviate significantly from the desired distribution. For *h*/*D* < 3:1, all these optimization methods turn out to be hard to obtain a satisfactory result or simply fail to converge, as the start point constructed based on point source assumption becomes much worse. To design a highly compact optical system with high performance, a more precise initial optical system is urgently desired.

In fact, to our best knowledge, for the case of *h*/*D* < 3:1, few methods have been reported to design rotationally symmetric single-surface illumination systems while meeting requirements for illuminance uniformity and light control efficiency. In this paper, a two-step optimization method is proposed for the design of such an optical system, to generate a uniform far-field illumination distribution with an illuminance uniformity exceeding 0.7 and a light control efficiency beyond 0.8. As mentioned above, the performance of the initial point plays a key role in the optimization strategy. Therefore, in the first step, a more accurate initial optical system is constructed by an improved TED method. The classical TED method is used to form a circular uniform illumination pattern, and then an optimization process is added to obtain a trade-off between illuminance uniformity and light control efficiency. In the second step, an iterative feedback modification process is employed to optimize the initial optical system, so that degradation due to insufficient control of skew rays is alleviated [3]. Because the initial optical system constructed in the first step is rather accurate, the second-step feedback modification can converge to a satisfactory result within several iterations. As an example, a compact single-surface lens is designed for a discoid LED surface light source with *h/D* = 2.5:1. After 3 iterations in the second-step optimization, an illuminance uniformity of 0.75 and a light control efficiency of 0.86 are simultaneously secured.

## 2. Design process

The purpose of the paper is to design a high performance compact rotationally symmetric single-surface optical system for a LED surface source to generate a uniform far-field illumination distribution within a prescribed target field with *h/D* < 3:1, while keeping the light control efficiency *η* > 0.8 and illuminance uniformity *U _{E}* > 0.7 [26]. Here, the light control efficiency

*η*is defined as the ratio of the flux within the desired field to the total flux emitted by the source, while the illuminance uniformity

*U*is defined as the ratio of minimum to average illuminance in the specified target region.

_{E}The proposed two-step optimization process is described by a flow diagram shown in Fig. 4.In the first step, an initial lens model is constructed based on an improved TED method, wherein the conventional TED method is utilized to solve a rotationally symmetric illumination distribution problem, and an optimization process is added to ensure a trade-off between illumination uniformity and light control efficiency. As a result, a more accurate initial lens model is constructed than that obtained with the conventional point source approximation. In the second step, an iterative feedback modification process is employed to optimize the initial lens model, so that the degradation of performance due to insufficient control of skew rays is alleviated. As the TED method is originally proposed to solve the two-dimensional (2D) light reshaping problem, it inherently ignores the impact of the skew rays in a three-dimensional (3D) optical system design. Although the initial lens model generated by the improved TED method performs much better than using point source assumption, it is still defective and a second-step optimization is necessary. A detailed description of the second-step optimization will be discussed later in Section 2.2. It is worth noting that, as a result of the high performance of the initial lens obtained by the improved TED method, the second-step feedback optimization is very fast and effective, usually requiring less than ten iterations.

#### 2.1 First step: constructing an initial lens model based on an improved TED method

The design process can be done in 2D condition because the light source, the lens, and the desired illuminance distribution are all rotationally symmetric. So in this paper only the 2D design process is taken into account. The 3D model of the lens can be obtained simply by rotating the designed 2D profile. The conventional TED method is used here to achieve an illumination distribution with rotational symmetry.

