Abstract

Designing an illumination system for a surface light source with a strict compactness requirement is quite challenging, especially for the general three-dimensional (3D) case. In accordance with the two key features of an expected illumination distribution, i.e., a well-controlled boundary and a precise illumination pattern, a two-step design method is proposed in this paper for highly compact 3D freeform illumination systems. In the first step, a target shape scaling strategy is combined with an iterative feedback modification algorithm to generate an optimized freeform optical system with a well-controlled boundary of the target distribution. In the second step, a set of selected radii of the system obtained in the first step are optimized to further improve the illuminating quality within the target region. The method is quite flexible and effective to design highly compact optical systems with almost no restriction on the shape of the desired target field. As examples, three highly compact freeform lenses with ratio of center height h of the lens and the maximum dimension D of the source ≤ 2.5:1 are designed for LED surface light sources to form a uniform illumination distribution on a rectangular, a cross-shaped and a complex cross pierced target plane respectively. High light control efficiency of η > 0.7 as well as low relative standard illumination deviation of RSD < 0.07 is obtained simultaneously for all the three design examples.

© 2014 Optical Society of America

1. Introduction

LED surface light sources are very promising in solid-state lighting [1]. However, the design of a beam shaping optical system for a surface light source with a strict compactness requirement is quite challenging and remains a topic of further investigation [2, 3]. Several methods have been proposed to handle this case [414], which can be divided into two major categories: one is considering the source extension inherently [46], and the other is combining an optimization strategy with the point source assumption [714].

The first category contains mainly two methods: the tailored edge-ray design (TED) method [4, 5] and the simultaneous multiple surface (SMS) design method [6]. The TED method generally focuses on reflectors with translational or rotational symmetry, where the reflectors are constructed on the basis of the edge-ray principle [4, 5]. However, it cannot be directly applied to designing three-dimensional (3D) systems. The SMS design method is quite effective to transport the energy of an extended source into a specified target region, by coupling multiple pairs of incoming and outgoing wavefronts with multiple surfaces calculated simultaneously [6]. However, our paper is focused on generating a prescribed distribution using only one surface rather than multiple surfaces. Therefore, the SMS design method is not considered in our case.

Most of the methods belong to the latter category which are fairly easy to handle [714], and they have been widely applied in designing various illumination optical systems. In this method, an initial optical system is calculated by the point source approximation before employing certain optimization algorithm to improve its performance. The ratio h/D, where h is the center height of the optical system and D is the maximal dimension of the source, is used herein to express the compactness of the optical system. High performance can be achieved using this method for h/D > 5:1, when the initially simulated distribution does not deviate too much from the expectation. However, when h/D < 3:1 (commonly demanded for LED surface light source to miniaturize the optical system and reduce the cost), it turns out to be hard to converge, because the initial system directly obtained with the point source approximation gets much worse. Therefore, in this case, a better initial optical system is urgently demanded to ensure the robustness and effectiveness of the optimization process. In our previous work [15], we proposed to construct the initial system by an improved TED method, which is more precise than that obtained using a point source. However, the method is restricted to the design of optical systems with translational or rotational symmetry. In this paper, we will demonstrate a new method that can be effectively applied to the general 3D cases.

In fact, for the design of an excellent illumination system, two critical points must always be taken into account, as illustrated in Fig. 1: firstly, energy of the source should be effectively transferred to the target region, i.e., the boundary of the target distribution must be precisely controlled; secondly, the illumination pattern within the target must also be ensured. In the above mentioned optimization methods, the two key issues are handled together using a certain optimization algorithm, i.e., the boundary shape and the illumination pattern are controlled simultaneously. When the optical system is highly compact, the performance of the initial system is very poor, and both the boundary shape and the pattern are far from ideal. Therefore, the optimization algorithm is very inefficient.

 figure: Fig. 1

Fig. 1 Two key points for illumination design and the corresponding two-step optimization strategy.

Download Full Size | PPT Slide | PDF

As a consequence, we propose in our method to divide the optimization design process into two sequential steps in consistence with the two concerns (see Fig. 1): the boundary shape of the simulated distribution is primarily controlled in the first step by an optimization strategy, then the illumination pattern within the target region is further optimized in the second step using another strategy. It is worth noting that, the aim of the first step in the two-step optimization method is also to acquire a more precise initial point for the second step. The design process is detailed in Section 2. To show the flexibility and efficiency of the method, a series of highly compact freeform lenses with h/D ≤ 2.5 are designed as examples in Section 3.

2. Highly compact 3D freeform lens design for LED surface light source

The flow diagram of the two-step optimization design process for a highly compact freeform lens is illustrated in Fig. 2.

 figure: Fig. 2

Fig. 2 A flow diagram of the two-step optimization process. The first step (in the red box) is a combination of two algorithms: iterative feedback modification of the illumination and target shape scaling. The second step (in the blue box) is realized by optimizing a set of selected radii of the initial optimal lens model obtained in the first step.

Download Full Size | PPT Slide | PDF

The purpose of the first step (Section 2.1) is to achieve a better match of the boundary of the simulated distribution to the desired target. Actually, the first step contains two algorithms: iterative feedback modification of the illumination (Section 2.1.3) and target shape scaling (Section 2.1.4). The two algorithms are combined to realize the first-step optimization, as shown in the red box of Fig. 2. The whole process of the first-step optimization (Section 2.1) can be summarized as follows:

  1. Prescribe the target shape and the illumination pattern.
  2. Based on the target shape and the illumination pattern, establish the source-target mapping using an assumed point source (Section 2.1.1). During this process, a simplified method is used as a replacement for the conventional method [1618] to calculate the mapping. Mapping between the point source and any complex-shaped target can be generated more easily and faster with this method, making the optimization routine much more efficient and practical.
  3. According to the mapping, construct the lens by a geometrical method and the NURBS method (Section 2.1.2), and simulate the performance of the lens with the surface light source.
  4. Define and calculate the merit function MF1 (a weighted combination of the target shape deviation and the illumination deviation, Section 2.1.3) for the first-step optimization.
  5. Based on the simulation result, iteratively modify the prescribed illumination pattern by a feedback function, and correspondingly repeat steps 2-4 using the modified illumination pattern until MF1 is optimal for the current target shape (Section 2.1.3).
  6. Change the target shape (by adjusting several coefficients), and correspondingly repeat steps 2-5 (Section 2.1.4) using the new target shape until MF1 is optimal for all the target shape, and then output the initially optimal lens model.