The far-field lighting system is shown in Fig.
5(a), where *H* is the distance between the light source and the target
plane, *θ* is the ray angle with respect to the light axis, and
*x* is the corresponding coordinate of the target plane. As detailed in the
Appendix, we can easily get:

*I*(

_{2D}*θ*) is the luminous intensity in the direction of

*θ*and

*E*(

_{2D}*x*) is the illuminance at point

*x*, both in 2D condition. We can also obtain Eq. (2) as described in the Appendix:where

*L*

_{2}

*is the surface luminance of the lens in 2D condition and*

_{D}*W*(

*θ*) is the apparent projection length of the LED source in the direction of

*θ*as seen through the lens shown in Fig. 5(b). The system is assumed lossless and the LED source is taken to be Lambertian. The LED illuminates the lens, and then the surface of the lens becomes a secondary Lambertian source with a uniform luminance according to the conservation of luminance. Therefore,

*L*

_{2}

*is constant and independent of the direction*

_{D}*θ*. For the design of generating a uniform illuminance distribution

*E*

_{2}

*(*

_{D}*x*) =

*E*

_{0}, through Eqs. (1) and (2), we can get:where

*W*

_{0}is the projected length in the axial direction. Therefore, in order to obtain a uniform illuminance distribution we need just to design a lens that can control the apparent-projected length of the LED source in prescribed directions in the form of Eq. (3). However, what calls for special attention is that only the meridian rays are taken into consideration in the 2D design process, while the skew rays are inherently ignored as shown in Fig. 5(c). This will bring a performance degradation of the 3D optical system generated simply by rotating the 2D profile. A second-step optimization is introduced later in Section 2.2 to alleviate the influence of the ignored skew rays.

Rabl and Gordon described 4 fundamental classes of 2D TED solutions in Ref [17]. All the 4 solutions performed very well for reflectors. The 4 classes of solutions can be transformed into 4 types of solutions for the lens design, as shown in Fig. (6). Figures 6(a) and 6(b) are both related to the far-edge diverging and the near-edge diverging solutions for reflector design, whereas Figs. 6(c) and 6(d) correspond to the far-edge converging and the near-edge converging solutions. It is obvious that, for the solutions illustrated in Figs. 6(b)-6(d), large deflection of the ray trajectories is desired and the total internal reflection inevitably happens at the bottom part of the lens, making the lens design impractical [3]. So in this paper, we just take the diverging solution shown in Fig. 6(a) into account.

An intermediate-points-auxiliary geometric configuration method is used here to construct the 2D
profile of the lens (see Fig. 7). The *y*-axis is assumed to be the rotational symmetry axis of the source
and the lens. *F* and *F’* are two edge points of the
source. Only the profile of the lens in the first quadrant needs to be designed because of the
symmetry of the system. The design process is described as follows.

*1). Presetting a quadratic polynomial curve A*_{0}*B*_{0}*as the central portion of the profile*

The profile of the system is constructed from the center to the border. The central portion *A*_{0}*B*_{0} of the profile is preset as the quadratic polynomial form of Eq. (4):

*A*

_{0}

*B*

_{0}described as follows.

Assuming that the incident ray *FA*_{0} emitted from the edge point *F* refracts at point *A*_{0} and the outgoing ray is parallel to the *y*-axis, while another ray *F’B*_{0} emitted from the edge point *F’* refracts at point *B*_{0} and also emits parallel to the *y*-axis. According to the edge-ray principle, the projection length *W*_{0} along the *y*-axis is determined by the outgoing rays of the edge incident rays *FA*_{0} and *F’B*_{0}. Given the position of *B*_{0}, the normal vector *N*_{0} at *B*_{0} is calculated with the Snell’s law. Then, the preset quadratic polynomial curve *A*_{0}*B*_{0} with symmetry about the *y*-axis is obtained. It is obvious that the curve *A*_{0}*B*_{0} is determined by the choice of point *B*_{0}. The selection of point *B*_{0} should follow two additional principles: Firstly, limitations and requirements of the size of the lens should be taken into account; Secondly, the projection angular range needs to be guaranteed to cover the specified illuminating region.