The purpose of the second step (Section 2.2) is to further optimize the illumination pattern within the target region. It is realized by optimizing a set of selected radii of the initial optimal lens model obtained in the first step, as shown in the blue box of Fig. 2 (Section 2.2). The whole process of the second-step optimization (Section 2.2) can be summarized as follows:

  1. Choose a series of radii in certain directions of the initial optimal lens model obtained in the first step as the optimization variables.
  2. Construct the lens by NURBS method using the radii, and simulate the performance of the lens with surface light source.
  3. Define and calculate the merit function MF2 (defined as a weighted combination of the illumination deviation and the light control efficiency) for the second-step optimization.
  4. Adjust the selected radii, and correspondingly repeat steps 2 and 3 using the new radii until MF2 is optimal, and then output the finally optimal lens model.

2.1 First-step optimization

As mentioned above, the process of the first-step optimization involves basically four parts: mapping establishment, surface construction, illumination feedback modification and target shape scaling, which will be detailed respectively in subsections 2.1.1-2.1.4.

2.1.1 Establishing source-target ray mapping

Suppose that the system is lossless, the flux conservation between the source and the target can be described as:

ΩSI(θ,φ)sinθdθdφ=ΩTE0(x,y)dxdy,
where (θ, φ) are spherical coordinates used to stand for the direction of the ray emitted from the source, I(θ, φ) is the luminous intensity distribution of the source, ΩS denotes the solid angle of the source ray directions, (x, y) are Cartesian coordinates used to specify the position on the target plane, E0(x, y) is the target illumination distribution, and ΩT is the corresponding target region to be illuminated. Equation (1) is a classical topology problem of transferring a pattern (source intensity distribution) into another pattern (target illumination distribution). Infinite possible mappings can be derived by solving the double integral along different integration paths:
(x,y)=(f0(θ,φ),g0(θ,φ)).
Several methods were proposed to generate the source-target ray mapping and proved quite convenient [1618]. However, considering both the complexity of the target shape and the considerable iteration number of the optimization process, we propose here another simplified mapping construction method to establish the mapping, which is much simpler and faster. Moreover, the ray mapping is usually constructed using an ideal point source, so a real source might make the accuracy of the ray mapping unnecessary, which means proper simplifications are allowed.

In our proposed method, the polar coordinate system (ρ, γ), whose origin is set as the geometrical center of the target region, is used to describe the target plane instead of the Cartesian coordinate system (x, y). The surface light source is approximated as a point light source at its geometric center with the same luminous intensity distribution, as shown in Fig. 3.The spherical coordinate system (θ, φ) is used to specify the ray directions. The ray mapping between the assumed point source and the target is simplified into several 2D mappings along different azimuth angles. The azimuth angle φ and the zenith angle θ of the source are divided equally into M + 1 and N + 1 parts respectively:

φi=φmin+φmaxφminMi(i=0,1,M),
θj=θmin+θmaxθminNj(j=0,1,N),
where φmin and φmax are the minimum and maximum azimuth angles of the source, while θmin and θmax are the minimum and maximum zenith angles of the source respectively. Then the polar angle of the target plane is correspondingly divided into M + 1 parts:
γi=φi(i=0,1,M).
The desired boundary shape of the target region can be described as:
S(ρB,γ)=0,
where ρB is the polar radius of the boundary, i.e., the maximum polar radius along the polar angle γ. The boundary radius ρBi along γi can be derived by solving Eq. (6) with γ = γi. For each azimuth angle φi and the corresponding polar angle γi, the 2D energy conservation between the source and the target can be described as:
θ0θjI(θ,φi)dθθ0θNI(θ,φi)dθ=ρi,0ρi,jE0(ρ,γi)dρρi,0ρBiE0(ρ,γi)dρ(i=0,1,M,j=0,1,N),
where ρi,0 is the minimum polar radius of the target in the direction of γi. According to Eq. (7) the polar radii grids ρi,j can be calculated as a function about φi and θj:
ρi,j=F(θj,φi).
From Eqs. (5) and (8), the mapping between the directions of the assumed point source and the positions of the target is established as follows, as shown in Fig. 3:

 figure: Fig. 3

Fig. 3 Establish the source-target mapping using an assumed point source which is at the geometric center of the surface light source with the same luminous intensity distribution as the surface source. For a point source, a point on the lens profile will generate a single point on the target plane.

Download Full Size | PPT Slide | PDF

(θj,φi)(ρi,j,γi){γi=φiρi,j=F(θj,φi)(i=0,1,M,j=0,1,N).

As shown in Eq. (9), the simplified mapping is generated under the assumption that γi = φi which is usually not correct for a point source unless the optical system is rotationally symmetric. However, the extension of the source will make the accuracy of the ray mapping for a point source unnecessary and the simplified mapping is acceptable. A further qualitative analysis of the simplified mapping is illustrated in Fig. 4.Suppose that the profiles of the lens have been constructed according to the point-source based ray mapping, a point on the profiles will create a finite-size image of a real source on the target plane, which appears as a small light spot as shown in Fig. 4. Then, by overlapping the small spots, for each profile with azimuth angle φi, it will generate a large light spot along the corresponding radial direction γi on the target plane. Constrained by the mapping established with point source assumption, the dimension of the large spot generated with the surface light source along γi will be around ρBi, considering the finite size of each small spot. Overall, through the overlapping of all the large light spots created by each of the profiles, a distribution with the boundary approaching the target is obtained. Moreover, the boundary of the overlapped distribution can be flexibly adjusted by scaling ρBi when constructing the mapping. Therefore, the simplified mapping is reasonable, and it is suitable for the first-step optimization.

 figure: Fig. 4

Fig. 4 For a surface light source, a point on the lens profile will create a finite-size light spot on the target plane. Through the overlapping of all the spots, a target distribution is obtained.

Download Full Size | PPT Slide | PDF

2.1.2 Constructing freeform lens model with geometrical construction method

Using the ray mapping obtained in Section 2.1.1, each profile of the lens with different azimuth angle could be calculated with a geometrical construction method as mentioned in [17], as shown in Fig. 5.Specifying an initial point P0 on each longitudinal profile, then the unit normal vector N0 at P0 can be calculated by the Snell’s Law based on the ray mapping:

N0=(n2Out0n1In0)/|n2Out0n1In0|,
where In0 and Out0 are respectively the unit vectors of the incident and refractive rays at P0 while n1 and n2 are respectively refractive indexes of the lens and the surrounding medium. Then the next point P1 on the profile can be calculated as the intersection of the next incident ray In1 with the tangent plane at P0, and the normal vector N1 at P1 can be calculated by Eq. (10) according to the ray mapping. Repeating this process, each profile with different azimuth angle is generated. Specially, if θ0 ≠ 0, a seed curve along latitude direction should be first constructed using the above-mentioned method. A smooth freeform surface of the lens could be generated by interpolating the profiles with a NURBS surface, and then the model of the lens is obtained.

 figure: Fig. 5

Fig. 5 Numerical configuration of the profile by geometrical construction method.

Download Full Size | PPT Slide | PDF

2.1.3 Iterative feedback modification

In 2.1.2, the lens model is generated, and the performance of the lens is simulated with the surface light source. However, the simulation result ES(ρ, γ) usually has a large deviation from the desired illumination distribution E0(ρ, γ), mainly caused by the non-ignorable dimension of the source. The feedback method [10, 15] is utilized to alleviate the deviation to some extent. Based on the deviation between ES(ρ, γ) and E0(ρ, γ), a specific feedback function β(ρ, γ) is defined to modify the currently preset illumination distribution E0(ρ, γ), and a new preset illumination distribution EM(ρ, γ) is calculated. Then the freeform surface lens is redesigned in the same method mentioned in Sections 2.1.1 and 2.1.2 using the newly prescribed illumination distribution EM(ρ, γ). Achieving an acceptable simulated distribution requires multiple iterations. The feedback function for the k-th iteration is defined based on the difference between the k-th simulated illumination distribution and the desired distribution:

βk(ρ,γ)={E0(ρ,γ)/[λ1ESk(ρ,γ)+(1λ1)E0(ρ,γ)]}λ2,
where 0 ≤ λ1 ≤ 1, λ2 ≥ 0 are the weighting parameters, and ESk(ρ, γ) is the simulated illumination of the k-th iteration. The correspondingly modified illumination of the k-th iteration EMk(ρ, γ) is:

EMk(ρ,γ)=Πi=1kβi(ρ,γ)E0(ρ,γ).

It is worth noting the iterative feedback modification is just one part of the first-step optimization process where a target shape scaling algorithm is also included which will be described in Section 2.1.4. The merit function for the first-step optimization is defined as:

MF1=ω1RSDshape+(1ω1)RSD,
where 0 < ω1 < 1 is the weight and ω1 is usually close to 0.5, RSDshape is the relative standard shape deviation, and RSD is the relative standard deviation of the simulated illumination from the desired. To give a more precise evaluation of the target distribution, more calculation points on the target plane are needed. Therefore, the mesh of the target for calculation should be reconstructed. The polar angle is redivided into M1 + 1 parts while the polar radius along each polar angle is redivided into N1 + 1 parts:
γi=φmin+φmaxφminM1i(i=0,1,,M1),
ρi,j=ρmini(γi)+ρmaxi(γi)ρmini(γi)N1j(j=0,1,,N1),
where ρmini(γ′i) and ρmaxi(γ′i) are respectively the minimum and maximum polar radius of the desired target plane along polar angle γ′i. Then RSDshape and RSD can be calculated as following:
RSD2shape=i=0M1(ρSi(γi)ρmaxi(γi)ρmaxi(γi))2M1,
RSD2=i=0M1j=0N1(ES(ρi,j,γi)E0(ρi,j,γi)E0(ρi,j,γi))2(M1+1)(N1+1)1,
where ρSi(γ′i) is the maximum polar radius of the simulated illumination pattern.

Usually 4~6 times of iterations are enough to obtain an approximately optimal result that MF1 is optimal for the current target shape which takes less than one minutes with a 3.4 GHz Intel(R) Core(TM) i7-3770 CPU.

2.1.4 Target shape scaling

By Section 2.1.3, the prescribed illumination pattern is optimized, and the performance of the lens is improved to some extent with a much better simulated distribution than the starting point source solution. However, as expressed in Fig. 4, directly using the desired target shape to design the lens, the boundary shape of the simulated illumination pattern usually has a large deviation from the expectation. Therefore, besides the iterative feedback modification algorithm, a target shape scaling strategy is also employed in the first-step optimization process as shown in Fig. 6.The two algorithms are combined together to obtain a better match of the simulated illumination boundary to ideal. The target shape optimization is realized by iteratively scaling the prescribed boundary shape of the target with several coefficients. Here, two parameters are introduced to adjust the prescribed target shape, with which the modified shape is defined as:

S(α1ρBα2,γ)=0,
where α1 and α2 are the shape modification factors and act as the optimization variables. It is worth noting that, a linear scaling of the target shape is adequate for most cases, i.e., α2 is usually near 1. And, the range of the values for α1 is usually 0.5 to 2. Moreover, Eq. (18) gives just one form to modify the target shape with two variables. Other forms that can apply to changing the target shape continuously may also be considered.

 figure: Fig. 6

Fig. 6 Optimize the preset shape of the target plane.