*2). Generating the rest part of the profile by an intermediate-points-auxiliary geometric construction method*

As described above, the projection length *W*_{0} in the *y*-axis direction can be obtained by presetting the curve *A*_{0}*B*_{0}. Considering the next ray *FA _{M}*

_{0}emitted from point

*F*which makes a very small angle with the ray

*FA*

_{0}, it intersects the curve

*A*

_{0}

*B*

_{0}at point

*A*

_{M}_{0}, as shown in Fig. 7. The normal vector at point

*A*

_{M}_{0}can be calculated from the analytical expression of the quadratic polynomial curve

*A*

_{0}

*B*

_{0}. The outgoing ray of

*FA*

_{M}_{0}forms an angle

*θ*

_{M}_{0}with the

*y*-axis, which can be calculated according to Snell’s law. The projection length

*W*(

*θ*

_{M}_{0}) of the source in the direction

*θ*

_{M}_{0}can be obtained according to Eq. (3). The point

*B*

_{M}_{0}is then determined as the intersection point between the tangent line at point

*B*

_{0}and the line in the direction

*θ*=

*θ*

_{M}_{0}whose distance to point

*A*

_{M}_{0}is

*W*(

*θ*

_{M}_{0}). The point

*B*

_{M}_{0}is treated as an intermediate auxiliary point to calculate a new point on the profile, thus the construction accuracy of the profile is guaranteed by replacing a sequence of tangent segments usually adopted.

Again, considering the next ray *FA*_{1} emitted from point *F* which makes a very small angle with the ray *FA _{M}*

_{0}, it intersects the curve at point

*A*

_{1}. The direction angle

*θ*

_{1}of the outgoing ray with the

*y*-axis and the projection length

*W*(

*θ*

_{1}) in the direction

*θ*

_{1}can also be calculated according to Eq. (3). Then the new point

*B*

_{1}of the lens profile is calculated as the intersection point between the straight line passing through the auxiliary point

*B*

_{M}_{0}with direction perpendicular to the normal vector

*N*

_{1}at point

*B*

_{1}, which relates to the position of point

*B*

_{1}and the refraction ray at point

*B*

_{1}in the direction

*θ*=

*θ*

_{1}with a distance of

*W*(

*θ*

_{1}) to point

*A*

_{1}. Repeat the above process till the point

*B*(

_{i}*i >*1) reaches the

*x*-axis, and the complete 2D profile of the lens is obtained.

It is worth noting that the projection length *W*(*θ _{i}*) is determined by the outgoing rays of

*FA*and

_{i}*F’B*, i.e., the rays

_{i}*FA*and

_{i}*F’B*which are used to define edge rays guarantee the mapping between the boundary of the source and target. This conclusion is confirmed by the fact that the output ray of the ray passing through any point, such as

_{i}*O’*, on

*OF’*and

*B*(

_{i}*i*= 0, 1, 2 …) will form an angle larger than

*θ*with the

_{i}*y*-axis according to the Snell’s law, as illustrated in Fig. 7.

*3). Optimizing the 3D model of the lens*

An optimization process is then added to the above-mentioned TED process to obtain a balance between the illumination uniformity and the light control efficiency.

The 3D model of the lens is obtained by rotating the 2D profile obtained above about the *y*-axis. The performance of the lens is simulated using Monte-Carlo ray tracing method. However, the simulation result usually deviates from the requirements to some extent due to somewhat arbitrary choice of the preset curve *A*_{0}*B*_{0}, which determines the profile of the lens. Therefore, the curve *A*_{0}*B*_{0} should be optimized to improve the performance. Moreover, as the curve *A*_{0}*B*_{0} is determined by point *B*_{0}, coordinates of point *B*_{0} are chosen as the optimization variables. Thus, the optimization problem can be described as:

*MF*is the merit function used to evaluate the overall lighting performance, and

*η*and

_{T}*U*are the minimum requirements for

_{ET}*η*and

*U*, respectively. For a single optical surface and a surface light source, it is hard to simultaneously control the edge falloff and the distribution. Namely, tradeoffs between

_{E}*η*and

*U*must be made. Thus the merit function

_{E}*MF*is defined as:where σ (0 < σ < 1) is the weight factor that controls the trade-off between

*η*and

*U*. Constantly modify the location of point

_{E}*B*

_{0}and repeat the design process mentioned above till

*MF*is optimal, and the initial optimal lens model is obtained.