Download Full Size | PPT Slide | PDF

When the scaling factors α1 and α2 are given, a modified boundary shape of the target is obtained. Based on the newly prescribed target shape, a new lens model is generated by repeating the process mentioned in Sections 2.1.1-2.1.3. At the meantime, the performance of the lens is simulated, and MF1 is calculated subsequently. Then, the optimization problem is established, where α1 and α2 are optimization parameters, MF1 is the object function, and the restriction on light control efficiency η is the constraint condition:

{minα1,α2MF1(α1,α2)Constraint:ηηT,
where the light control efficiency η is the ratio of the flux contained in the target region to the total flux emitted from the source, and ηT is the minimum requirement for η. Local optimization algorithm (such as pattern search method or simplex method) can be used to solve this optimization problem, considering the time-consuming ray tracing process to give an accurate description of the target distribution. In this paper, the simplex based optimization method is chosen to solve the optimization problem, which proves to be very effective in the optical system optimization [7].

2.2 Second-step optimization

After section 2.1, an optimal initial lens model is obtained and the corresponding boundary shape of the simulated illumination pattern is well controlled. However, the illumination uniformity within the target region still needs to be improved, which will be tackled in the second-step optimization. The optimization is carried out by modifying the radii of the initial lens model in a set of given directions and reconstructing the corresponding lens model using the NURBS method, as shown in Fig. 7.Suppose the selected directions are (θ′l, φ′m) and the corresponding radii are rm,l, where l = 0, 1, …, Nl and m = 0, 1, …, Nm. The corresponding points on the lens surface can be calculated as:

P'm,l=rm,l(cosφmsinθl,sinφmsinθl,cosθl).
Then, a smooth freeform surface is constructed using the NURBS method through the set of points Pm,l.

 figure: Fig. 7

Fig. 7 Second-step optimization: optimize the polar radii of the freeform lens obtained in the first-step optimization in selected directions.

Download Full Size | PPT Slide | PDF

The merit function for the second-step optimization is defined as:

MF2=υ1RSD+(1υ1)(1η),
where 0 < υ1 < 1 is the weight. Then, the optimization problem is established with rm,l as optimization variables and MF2 as the object function:

minrm,lMF2(rm,l)(m=0,1,,Nm;l=0,1,,Nl).

Again, the simplex optimization algorithm is employed and turns out to be very effective as shown later. And it is worth noting that, the convergence time of the simplex optimization tends to grow exponentially with the number of the optimization variables, i.e., the value of Nm × Nl. To make a trade-off between the time and the quality of the result, the value of Nm × Nl is normally about 100. At this level, it will take about 3 hours for the convergence of the simplex optimization with 105 rays traced using a 3.4 GHz Intel(R) Core(TM) i7-3770 CPU.

3. Design examples

3.1 Generating a rectangular uniform illumination pattern for a square LED source

Usually, a uniform illumination system is desired for the rectangular Type I road lighting, which is specified in the IESNA (Illuminating Engineering Society of North America) Lighting Handbook [19]. Therefore, as the first example, a freeform lens with central height of h = 2.5 cm is constructed for a square LED surface light source with a length of L = 1 cm to form a uniform illumination pattern within a 40 m × 20 m rectangular target plane, as specified in Fig. 8.The source is a Lambertian source, which has a divergence angle of ± 90° and a total flux of 2000 lumens. The target plane is 10 m from the source. The refractive index of the lens is n1 = 1.59 where the source is immersed. In the example, the maximal dimension D of the source is 21/2L, i.e., h/D is restricted to about 1.77:1, indicating that the optical system is highly compact.

 figure: Fig. 8

Fig. 8 Sketch of the setting for designing freeform lens for LED square surface light source to form a uniform illumination pattern within a far-field rectangular region.

Download Full Size | PPT Slide | PDF

For the first-step optimization, the superellipse in the form of Eq. (23) is taken as the modified boundary shape of the target plane:

|ρcosγα1|α3+|ρsinγα2|α3=1,
where α1 > 10 m, α2 > 20 m and α3 > 2 are the optimization variables. The shape approaches a rectangle when α3 tends to infinity. As for the design to achieve a uniform illumination distribution, the relative standard deviation of the simulated illumination is redefined as Eq. (24) to quantify the spatial uniformity of the simulated distribution itself:
RSD2=i=0M1j=0N1(ES(ρi,j,γi)E¯S1)2(M1+1)(N1+1)1,
where E¯S is the average illumination of all the calculation points:
E¯S=i=0M1j=0N1ES(ρi,j,γi)(M1+1)(N1+1).
We take ω1 = 2/3 and υ1 = 4/5 for the merit functions MF1 [in Eq. (13)] and MF2 [in Eq. (21)] respectively, and ηT is set as 0.7. Moreover, only the design in the first quadrant is taken into account due to the symmetry of the optical system. All the other parameters in this example are shown in Table 1.:

Tables Icon

Table 1. Parameters set for the first example

The optimized target distributions of the first- and second-step optimization are compared with the simulation result of the optical system which is generated using the point source assumption, as shown in Fig. 9.As for the latter [see Fig. 9(a)], there is an obvious energy aggregation at the target center, which causes a considerable inhomogeneity of the illumination distribution. For this case, η is 0.82, but RSD reaches 0.32, which makes the design unpractical. As for the first-step optimization, the boundary shape of the illumination pattern approaches the desired rectangular shape [as shown in Fig. 9(b)], however it exhibits an obvious depression around the two ends along y-axis. By the second-step optimization, the depression along y-axis is eliminated and a uniform target distribution is obtained [see Fig. 9(c)]. Taking the location where the illumination drops down to 60% of the maximum illumination as the edge point of the target distribution, the lighting parameters are calculated and shown in Table 2. A rather high light control efficiency of η = 0.73 and a low relative standard illumination deviation of RSD = 0.060 are obtained simultaneously for the final optimal result.

 figure: Fig. 9

Fig. 9 Simulated illumination distributions of the optical systems generated by (a) point source assumption, (b) the first-step optimization, and (c) the second-step optimization.

Download Full Size | PPT Slide | PDF

Tables Icon

Table 2. Lighting parameters calculated for the two-step optimization

Model and profiles of the optimal lens with a center height of 2.5 cm are shown in Figs. 10(a) and 10(b) respectively.

 figure: Fig. 10

Fig. 10 (a) Models and (b) cross-sectional profiles of the optimal freeform lens.