A more detailed explanation of the effect of optimization is illustrated in Fig. 8. If the position of *B*_{0} is selected to make the exit ray of
ray *FB*_{0} take an angle of *θ _{max}*,
the exit angle

*θ*’ of the ray

_{i}*FB*will be larger than the prescribed maximal emergence angle

_{i}*θ*Therefore, if we choose the position of

_{max}.*B*

_{0}to make

*θ*right corresponding to the edge of the target field as the conventional TED method does, the illumination uniformity within the target region may be ensured, but a large portion of light may be thrown outside of the target. On the hand, if the position of

_{max}*B*

_{0}is selected to make

*θ*much smaller, an improvement of the light control efficiency can be realized. However, the illumination uniformity will deteriorate. As a result, the preset curve

_{max}*A*

_{0}

*B*

_{0}must be optimized to get a balance between the light control efficiency and the illumination uniformity.

#### 2.2 Second step: iterative feedback modification of the lens model

After the first step, the initial optimal lens model is obtained and the overall *MF* is optimized. The corresponding simulation result of the initial lens model obtained above is usually much better than that based on either point source assumption or conventional TED method. However, the resulted illuminance distribution may still deviate somewhat from the desired one, especially for the central region. One reason is the insufficient control of skew rays, which exist inherent in the TED method [16–20], and the other may be the errors occurred in the numerical construction process. An iterative feedback method [24] is introduced to compensate for this deviation, as illustrated in Fig. 4. Assuming the desired illuminance distribution along *x*-axis is *E*_{0}(*x*) and the simulation result is *E _{S}*(

*x*), a new illuminance distribution

*E*(

_{M}*x*) is first calculated by modifying

*E*

_{0}(

*x*) with a specific feedback function

*β*(

*x*) based on the deviation between

*E*

_{0}(

*x*) and

*E*(

_{S}*x*). Then the projection length

*W*(

*θ*) of the light source through the lens is adjusted using the new illuminance distribution

*E*(

_{M}*x*) according to Eqs. (1) and (2). Finally, the lens model is redesigned using the new

*W*(

*θ*) and the optimized preset curve

*A*

_{0}

*B*

_{0}based on the method described in section 2.1. Multiple iterations should be performed until the final simulation result converges or becomes acceptable. The feedback function used here is defined as:

*α*

_{1}≤ 1 and

*α*

_{2}> 0 are the adjustable parameters, and

*E*(

_{Sj}*x*) is the simulation result of the

*j-*th iteration. After

*J*times of iterations, the modified illuminance distribution

*E*(

_{MJ}*x*) becomes:Usually several iterations are enough for the second-step optimization to converge to an optimal result.

## 3. Design example

#### 3.1 Parameters setting

As an application example of the proposed method, a rotationally symmetric lens with a center
height of 2.5 cm is designed for a discoidal LED surface light source with a diameter of 1 cm
to generate a uniform illuminance distribution within a given circular target field, as
depicted in Fig. 9. The dimension-ratio *h*/*D* is 2.5 for this optical
system, which represents a high compactness. The LED surface source is assumed to be a
Lambertian source with the maximum divergence half-angle of 90°. The circular target
plane with a radius of *R* = 10 m is placed *H* = 10 m away from
the source. The LED source is assumed to be immersed in the lens. The refractive index of the
lens material is set as *n* = 1.59.