Download Full Size | PPT Slide | PDF

3.2 Generating a cross-shaped uniform illumination pattern with a discoid LED source

Sometimes, a Type I four-way road illuminating system is required which is also specified in the IESNA Lighting Handbook [19]. Hence, as the second example, a freeform surface lens is designed for a discoid LED source to generate a cross-shaped uniform illumination distribution on the target. Design parameters are specified in Fig. 11.The diameter of the discoid LED source is D = 1 cm, and it also has a divergence angle of ± 90° and a total flux of 2000 lumens. The desired target is a cross, consisting of two 40 m × 20 m rectangles perpendicular to each other. All the other parameters are kept the same as in example 3.1. Therefore, h/D is 2.5:1 in this example. Moreover, the cross shape formed by two orthogonal superellipses in the form of Eq. (23) is chosen as the shape to be optimized.

 figure: Fig. 11

Fig. 11 Sketch of the setting for designing freeform lens for discoid LED surface light source to form a uniform illumination pattern within a far-field cross-shaped region.

Download Full Size | PPT Slide | PDF

The corresponding simulation results of the two steps are respectively shown in Fig. 12.A uniform cross-shaped illumination distribution is generated after the two-step optimization. However, a slightly rounded corner is formed by the extension of the source, which cannot be eliminated using a single freeform surface. The lighting parameters are calculated and shown in Table 3.A rather high light control efficiency of η = 0.78 and a low relative standard illumination deviation of RSD = 0.061 are obtained simultaneously for the final optimal result.

 figure: Fig. 12

Fig. 12 Simulated illumination distributions of the optical systems obtained after (a) the first-step optimization, and (b) the second-step optimization.

Download Full Size | PPT Slide | PDF

Tables Icon

Table 3. Lighting parameters calculated for the two-step optimization

Model and profiles of the optimal lens with a center height of 2.5 cm are shown in Figs. 13(a) and 13(b) respectively.

 figure: Fig. 13

Fig. 13 (a) Models and (b) cross-sectional profiles of the optimal freeform lens.

Download Full Size | PPT Slide | PDF

3.3 Generating a more complicated uniform illumination pattern with a discoid LED source

To show the capability of the method in handling target distribution with more complex boundary shape, a freeform lens is also designed for a discoid LED surface light source to form a uniform illumination distribution on a cross-shaped target region, which meanwhile has a circular void at the intersection (as illustrated in Fig. 14). The diameter of the central hole is 10 m, while all the other parameters are kept the same as prescribed in example 3.2. So, h/D is also 2.5:1 in this example.

 figure: Fig. 14

Fig. 14 Sketch of the setting for designing a freeform lens for discoid LED surface light source to form a uniform illumination pattern within a complex-shaped target region.

Download Full Size | PPT Slide | PDF

The optimum result is obtained after the two-step optimization, as shown in Fig. 15.The boundary shape of the target distribution is precisely controlled with some small local deformations. A rather high light control efficiency of η = 0.71 and a low relative standard illumination deviation of RSD = 0.065 are obtained simultaneously for this design.

 figure: Fig. 15

Fig. 15 Final simulation result after the two-step optimization.

Download Full Size | PPT Slide | PDF

Model and profiles of the optimal lens with a center height of 2.5 cm are shown in Figs. 16(a) and 16(b) respectively.

 figure: Fig. 16

Fig. 16 (a) Models and (b) cross-sectional profiles of the optimal freeform lens.

Download Full Size | PPT Slide | PDF

As shown in the examples 3.1-3.3, using the proposed method, highly compact freeform illumination systems (h/D ≤ 2.5:1) with high performance can be effectively designed to generate a prescribed light distribution with almost no restriction to the boundary shape of the source or the target. All the results show that the boundary shape of the target distribution is properly guaranteed with a relative standard boundary shape deviation RSDshape less than 0.1, and the illumination uniformity within the target region is also enhanced with a relative standard illumination deviation RSD smaller than 0.07.

4. Conclusions

In summary, we proposed a two-step optimization method together with a simplified source-target ray mapping establishing method to design highly compact three-dimensional freeform lenses for LED surface light sources. The boundary shape of the simulated target distribution is well controlled using the first-step optimization by modifying the shape of the prescribed target region as well as the prescribed illumination pattern. The lighting quality within the target region is further improved using the second-step optimization through optimizing a series of selected radii of the lens model obtained after the first-step optimization. As examples, three highly compact lenses (h/D ≤ 2.5:1) are designed for LED surface light sources to generate a uniform illumination distribution on a rectangular, a cross-shaped and a complex cross pierced target plane respectively. High light control efficiency of η > 0.7 as well as low relative standard illumination deviation of RSD < 0.07 is obtained simultaneously in all the three examples. As the examples show, the method is quite flexible and effective to design highly compact 3D optical systems, with almost no restriction on the shapes of both the source and the target field.

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).

References and links

1. Cree LED products, “Cree XLamp LEDs” (CREE 2013). http://www.cree.com/led-components-and-modules/products/xlamp

2. R. Winston, J. C. Miñano, P. Benítez, with contributions by N. Shatz, and J. C. Bortz, Nonimaging Optics (Elsevier, 2005), Chap. 7.

3. F. R. Fournier, “A review of beam shaping strategies for LED lighting,” Proc. SPIE 8170, 817007 (2011). [CrossRef]  

4. H. R. Ries and R. Winston, “Tailored edge-ray reflectors for illumination,” J. Opt. Soc. Am. A 11(4), 1260–1264 (1994). [CrossRef]  

5. A. Rabl and J. M. Gordon, “Reflector design for illumination with extended sources: the basic solutions,” Appl. Opt. 33(25), 6012–6021 (1994). [CrossRef]   [PubMed]  

6. P. Benítez, J. C. Miňano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1502 (2004). [CrossRef]  

7. R. J. Koshel, “Simplex optimization method for illumination design,” Opt. Lett. 30(6), 649–651 (2005). [CrossRef]   [PubMed]  

8. F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Optimization of single reflectors for extended sources,” Proc. SPIE 7103, 71030I (2008). [CrossRef]  

9. P. Liu, R. M. Wu, Z. R. Zheng, H. F. Li, and X. Liu, “Optimized design of LED freeform lens for uniform circular illumination,” J. Zhejiang Univ. Sci. C 13(12), 929–936 (2012). [CrossRef]  

10. Y. Luo, Z. X. Feng, Y. J. Han, and H. T. Li, “Design of compact and smooth free-form optical system with uniform illuminance for LED source,” Opt. Express 18(9), 9055–9063 (2010). [CrossRef]   [PubMed]  

11. K. Wang, F. Chen, Z. Y. Liu, X. B. Luo, and S. Liu, “Design of compact freeform lens for application specific light-emitting diode packaging,” Opt. Express 18(2), 413–425 (2010). [CrossRef]   [PubMed]  

12. H. T. Li, S. C. Chen, Y. J. Han, and Y. Luo, “A fast feedback method to design easy-molding freeform optical system with uniform illuminance and high light control efficiency,” Opt. Express 21(1), 1258–1269 (2013).