Other design parameters are selected as follows: *σ* = 0.5, *α*_{1} = 0.5, *α*_{2} = 1, *η _{T}* = 0.8 and

*U*= 0.7. The restraint on

_{ET}*U*follows the CIE standard for lighting of work places [26]. As a result, the merit function

_{E}*MF*in Eq. (6) becomes:

Using these functions, an initial lens model is first constructed by the improved TED method. And then, the second-step iterative feedback modification is employed to further improve the performance of the initial lens. It takes 3 iterations in less than 2 minutes for the second-step optimization to converge. Therefore, the second-step is very fast and effective, owing to the accuracy of the initial lens model.

#### 3.2 Results and discussion

The trade-off between light spillage and illumination uniformity is made during the first-step
optimization by adjusting the position of *B*_{0} and thus the maximal
exit angle *θ _{max}* as well as the entire curve

*A*

_{0}

*B*

_{0}. The optimized result (black line) is compared with the conventional TED design (red line) in Fig. 10.Here, the input ray angle refers to the angle formed by the ray emitted from

*F*passing through

*A*

_{0}

*B*

_{0}and the newly generated curve on the right hand with the

*y*-axis, while the output ray angle represents the angle made by the corresponding exit ray with the

*y*-axis, as shown in Fig. 8. In this example

*H*=

*R*, so the ideal maximal exit angle is

*π*/4, marked as the blue line in Fig. 10. However, as mentioned in Section 2.1, if we choose the position of

*B*

_{0}to make the maximal exit angle

*θ*right corresponding to the edge of the target illuminating field as conventional TED method does, a large portion of light will fall outside of the target. Conversely, if

_{max}*θ*is made much smaller, the light spillage effect will be reduced, unfortunately at the cost of degraded illumination uniformity. The optimized maximal emerging angle

_{max}*θ*in this example is about 34 degrees, as shown in Fig. 10 by the black line, and an evident reduction of the light spillage is achieved.

_{max}The simulated performances of the lens after the first- and the second-step optimization
are shown in Figs. 11(c) and 11(d) respectively. As a comparison, the simulation results of lens designed
using point source assumption with and without feedback modification are also shown in Figs. 11(a) and 11(b),
respectively. None of the results shows a clear edge due to the intrinsic limitations of a
single optical surface and a surface light source. Result of the initial design based on point
source approximation shows a highly concentrated energy distribution in the central portion of
the target plane, which can be improved by employing feedback modification, as shown in Figs. 11(a) and 11(b).
But there is still considerable fluctuation in the central part after feedback modification,
because it is hard for the feedback modification process to converge or obtain an optimal
result from a bad start point obtained with point source assumption. The initial optical system
constructed with point source assumption has a large light control efficiency of 0.95, while
the illuminance uniformity is only 0.40, which is far below the minimum requirement of
*U _{ET}* = 0.7. By feedback modification, the final optical system
designed by point source approximation has an illuminance uniformity of 0.70, just passing the
minimum requirement, but the light control efficiency is only 0.78, short of the minimum
requirement of

*η*= 0.8.

_{T}When the proposed two-step optimization is used, the initial simulation result is much better except for a small depression in the center, which is almost completely eliminated by the second-step feedback modification, as shown in Figs. 11(c) and 11(d). The optimized optical system exhibits a high illuminance uniformity of 0.75 as well as a high light control efficiency of 0.86, satisfying both requirements simultaneously. It is worth noting that, the illuminance uniformity in central regions with radius < 8.5 m is even higher than 0.90, as can be seen in Fig. 11(d).

The lens model obtained with the proposed two-step optimization is shown in Fig. 12.The height of the lens is 25 mm, while the diameter is about 33.5 mm.