13. A. Bruneton, A. Bäuerle, R. Wester, J. Stollenwerk, and P. Loosen, “High resolution irradiance tailoring using multiple freeform surfaces,” Opt. Express 21(9), 10563–10571 (2013). [CrossRef]   [PubMed]  

14. K. Wang, Y. J. Han, H. T. Li, and Y. Luo, “Overlapping-based optical freeform surface construction for extended lighting source,” Opt. Express 21(17), 19750–19761 (2013). [CrossRef]   [PubMed]  

15. X. L. Mao, H. T. Li, Y. J. Han, and Y. Luo, “A two-step design method for high compact rotationally symmetric optical system for LED surface light source,” Opt. Express 22(S2Suppl 2), A233–A247 (2014). [CrossRef]   [PubMed]  

16. W. A. Parkyn, “The design of illumination lenses via extrinsic differential geometry,” Proc. SPIE 3482, 191–193 (1998). [CrossRef]  

17. L. Wang, K. Qian, and Y. Luo, “Discontinuous free-form lens design for prescribed irradiance,” Appl. Opt. 46(18), 3716–3723 (2007). [CrossRef]   [PubMed]  

18. Y. Ding, X. Liu, Z. R. Zheng, and P. F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16(17), 12958–12966 (2008). [CrossRef]   [PubMed]  

19. M. S. Rea, The IESNA Lighting Handbook: Reference and Application, 9th ed. (IESNA, 2000).

References

  • View by:
  • |
  • |
  • |

  1. Cree LED products, “Cree XLamp LEDs” (CREE 2013). http://www.cree.com/led-components-and-modules/products/xlamp
  2. R. Winston, J. C. Miñano, P. Benítez, with contributions by N. Shatz, and J. C. Bortz, Nonimaging Optics (Elsevier, 2005), Chap. 7.
  3. F. R. Fournier, “A review of beam shaping strategies for LED lighting,” Proc. SPIE 8170, 817007 (2011).
    [Crossref]
  4. H. R. Ries and R. Winston, “Tailored edge-ray reflectors for illumination,” J. Opt. Soc. Am. A 11(4), 1260–1264 (1994).
    [Crossref]
  5. A. Rabl and J. M. Gordon, “Reflector design for illumination with extended sources: the basic solutions,” Appl. Opt. 33(25), 6012–6021 (1994).
    [Crossref] [PubMed]
  6. P. Benítez, J. C. Miňano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1502 (2004).
    [Crossref]
  7. R. J. Koshel, “Simplex optimization method for illumination design,” Opt. Lett. 30(6), 649–651 (2005).
    [Crossref] [PubMed]
  8. F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Optimization of single reflectors for extended sources,” Proc. SPIE 7103, 71030I (2008).
    [Crossref]
  9. P. Liu, R. M. Wu, Z. R. Zheng, H. F. Li, and X. Liu, “Optimized design of LED freeform lens for uniform circular illumination,” J. Zhejiang Univ. Sci. C 13(12), 929–936 (2012).
    [Crossref]
  10. Y. Luo, Z. X. Feng, Y. J. Han, and H. T. Li, “Design of compact and smooth free-form optical system with uniform illuminance for LED source,” Opt. Express 18(9), 9055–9063 (2010).
    [Crossref] [PubMed]
  11. K. Wang, F. Chen, Z. Y. Liu, X. B. Luo, and S. Liu, “Design of compact freeform lens for application specific light-emitting diode packaging,” Opt. Express 18(2), 413–425 (2010).
    [Crossref] [PubMed]
  12. H. T. Li, S. C. Chen, Y. J. Han, and Y. Luo, “A fast feedback method to design easy-molding freeform optical system with uniform illuminance and high light control efficiency,” Opt. Express 21(1), 1258–1269 (2013).
  13. A. Bruneton, A. Bäuerle, R. Wester, J. Stollenwerk, and P. Loosen, “High resolution irradiance tailoring using multiple freeform surfaces,” Opt. Express 21(9), 10563–10571 (2013).
    [Crossref] [PubMed]
  14. K. Wang, Y. J. Han, H. T. Li, and Y. Luo, “Overlapping-based optical freeform surface construction for extended lighting source,” Opt. Express 21(17), 19750–19761 (2013).
    [Crossref] [PubMed]
  15. X. L. Mao, H. T. Li, Y. J. Han, and Y. Luo, “A two-step design method for high compact rotationally symmetric optical system for LED surface light source,” Opt. Express 22(S2Suppl 2), A233–A247 (2014).
    [Crossref] [PubMed]
  16. W. A. Parkyn, “The design of illumination lenses via extrinsic differential geometry,” Proc. SPIE 3482, 191–193 (1998).
    [Crossref]
  17. L. Wang, K. Qian, and Y. Luo, “Discontinuous free-form lens design for prescribed irradiance,” Appl. Opt. 46(18), 3716–3723 (2007).
    [Crossref] [PubMed]
  18. Y. Ding, X. Liu, Z. R. Zheng, and P. F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16(17), 12958–12966 (2008).
    [Crossref] [PubMed]
  19. M. S. Rea, The IESNA Lighting Handbook: Reference and Application, 9th ed. (IESNA, 2000).