In order to demonstrate the effectiveness of the proposed method in designing highly compact
optical systems, a series of optical system models with *h/D* = 3, 2, 1.5 and 1
are designed gradually shrinking the central height of the lens while keeping the diameter of
light source and all the other settings the same as described in Section 3.1. The corresponding
simulation results are shown in Fig. 13.All the three lighting parameters, *U _{E}*,

*η*and

*MF*, decrease with the dimension ratio

*h/D*. The illuminance uniformity

*U*> 0.70 and light control efficiency

_{E}*η*> 0.80 are secured for all dimension ratios except the case with

*h/D*= 1. Even for

*h/D*= 1,

*U*= 0.68 and

_{E}*η*= 0.62 could still be obtained. As a whole, the merit function

*MF*is used to indicate the performance of the design, and it changes from 0.85 to 0.65 as

*h/D*decreases.

In fact, this method can also be used to form more complex non-uniform illumination distribution,
such as linear illumination distribution. As an example, an optical system with
*h*/*D* = 2.5:1 is also designed to form a circular linear
illumination distribution. Apart from the target distribution, all the other parameters are the
same as those in the example mentioned in Section 3.1. The target distribution is described as
Eq. (11):

*θ*:

*N*is the total number of the positions to be calculated while

*E*(

_{S}*x*) and

_{i}*E*(

*x*) are the measured and the desired illuminance at point

_{i}*x*, respectively.

_{i}## 4. Conclusions

In practical applications, highly compact optical system is usually desired for LED surface light source to miniaturize the overall illumination system. For example, the height of an optical lens is usually expected to be less than 3 cm for a 1-cm-diameter LED source, i.e., the dimension-ratio *h*/*D* is usually desired to be limited to less than 3:1. In addition, a practical optical system should provide a high illuminance uniformity *U _{E}* up to 0.7 as well as a high light control efficiency

*η*up to 0.8. However, few effective methods have been reported yet to design such an optical system with high performance. We propose a two-step optimization method to design compact free-form rotationally symmetric single-surface optical system with

*h*/

*D*< 3:1. In the first step, an accurate initial optical system model is constructed by an improved TED method, where an optimization process is added to the conventional TED method to get a balance between

*η*and

*U*. In the second step, an iterative feedback modification strategy is adopted to mitigate the performance degradation due to insufficient control of skew rays. It takes only a limited number of iterations for the second-step feedback modification to converge to a satisfactory result, thanks to the high accuracy of the initial optical system constructed in the first step. As an example, a free-form rotationally symmetric lens with a height of

_{E}*h*= 25 mm is designed for a discoidal LED source which has a diameter of

*D*= 10 mm. High illuminance uniformity of 0.75 and high light control efficiency of 0.86 are obtained simultaneously. The proposed method can also be used to form more complex non-uniform illumination distribution. An optical system with

*h*/

*D*= 2.5:1 is also designed to form a circular linear illumination distribution using the method, where a light control efficiency of 0.85 and a relative standard deviation of 0.039 are obtained. It is illustrated that this method is very effective in designing highly compact optical system with high performance. However, this method is limited to designing rotationally or translationally symmetric optical systems at present, and how to extend it to design more complex free-form optical system remains the topic for further investigation.

## Appendix

As illustrated in Fig. 5, the luminous intensity *I _{2D}* (

*θ*) in the direction of

*θ*in 2D condition is defined as [27]:

*dΦ*and

*dθ*are the differentials of the flux

*Φ*and the ray angle

*θ*, respectively. The illuminance

*E*(

_{2D}*x*) at point

*x*is defined as [27]:where

*dx*is the differential of

*x*. The differential relationship between

*θ*and

*x*can be easily derived from Fig. 5(a):According to Eqs. (14)-(16), we can get:The definition of luminance

*L*

_{2}

*in 2D condition is described as [27]:From Eqs. (14) and (18), we can get:*

_{D}*W*(

*θ*) is the apparent projection length of the source in the direction of

*θ*as seen through the lens.

## Acknowledgment

This work was supported by the National Natural Science Foundation of China (Grant Nos. 61307024, 61176015, and 61176059), the National Key Technology Research and Development Program of the Ministry of Science and Technology of China (Grant Nos. 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).

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