2014 (1)

2013 (3)

2012 (1)

P. Liu, R. M. Wu, Z. R. Zheng, H. F. Li, and X. Liu, “Optimized design of LED freeform lens for uniform circular illumination,” J. Zhejiang Univ. Sci. C 13(12), 929–936 (2012).
[Crossref]

2011 (1)

F. R. Fournier, “A review of beam shaping strategies for LED lighting,” Proc. SPIE 8170, 817007 (2011).
[Crossref]

2010 (2)

2008 (2)

F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Optimization of single reflectors for extended sources,” Proc. SPIE 7103, 71030I (2008).
[Crossref]

Y. Ding, X. Liu, Z. R. Zheng, and P. F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16(17), 12958–12966 (2008).
[Crossref] [PubMed]

2007 (1)

2005 (1)

2004 (1)

P. Benítez, J. C. Miňano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1502 (2004).
[Crossref]

1998 (1)

W. A. Parkyn, “The design of illumination lenses via extrinsic differential geometry,” Proc. SPIE 3482, 191–193 (1998).
[Crossref]

1994 (2)

Bäuerle, A.

Benítez, P.

P. Benítez, J. C. Miňano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1502 (2004).
[Crossref]

Blen, J.

P. Benítez, J. C. Miňano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1502 (2004).
[Crossref]

Bruneton, A.

Cassarly, W. J.

F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Optimization of single reflectors for extended sources,” Proc. SPIE 7103, 71030I (2008).
[Crossref]

Chaves, J.

P. Benítez, J. C. Miňano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1502 (2004).
[Crossref]

Chen, F.

Chen, S. C.

Ding, Y.

Dross, O.

P. Benítez, J. C. Miňano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1502 (2004).
[Crossref]

Falicoff, W.

P. Benítez, J. C. Miňano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1502 (2004).
[Crossref]

Feng, Z. X.

Fournier, F. R.

F. R. Fournier, “A review of beam shaping strategies for LED lighting,” Proc. SPIE 8170, 817007 (2011).
[Crossref]

F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Optimization of single reflectors for extended sources,” Proc. SPIE 7103, 71030I (2008).
[Crossref]

Gordon, J. M.

Gu, P. F.

Han, Y. J.

Hernández, M.

P. Benítez, J. C. Miňano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1502 (2004).
[Crossref]

Koshel, R. J.

Li, H. F.

P. Liu, R. M. Wu, Z. R. Zheng, H. F. Li, and X. Liu, “Optimized design of LED freeform lens for uniform circular illumination,” J. Zhejiang Univ. Sci. C 13(12), 929–936 (2012).
[Crossref]

Li, H. T.

Liu, P.

P. Liu, R. M. Wu, Z. R. Zheng, H. F. Li, and X. Liu, “Optimized design of LED freeform lens for uniform circular illumination,” J. Zhejiang Univ. Sci. C 13(12), 929–936 (2012).
[Crossref]

Liu, S.

Liu, X.

P. Liu, R. M. Wu, Z. R. Zheng, H. F. Li, and X. Liu, “Optimized design of LED freeform lens for uniform circular illumination,” J. Zhejiang Univ. Sci. C 13(12), 929–936 (2012).
[Crossref]

Y. Ding, X. Liu, Z. R. Zheng, and P. F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16(17), 12958–12966 (2008).
[Crossref] [PubMed]

Liu, Z. Y.

Loosen, P.

Luo, X. B.

Luo, Y.

Mao, X. L.

Minano, J. C.

P. Benítez, J. C. Miňano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1502 (2004).
[Crossref]

Mohedano, R.

P. Benítez, J. C. Miňano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1502 (2004).
[Crossref]

Parkyn, W. A.

W. A. Parkyn, “The design of illumination lenses via extrinsic differential geometry,” Proc. SPIE 3482, 191–193 (1998).
[Crossref]

Qian, K.

Rabl, A.

Ries, H. R.

Rolland, J. P.

F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Optimization of single reflectors for extended sources,” Proc. SPIE 7103, 71030I (2008).
[Crossref]

Stollenwerk, J.

Wang, K.

Wang, L.

Wester, R.

Winston, R.

Wu, R. M.

P. Liu, R. M. Wu, Z. R. Zheng, H. F. Li, and X. Liu, “Optimized design of LED freeform lens for uniform circular illumination,” J. Zhejiang Univ. Sci. C 13(12), 929–936 (2012).
[Crossref]

Zheng, Z. R.

P. Liu, R. M. Wu, Z. R. Zheng, H. F. Li, and X. Liu, “Optimized design of LED freeform lens for uniform circular illumination,” J. Zhejiang Univ. Sci. C 13(12), 929–936 (2012).
[Crossref]

Y. Ding, X. Liu, Z. R. Zheng, and P. F. Gu, “Freeform LED lens for uniform illumination,” Opt. Express 16(17), 12958–12966 (2008).
[Crossref] [PubMed]

Appl. Opt. (2)

J. Opt. Soc. Am. A (1)

J. Zhejiang Univ. Sci. C (1)

P. Liu, R. M. Wu, Z. R. Zheng, H. F. Li, and X. Liu, “Optimized design of LED freeform lens for uniform circular illumination,” J. Zhejiang Univ. Sci. C 13(12), 929–936 (2012).
[Crossref]

Opt. Eng. (1)

P. Benítez, J. C. Miňano, J. Blen, R. Mohedano, J. Chaves, O. Dross, M. Hernández, and W. Falicoff, “Simultaneous multiple surface optical design method in three dimensions,” Opt. Eng. 43(7), 1489–1502 (2004).
[Crossref]

Opt. Express (7)

Opt. Lett. (1)

Proc. SPIE (3)

F. R. Fournier, W. J. Cassarly, and J. P. Rolland, “Optimization of single reflectors for extended sources,” Proc. SPIE 7103, 71030I (2008).
[Crossref]

F. R. Fournier, “A review of beam shaping strategies for LED lighting,” Proc. SPIE 8170, 817007 (2011).
[Crossref]

W. A. Parkyn, “The design of illumination lenses via extrinsic differential geometry,” Proc. SPIE 3482, 191–193 (1998).
[Crossref]

Other (3)

Cree LED products, “Cree XLamp LEDs” (CREE 2013). http://www.cree.com/led-components-and-modules/products/xlamp

R. Winston, J. C. Miñano, P. Benítez, with contributions by N. Shatz, and J. C. Bortz, Nonimaging Optics (Elsevier, 2005), Chap. 7.

M. S. Rea, The IESNA Lighting Handbook: Reference and Application, 9th ed. (IESNA, 2000).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (16)

Fig. 1
Fig. 1 Two key points for illumination design and the corresponding two-step optimization strategy.
Fig. 2
Fig. 2 A flow diagram of the two-step optimization process. The first step (in the red box) is a combination of two algorithms: iterative feedback modification of the illumination and target shape scaling. The second step (in the blue box) is realized by optimizing a set of selected radii of the initial optimal lens model obtained in the first step.
Fig. 3
Fig. 3 Establish the source-target mapping using an assumed point source which is at the geometric center of the surface light source with the same luminous intensity distribution as the surface source. For a point source, a point on the lens profile will generate a single point on the target plane.
Fig. 4
Fig. 4 For a surface light source, a point on the lens profile will create a finite-size light spot on the target plane. Through the overlapping of all the spots, a target distribution is obtained.
Fig. 5
Fig. 5 Numerical configuration of the profile by geometrical construction method.
Fig. 6
Fig. 6 Optimize the preset shape of the target plane.
Fig. 7
Fig. 7 Second-step optimization: optimize the polar radii of the freeform lens obtained in the first-step optimization in selected directions.
Fig. 8
Fig. 8 Sketch of the setting for designing freeform lens for LED square surface light source to form a uniform illumination pattern within a far-field rectangular region.
Fig. 9
Fig. 9 Simulated illumination distributions of the optical systems generated by (a) point source assumption, (b) the first-step optimization, and (c) the second-step optimization.
Fig. 10
Fig. 10 (a) Models and (b) cross-sectional profiles of the optimal freeform lens.
Fig. 11
Fig. 11 Sketch of the setting for designing freeform lens for discoid LED surface light source to form a uniform illumination pattern within a far-field cross-shaped region.
Fig. 12
Fig. 12 Simulated illumination distributions of the optical systems obtained after (a) the first-step optimization, and (b) the second-step optimization.
Fig. 13
Fig. 13 (a) Models and (b) cross-sectional profiles of the optimal freeform lens.
Fig. 14
Fig. 14 Sketch of the setting for designing a freeform lens for discoid LED surface light source to form a uniform illumination pattern within a complex-shaped target region.
Fig. 15
Fig. 15 Final simulation result after the two-step optimization.
Fig. 16
Fig. 16 (a) Models and (b) cross-sectional profiles of the optimal freeform lens.

Tables (3)

Tables Icon

Table 1 Parameters set for the first example

Tables Icon

Table 2 Lighting parameters calculated for the two-step optimization

Tables Icon

Table 3 Lighting parameters calculated for the two-step optimization

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

Ω S I(θ,φ)sinθdθdφ = Ω T E 0 (x,y)dxdy ,
(x,y)=( f 0 (θ,φ), g 0 (θ,φ)).
φ i = φ min + φ max φ min M i ( i = 0 , 1 , M ) ,
θ j = θ min + θ max θ min N j ( j = 0 , 1 , N ) ,
γ i = φ i ( i = 0 , 1 , M ) .
S ( ρ B , γ ) = 0 ,
θ 0 θ j I ( θ , φ i ) d θ θ 0 θ N I ( θ , φ i ) d θ = ρ i , 0 ρ i , j E 0 ( ρ , γ i ) d ρ ρ i , 0 ρ B i E 0 ( ρ , γ i ) d ρ ( i = 0 , 1 , M , j = 0 , 1 , N ) ,
ρ i , j = F ( θ j , φ i ) .
( θ j , φ i )( ρ i,j , γ i ){ γ i = φ i ρ i,j =F( θ j , φ i ) (i=0,1,M,j=0,1,N).
N 0 = ( n 2 O u t 0 n 1 I n 0 ) / | n 2 O u t 0 n 1 I n 0 | ,
β k (ρ,γ)= { E 0 (ρ,γ)/[ λ 1 E Sk (ρ,γ)+(1 λ 1 ) E 0 (ρ,γ)] } λ 2 ,
E Mk (ρ,γ)= Π i=1 k β i (ρ,γ) E 0 (ρ,γ).
M F 1 = ω 1 RS D shape +(1 ω 1 )RSD,
γ i = φ min + φ max φ min M 1 i(i=0,1,, M 1 ),
ρ i,j = ρ mini ( γ i )+ ρ maxi ( γ i ) ρ mini ( γ i ) N 1 j(j=0,1,, N 1 ),
RS D 2 shape = i=0 M 1 ( ρ Si ( γ i ) ρ maxi ( γ i ) ρ maxi ( γ i ) ) 2 M 1 ,
RS D 2 = i=0 M 1 j=0 N 1 ( E S ( ρ i,j , γ i ) E 0 ( ρ i,j , γ i ) E 0 ( ρ i,j , γ i ) ) 2 ( M 1 +1)( N 1 +1)1 ,
S ( α 1 ρ B α 2 , γ ) = 0 ,
{ min α 1 , α 2 M F 1 ( α 1 , α 2 ) Constraint:η η T ,
P ' m , l = r m , l ( cos φ m sin θ l , sin φ m sin θ l , cos θ l ) .
M F 2 = υ 1 RSD+(1 υ 1 )(1η),
min r m,l M F 2 ( r m,l )(m=0,1,, N m ;l=0,1,, N l ).
| ρ cos γ α 1 | α 3 + | ρ sin γ α 2 | α 3 = 1 ,
R S D 2 = i = 0 M 1 j = 0 N 1 ( E S ( ρ i , j , γ i ) E ¯ S 1 ) 2 ( M 1 + 1 ) ( N 1 + 1 ) 1 ,
E ¯ S = i = 0 M 1 j = 0 N 1 E S ( ρ i , j , γ i ) ( M 1 + 1 ) ( N 1 + 1 ) .

Metrics